Abstract group 4608.rp: $A_4.(S_3\times D_4^2)$

• Label: 4608.rp
• Order: \( 2^{9} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: 4
• $\card{\Aut(G)}$: \( 2^{17} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{10} \)
• Perm deg.: not computed
• Trans deg.: not computed
• Rank: $5$

Group information

Description:	$A_4.(S_3\times D_4^2)$
Order:	\(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_5^4.C_4^2.C_2$, of order \(1179648\)\(\medspace = 2^{17} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$ x 2
Derived length:	$2$

This group is nonabelian and metabelian (hence solvable). Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	575	26	448	1990	1568	4608
Conjugacy classes	1	63	5	36	123	72	300
Divisions	1	63	3	36	77	45	225
Autjugacy classes	1	15	3	12	20	15	66

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	96	96	32	30	32	3	10	0	1	300
Irr. rational chars.	32	64	32	42	32	11	10	1	1	225

Minimal presentations

Permutation degree:	not computed
Transitive degree:	not computed
Rank:	$5$
Inequivalent generating 5-tuples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{6}=c^{4}=d^{2}=e^{4}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,7,6)(8,15)(12,14), (1,2), (1,3)(4,5)(6,7)(8,12)(11,13), (4,5,7) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_4\times D_4^2)$ . $S_3$	$(S_3\times D_4^2)$ . $A_4$	$(S_3\times A_4)$ . $D_4^2$ (4)	$D_4^2$ . $(S_3\times A_4)$	all 289

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{4} \times C_{6} \simeq C_{2}^{5} \times C_{3}$
Schur multiplier:	$C_{2}^{11}$
Commutator length:	$1$

Subgroups

There are 1725 normal subgroups (69 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	a subgroup isomorphic to $C_2^2$
Commutator:	a subgroup isomorphic to $C_2^3\times C_6$
Frattini:	a subgroup isomorphic to $C_2^2$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3\times D_4^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 9 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 4608: $A_4.(S_3\times D_4^2)$

Order 2304: $C_2^5.(C_6\times D_6)$ x 4, $A_4\times C_2^4:D_6$ x 4, $A_4\times D_{12}:D_4$ x 4, $A_4\times C_2^4:D_6$ x 4, $A_4\times C_2^4:D_6$ x 4, $C_{12}.(A_4\times C_2^4)$ x 4, $A_4\times C_4^2:D_6$ x 2, $D_4\times A_4\times D_{12}$ x 2, $C_6.(C_2^4\times A_4).C_2$, $A_4\times C_3:D_4^2$, $C_2^6:C_6^2$

Order 1536: $C_2\times A_4\times D_4^2$, $C_2\times D_6\times D_4^2$

Order 1152: $D_4\times A_4\times D_6$ x 64, $A_4\times D_6:D_4$ x 12, $A_4\times C_2^2:D_{12}$ x 12, $C_2^3:D_6\times A_4$ x 8, $A_4\times C_2^3.D_6$ x 8, $A_4\times D_6:D_4$ x 8, $S_3\times C_2^4:C_{12}$ x 8, $A_4\times C_{12}:D_4$ x 6, $C_2^3\times A_4\times D_6$ x 4, $C_2^2\times A_4\times D_{12}$ x 4, $C_2^3:C_{12}\times D_6$ x 4, $A_4\times C_2^4:S_3$ x 4, $A_4\times C_{12}:D_4$ x 4, $A_4\times C_{12}:D_4$ x 4, $C_{12}:D_4\times A_4$ x 4, $A_4\times C_4:D_{12}$ x 4, $C_2^4:(S_3\times C_{12})$ x 4, $C_2^5:C_6^2$ x 4, $C_2^5.C_6^2$ x 4, $C_2^5:C_6^2$ x 4, $C_2^3.D_6\times A_4$ x 2, $A_4\times D_{12}:C_4$ x 2, $A_4\times D_6.D_4$ x 2, $C_4\times A_4\times D_{12}$ x 2, $C_2^5.C_6^2$ x 2, $A_4\times C_4:D_{12}$, $S_3\times A_4\times C_4^2$, $C_2^5.C_6^2$, $C_{12}^2:C_2^3$

Order 768: $D_6\times D_4^2$ x 32, $A_4\times D_4^2$ x 17, $C_2^7:C_6$ x 4, $C_2^3:D_4\times A_4$ x 4, $C_2^7:C_6$ x 4, $C_{12}:C_2^6$ x 4, $C_2\times C_6:D_4^2$ x 4, $C_2^6:D_6$ x 4, $C_2^6:D_6$ x 4, $C_2^3:D_4\times D_6$ x 4, $C_2^6:D_6$ x 4, $D_4\times C_2^3:C_{12}$ x 2, $C_2^2\times D_4\times D_{12}$ x 2, $C_4\times C_{12}:C_2^4$ x 2, $C_2\times C_6\times D_4^2$, $A_4\times C_4^2:C_2^2$, $C_2\times C_6:D_4^2$, $C_2^2\times C_4^2:D_6$

Order 576: $S_3\times D_4\times A_4$ x 132, $C_6:D_4\times A_4$ x 64, $C_2^2\times A_4\times D_6$ x 60, $C_2^4:C_6^2$ x 32, $C_2\times A_4\times D_{12}$ x 32, $C_4\times A_4\times D_6$ x 31, $A_4\times D_6:C_4$ x 16, $C_2^4.C_6^2$ x 8, $A_4\times C_6.D_4$ x 8, $C_2^4:C_6^2$ x 4, $C_6^2:C_2^4$ x 4, $C_2^4.C_6^2$ x 4, $C_6.C_2^3\times A_4$ x 4, $C_3^2.D_4^2$ x 4, $C_3^2.D_4^2$ x 4, $C_6^2.C_2^4$ x 4, $C_3^2:D_4^2$ x 4, $C_6^2.C_2^4$ x 4, $A_4\times C_6.D_4$ x 4, $C_{12}:C_4\times A_4$ x 3, $C_{12}^2:C_2^2$ x 2, $C_{12}^2:C_2^2$ x 2, $C_2^4.C_6^2$ x 2, $A_4\times C_{12}:C_4$ x 2, $C_3^2\times D_4^2$, $C_{12}^2:C_2^2$, $C_{12}^2:C_2^2$, $C_2^2:C_{12}^2$

Order 512: $C_2^3\times D_4^2$

Order 384: $C_{12}:C_2^5$ x 32, $C_2^5:D_6$ x 12, $C_2^4:D_{12}$ x 12, $D_6:C_2^5$ x 8, $C_2^5.D_6$ x 8, $C_2^5.D_6$ x 8, $C_2^5.D_6$ x 8, $C_2^5.D_6$ x 6, $D_6\times C_2^5$ x 4, $C_{12}:C_2^5$ x 4, $C_2^4\times D_{12}$ x 4, $C_{12}:C_2^5$ x 4, $C_2^6:C_6$ x 4, $C_2^2\times C_{12}:D_4$ x 4, $C_2^6:C_6$ x 4, $C_2^5:D_6$ x 4, $C_2^5.D_6$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $C_2^2\times C_4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $A_4\times C_4:D_4$ x 4, $C_2^6:C_6$ x 4, $C_2^2\times D_4\times C_{12}$ x 2, $C_2^5.D_6$ x 2, $C_{12}:(C_2^3\times C_4)$ x 2, $C_2^2\times D_{12}:C_4$ x 2, $C_4\times C_2^2\times D_{12}$ x 2, $C_4\times D_4\times A_4$ x 2, $C_2^2\times C_{12}:D_4$, $S_3\times D_4^2$, $C_2^2\times C_4:D_{12}$, $C_2\times D_6\times C_4^2$, $A_4\times C_4:D_4$

Order 288: $C_2\times A_4\times D_6$ x 25, $A_4\times C_3:D_4$ x 16, $C_2^3.C_6^2$ x 8, $A_4\times D_{12}$ x 8, $C_4\times S_3\times A_4$ x 8, $C_2^3:C_6^2$ x 8, $C_2^3.C_6^2$ x 7, $C_6:C_4\times A_4$ x 7

Order 256: $C_2^2\times D_4^2$

Order 192: $D_6:C_2^4$ x 32, $D_6\times C_2^4$ x 20, $C_{12}:C_2^4$ x 20, $C_{12}:C_2^4$ x 16, $C_2^3\times D_{12}$ x 16, $C_2^5:C_6$ x 16, $C_2^3.D_{12}$ x 16, $C_{12}:C_2^4$ x 15, $C_2^4:C_{12}$ x 8, $C_2^4:C_{12}$ x 8, $C_2^4.D_6$ x 8, $C_2^5\times C_6$ x 4, $A_4\times C_2^4$ x 4, $C_2^4\times C_{12}$ x 4, $C_6.C_2^5$ x 4, $C_2^4:C_{12}$ x 4, $C_2^4:D_6$ x 4, $C_2^4.D_6$ x 4, $D_{12}:D_4$ x 4, $S_3\times C_4:D_4$ x 4, $C_2^4:D_6$ x 4, $C_2^4:D_6$ x 4, $C_2^4.D_6$ x 3, $A_4\times C_4:C_4$ x 2, $C_2^2\times C_4:C_{12}$ x 2, $C_2^3.D_{12}$ x 2, $D_4\times D_{12}$ x 2, $C_4^2:D_6$ x 2, $A_4\times C_4^2$, $C_3\times D_4^2$, $C_2^2\times C_4\times C_{12}$, $C_3:D_4^2$, $C_4^2:D_6$

Order 144: $A_4\times D_6$ x 14, $C_2^2:C_6^2$ x 9, $C_{12}\times A_4$ x 4, $A_4\times C_3:C_4$ x 4

Order 128: $D_4\times C_2^4$ x 4, $C_2^4:D_4$ x 4, $C_2^4:D_4$ x 4, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$

Order 96: $D_4\times D_6$ x 32, $C_2^3\times D_6$ x 29, $C_2^3:D_6$ x 24, $C_2^2:D_{12}$ x 12, $C_{12}:C_2^3$ x 12, $C_2^2\times D_{12}$ x 12, $C_{12}:C_2^3$ x 12, $D_6:D_4$ x 12, $D_6:D_4$ x 8, $D_6.D_4$ x 8, $C_2^4\times C_6$ x 8, $C_2^3\times A_4$ x 8, $D_4\times A_4$ x 8, $C_2^3.D_6$ x 8, $C_2^3\times C_{12}$ x 7, $C_6.C_2^4$ x 7, $C_2^3:C_{12}$ x 7, $C_{12}:D_4$ x 6, $C_2^3.D_6$ x 4, $C_{12}:D_4$ x 4, $C_2^4:C_6$ x 4, $C_2^4:S_3$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_{12}:D_4$ x 4, $C_4:D_{12}$ x 4, $D_6.D_4$ x 2, $C_4\times D_{12}$ x 2, $D_4\times C_{12}$ x 2, $C_2^3.D_6$ x 2, $D_{12}:C_4$ x 2, $C_4:D_{12}$, $S_3\times C_4^2$, $C_{12}:D_4$

Order 72: $S_3\times A_4$ x 4, $C_6\times A_4$ x 3

Order 64: $D_4\times C_2^3$ x 16, $C_2^4:C_4$ x 8, $C_2^6$ x 4, $C_2^4\times C_4$ x 4, $C_2^3.D_4$ x 2, $D_4^2$, $C_2^2\times C_4^2$

Order 48: $C_2^2\times D_6$ x 34, $C_6:D_4$ x 32, $C_6\times D_4$ x 16, $S_3\times D_4$ x 16, $C_2\times D_{12}$ x 16, $D_6:C_4$ x 16, $C_4\times D_6$ x 15, $C_2^3\times C_6$ x 13, $C_2^2\times A_4$ x 9, $C_2^2\times C_{12}$ x 8, $C_6.C_2^3$ x 8, $C_2^2:C_{12}$ x 8, $C_6.D_4$ x 8, $C_4\times A_4$ x 4, $C_6.D_4$ x 4, $C_{12}:C_4$ x 3, $C_4:C_{12}$ x 2, $C_{12}:C_4$ x 2, $C_4\times C_{12}$

Order 36: $C_3\times A_4$

Order 32: $C_2^2\times D_4$ x 12, $C_2^5$ x 8, $C_2^3\times C_4$ x 7, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 24: $C_2\times D_6$ x 29, $C_3:D_4$ x 16, $C_2^2\times C_6$ x 11, $D_{12}$ x 8, $C_4\times S_3$ x 8, $C_3\times D_4$ x 8, $C_2\times C_{12}$ x 7, $C_6:C_4$ x 7, $C_2\times A_4$ x 3

Order 16: $C_2\times D_4$ x 16, $C_2^4$ x 13, $C_2^2:C_4$ x 8, $C_2^2\times C_4$ x 8, $C_4:C_4$ x 2, $C_4^2$

Order 12: $D_6$ x 14, $C_2\times C_6$ x 10, $C_{12}$ x 4, $C_3:C_4$ x 4, $A_4$

Order 8: $C_2^3$ x 11, $D_4$ x 8, $C_2\times C_4$ x 7

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$ x 10, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4608: $A_4.(S_3\times D_4^2)$ ($C_1$)

Order 2304: $C_2^5.(C_6\times D_6)$ ($C_2$) x 4, $A_4\times C_2^4:D_6$ ($C_2$) x 4, $A_4\times D_{12}:D_4$ ($C_2$) x 4, $A_4\times C_2^4:D_6$ ($C_2$) x 4, $A_4\times C_2^4:D_6$ ($C_2$) x 4, $C_{12}.(A_4\times C_2^4)$ ($C_2$) x 4, $A_4\times C_4^2:D_6$ ($C_2$) x 2, $D_4\times A_4\times D_{12}$ ($C_2$) x 2, $C_6.(C_2^4\times A_4).C_2$ ($C_2$), $A_4\times C_3:D_4^2$ ($C_2$), $C_2^6:C_6^2$ ($C_2$)

Order 1536: $C_2\times D_6\times D_4^2$ ($C_3$)

Order 1152: $D_4\times A_4\times D_6$ ($C_2^2$) x 32, $A_4\times D_6:D_4$ ($C_2^2$) x 12, $A_4\times C_2^2:D_{12}$ ($C_2^2$) x 12, $C_2^3:D_6\times A_4$ ($C_2^2$) x 8, $A_4\times C_2^3.D_6$ ($C_2^2$) x 8, $A_4\times D_6:D_4$ ($C_2^2$) x 8, $S_3\times C_2^4:C_{12}$ ($C_2^2$) x 8, $A_4\times C_{12}:D_4$ ($C_2^2$) x 6, $C_2^3\times A_4\times D_6$ ($C_2^2$) x 4, $C_2^2\times A_4\times D_{12}$ ($C_2^2$) x 4, $C_2^3:C_{12}\times D_6$ ($C_2^2$) x 4, $A_4\times C_2^4:S_3$ ($C_2^2$) x 4, $A_4\times C_{12}:D_4$ ($C_2^2$) x 4, $A_4\times C_{12}:D_4$ ($C_2^2$) x 4, $C_{12}:D_4\times A_4$ ($C_2^2$) x 4, $A_4\times C_4:D_{12}$ ($C_2^2$) x 4, $C_2^4:(S_3\times C_{12})$ ($C_2^2$) x 4, $C_2^5:C_6^2$ ($C_2^2$) x 4, $C_2^5.C_6^2$ ($C_2^2$) x 4, $C_2^5:C_6^2$ ($C_2^2$) x 4, $C_2^3.D_6\times A_4$ ($C_2^2$) x 2, $A_4\times D_{12}:C_4$ ($C_2^2$) x 2, $A_4\times D_6.D_4$ ($C_2^2$) x 2, $C_4\times A_4\times D_{12}$ ($C_2^2$) x 2, $C_2^5.C_6^2$ ($C_2^2$) x 2, $A_4\times C_4:D_{12}$ ($C_2^2$), $S_3\times A_4\times C_4^2$ ($C_2^2$), $C_2^5.C_6^2$ ($C_2^2$)

Order 768: $C_{12}:C_2^6$ ($C_6$) x 4, $C_2\times C_6:D_4^2$ ($C_6$) x 4, $C_2^6:D_6$ ($C_6$) x 4, $C_2^6:D_6$ ($C_6$) x 4, $C_2^3:D_4\times D_6$ ($C_6$) x 4, $C_2^6:D_6$ ($C_6$) x 4, $C_2^2\times D_4\times D_{12}$ ($C_6$) x 2, $C_4\times C_{12}:C_2^4$ ($C_6$) x 2, $C_2\times C_6\times D_4^2$ ($C_6$), $A_4\times D_4^2$ ($S_3$), $C_2\times C_6:D_4^2$ ($C_6$), $C_2^2\times C_4^2:D_6$ ($C_6$)

Order 576: $C_6:D_4\times A_4$ ($C_2^3$) x 32, $C_2^2\times A_4\times D_6$ ($C_2^3$) x 20, $C_2^4:C_6^2$ ($C_2^3$) x 16, $S_3\times D_4\times A_4$ ($D_4$) x 16, $C_2\times A_4\times D_{12}$ ($C_2^3$) x 16, $A_4\times D_6:C_4$ ($C_2^3$) x 16, $C_4\times A_4\times D_6$ ($C_2^3$) x 15, $C_2^4.C_6^2$ ($C_2^3$) x 8, $A_4\times C_6.D_4$ ($C_2^3$) x 8, $C_2^4:C_6^2$ ($C_2^3$) x 4, $C_2^4.C_6^2$ ($C_2^3$) x 4, $C_6.C_2^3\times A_4$ ($C_2^3$) x 4, $A_4\times C_6.D_4$ ($C_2^3$) x 4, $C_{12}:C_4\times A_4$ ($C_2^3$) x 3, $C_2^4.C_6^2$ ($C_2^3$) x 2, $A_4\times C_{12}:C_4$ ($C_2^3$) x 2, $C_2^2:C_{12}^2$ ($C_2^3$)

Order 384: $C_{12}:C_2^5$ ($C_2\times C_6$) x 32, $C_2^5:D_6$ ($C_2\times C_6$) x 12, $C_2^4:D_{12}$ ($C_2\times C_6$) x 12, $D_6:C_2^5$ ($C_2\times C_6$) x 8, $C_2^5.D_6$ ($C_2\times C_6$) x 8, $C_2^5.D_6$ ($C_2\times C_6$) x 8, $C_2^5.D_6$ ($C_2\times C_6$) x 8, $C_2^5.D_6$ ($C_2\times C_6$) x 6, $D_6\times C_2^5$ ($C_2\times C_6$) x 4, $C_{12}:C_2^5$ ($C_2\times C_6$) x 4, $C_2^4\times D_{12}$ ($C_2\times C_6$) x 4, $C_{12}:C_2^5$ ($C_2\times C_6$) x 4, $C_2^6:C_6$ ($D_6$) x 4, $C_2^2\times C_{12}:D_4$ ($C_2\times C_6$) x 4, $C_2^6:C_6$ ($C_2\times C_6$) x 4, $C_2^5:D_6$ ($C_2\times C_6$) x 4, $C_2^5.D_6$ ($C_2\times C_6$) x 4, $C_2^4:D_{12}$ ($C_2\times C_6$) x 4, $C_2^5.D_6$ ($C_2\times C_6$) x 4, $C_2^2\times C_4:D_{12}$ ($C_2\times C_6$) x 4, $C_2^5.D_6$ ($C_2\times C_6$) x 4, $A_4\times C_4:D_4$ ($D_6$) x 4, $C_2^6:C_6$ ($D_6$) x 4, $C_2^2\times D_4\times C_{12}$ ($C_2\times C_6$) x 2, $C_2^5.D_6$ ($C_2\times C_6$) x 2, $C_{12}:(C_2^3\times C_4)$ ($C_2\times C_6$) x 2, $C_2^2\times D_{12}:C_4$ ($C_2\times C_6$) x 2, $C_4\times C_2^2\times D_{12}$ ($C_2\times C_6$) x 2, $C_4\times D_4\times A_4$ ($D_6$) x 2, $C_2^2\times C_{12}:D_4$ ($C_2\times C_6$), $S_3\times D_4^2$ ($A_4$), $C_2^2\times C_4:D_{12}$ ($C_2\times C_6$), $C_2\times D_6\times C_4^2$ ($C_2\times C_6$), $A_4\times C_4:D_4$ ($D_6$)

Order 288: $A_4\times C_3:D_4$ ($C_2\times D_4$) x 16, $C_2\times A_4\times D_6$ ($C_2\times D_4$) x 16, $C_2\times A_4\times D_6$ ($C_2^4$) x 9, $C_2^3.C_6^2$ ($C_2\times D_4$) x 8, $A_4\times D_{12}$ ($C_2\times D_4$) x 8, $C_4\times S_3\times A_4$ ($C_2\times D_4$) x 8, $C_2^3:C_6^2$ ($C_2^4$) x 8, $C_2^3.C_6^2$ ($C_2^4$) x 7, $C_6:C_4\times A_4$ ($C_2^4$) x 7

Order 256: $C_2^2\times D_4^2$ ($C_3\times S_3$)

Order 192: $D_6:C_2^4$ ($C_2^2\times C_6$) x 32, $D_6\times C_2^4$ ($C_2^2\times C_6$) x 20, $C_{12}:C_2^4$ ($C_2^2\times C_6$) x 16, $C_{12}:C_2^4$ ($C_3\times D_4$) x 16, $C_2^3\times D_{12}$ ($C_2^2\times C_6$) x 16, $C_2^5:C_6$ ($C_2\times D_6$) x 16, $C_2^3.D_{12}$ ($C_2^2\times C_6$) x 16, $C_{12}:C_2^4$ ($C_2^2\times C_6$) x 15, $C_2^4:C_{12}$ ($C_2\times D_6$) x 8, $C_2^4:C_{12}$ ($C_2^2\times C_6$) x 8, $C_2^4.D_6$ ($C_2^2\times C_6$) x 8, $C_2^5\times C_6$ ($C_2^2\times C_6$) x 4, $A_4\times C_2^4$ ($C_2\times D_6$) x 4, $C_2^4\times C_{12}$ ($C_2^2\times C_6$) x 4, $C_6.C_2^5$ ($C_2^2\times C_6$) x 4, $C_{12}:C_2^4$ ($C_2\times A_4$) x 4, $C_2^4:C_{12}$ ($C_2\times D_6$) x 4, $C_2^4:D_6$ ($C_2\times A_4$) x 4, $C_2^4.D_6$ ($C_2^2\times C_6$) x 4, $D_{12}:D_4$ ($C_2\times A_4$) x 4, $S_3\times C_4:D_4$ ($C_2\times A_4$) x 4, $C_2^4:D_6$ ($C_2\times A_4$) x 4, $C_2^4:D_6$ ($C_2\times A_4$) x 4, $C_2^4.D_6$ ($C_2^2\times C_6$) x 3, $A_4\times C_4:C_4$ ($C_2\times D_6$) x 2, $C_2^2\times C_4:C_{12}$ ($C_2^2\times C_6$) x 2, $C_2^3.D_{12}$ ($C_2^2\times C_6$) x 2, $D_4\times D_{12}$ ($C_2\times A_4$) x 2, $C_4^2:D_6$ ($C_2\times A_4$) x 2, $A_4\times C_4^2$ ($C_2\times D_6$), $C_3\times D_4^2$ ($C_2\times A_4$), $C_2^2\times C_4\times C_{12}$ ($C_2^2\times C_6$), $C_3:D_4^2$ ($C_2\times A_4$), $C_4^2:D_6$ ($C_2\times A_4$)

Order 144: $A_4\times D_6$ ($C_2^2\times D_4$) x 12, $C_2^2:C_6^2$ ($C_2^2\times D_4$) x 8, $C_{12}\times A_4$ ($C_2^2\times D_4$) x 4, $A_4\times C_3:C_4$ ($C_2^2\times D_4$) x 4, $A_4\times D_6$ ($D_4:C_2^2$) x 2, $C_2^2:C_6^2$ ($C_2^5$)

Order 128: $D_4\times C_2^4$ ($C_6\times S_3$) x 4, $C_2^4:D_4$ ($C_6\times S_3$) x 4, $C_2^4:D_4$ ($C_6\times S_3$) x 4, $C_4^2:C_2^3$ ($C_6\times S_3$) x 2, $C_4^2:C_2^3$ ($C_6\times S_3$)

Order 96: $D_4\times D_6$ ($C_2^2\times A_4$) x 32, $C_2^3\times D_6$ ($C_6\times D_4$) x 16, $C_2^3:D_6$ ($C_6\times D_4$) x 16, $C_2^2:D_{12}$ ($C_2^2\times A_4$) x 12, $D_6:D_4$ ($C_2^2\times A_4$) x 12, $C_2^3\times D_6$ ($C_2^3\times C_6$) x 9, $D_6:D_4$ ($C_2^2\times A_4$) x 8, $D_6.D_4$ ($C_2^2\times A_4$) x 8, $C_2^4\times C_6$ ($C_2^3\times C_6$) x 8, $C_2^3\times A_4$ ($C_2^2\times D_6$) x 8, $C_{12}:C_2^3$ ($C_6\times D_4$) x 8, $C_2^3:D_6$ ($C_2^2\times A_4$) x 8, $C_2^2\times D_{12}$ ($C_6\times D_4$) x 8, $C_{12}:C_2^3$ ($C_6\times D_4$) x 8, $D_4\times A_4$ ($S_3\times D_4$) x 8, $C_2^3.D_6$ ($C_2^2\times A_4$) x 8, $C_2^3\times C_{12}$ ($C_2^3\times C_6$) x 7, $C_6.C_2^4$ ($C_2^3\times C_6$) x 7, $C_2^3:C_{12}$ ($C_2^2\times D_6$) x 7, $C_{12}:D_4$ ($C_2^2\times A_4$) x 6, $C_2^3.D_6$ ($C_2^2\times A_4$) x 4, $C_2^3\times D_6$ ($C_2^2\times A_4$) x 4, $C_{12}:C_2^3$ ($C_2^2\times A_4$) x 4, $C_2^2\times D_{12}$ ($C_2^2\times A_4$) x 4, $C_{12}:C_2^3$ ($C_2^2\times A_4$) x 4, $C_{12}:D_4$ ($C_2^2\times A_4$) x 4, $C_2^4:C_6$ ($C_2^2\times A_4$) x 4, $C_2^4:S_3$ ($C_2^2\times A_4$) x 4, $C_{12}:D_4$ ($C_2^2\times A_4$) x 4, $C_{12}:D_4$ ($C_2^2\times A_4$) x 4, $C_{12}:D_4$ ($C_2^2\times A_4$) x 4, $C_4:D_{12}$ ($C_2^2\times A_4$) x 4, $D_6.D_4$ ($C_2^2\times A_4$) x 2, $C_4\times D_{12}$ ($C_2^2\times A_4$) x 2, $D_4\times C_{12}$ ($C_2^2\times A_4$) x 2, $C_2^3.D_6$ ($C_2^2\times A_4$) x 2, $D_{12}:C_4$ ($C_2^2\times A_4$) x 2, $C_4:D_{12}$ ($C_2^2\times A_4$), $S_3\times C_4^2$ ($C_2^2\times A_4$), $C_{12}:D_4$ ($C_2^2\times A_4$)

Order 72: $S_3\times A_4$ ($D_4^2$) x 4, $C_6\times A_4$ ($D_4\times C_2^3$) x 2, $C_6\times A_4$ ($D_4:C_2^3$)

Order 64: $D_4\times C_2^3$ ($C_6\times D_6$) x 16, $C_2^4:C_4$ ($C_6\times D_6$) x 8, $C_2^6$ ($C_6\times D_6$) x 4, $C_2^4\times C_4$ ($C_6\times D_6$) x 4, $C_2^3.D_4$ ($C_6\times D_6$) x 2, $D_4^2$ ($S_3\times A_4$), $C_2^2\times C_4^2$ ($C_6\times D_6$)

Order 48: $C_6:D_4$ ($C_2^3\times A_4$) x 32, $C_2^2\times D_6$ ($C_2^3\times A_4$) x 20, $C_6\times D_4$ ($C_2^3\times A_4$) x 16, $S_3\times D_4$ ($D_4\times A_4$) x 16, $C_2\times D_{12}$ ($C_2^3\times A_4$) x 16, $D_6:C_4$ ($C_2^3\times A_4$) x 16, $C_4\times D_6$ ($C_2^3\times A_4$) x 15, $C_2^2\times D_6$ ($C_{12}:C_2^3$) x 12, $C_2^3\times C_6$ ($C_{12}:C_2^3$) x 8, $C_2^2\times A_4$ ($D_4\times D_6$) x 8, $C_2^2:C_{12}$ ($C_2^3\times A_4$) x 8, $C_6.D_4$ ($C_2^3\times A_4$) x 8, $C_2^3\times C_6$ ($C_2^3\times A_4$) x 4, $C_2^2\times C_{12}$ ($C_2^3\times A_4$) x 4, $C_2^2\times C_{12}$ ($C_{12}:C_2^3$) x 4, $C_6.C_2^3$ ($C_2^3\times A_4$) x 4, $C_6.C_2^3$ ($C_{12}:C_2^3$) x 4, $C_4\times A_4$ ($D_4\times D_6$) x 4, $C_6.D_4$ ($C_2^3\times A_4$) x 4, $C_{12}:C_4$ ($C_2^3\times A_4$) x 3, $C_2^2\times D_6$ ($C_6.C_2^4$) x 2, $C_4:C_{12}$ ($C_2^3\times A_4$) x 2, $C_{12}:C_4$ ($C_2^3\times A_4$) x 2, $C_2^3\times C_6$ ($C_2^4\times C_6$), $C_2^2\times A_4$ ($C_2^3\times D_6$), $C_4\times C_{12}$ ($C_2^3\times A_4$)

Order 36: $C_3\times A_4$ ($C_2\times D_4^2$)

Order 32: $C_2^5$ ($C_6^2:C_2^2$) x 8, $C_2^2\times D_4$ ($C_{12}:D_6$) x 8, $C_2^3\times C_4$ ($C_6^2:C_2^2$) x 7, $C_2^2\times D_4$ ($A_4\times D_6$) x 4, $C_4:D_4$ ($A_4\times D_6$) x 4, $C_2^2\wr C_2$ ($A_4\times D_6$) x 4, $C_4\times D_4$ ($A_4\times D_6$) x 2, $C_4:D_4$ ($A_4\times D_6$)

Order 24: $C_3:D_4$ ($C_2^5:C_6$) x 16, $C_2\times D_6$ ($C_2^5:C_6$) x 16, $C_2\times D_6$ ($A_4\times C_2^4$) x 9, $D_{12}$ ($C_2^5:C_6$) x 8, $C_4\times S_3$ ($C_2^5:C_6$) x 8, $C_2^2\times C_6$ ($A_4\times C_2^4$) x 8, $C_3\times D_4$ ($C_2^5:C_6$) x 8, $C_2\times C_{12}$ ($A_4\times C_2^4$) x 7, $C_6:C_4$ ($A_4\times C_2^4$) x 7, $C_2\times D_6$ ($C_3\times D_4^2$) x 4, $C_2^2\times C_6$ ($C_{12}:C_2^4$) x 2, $C_2\times A_4$ ($C_{12}:C_2^4$) x 2, $C_2^2\times C_6$ ($C_{12}.C_2^4$), $C_2\times A_4$ ($D_{12}:C_2^3$)

Order 16: $C_2\times D_4$ ($C_2\times A_4\times D_6$) x 16, $C_2^2:C_4$ ($C_2\times A_4\times D_6$) x 8, $C_2^4$ ($C_6^2:C_2^3$) x 8, $C_2^4$ ($C_2\times A_4\times D_6$) x 4, $C_2^2\times C_4$ ($C_6^2:C_2^3$) x 4, $C_2^2\times C_4$ ($C_2\times A_4\times D_6$) x 4, $C_4:C_4$ ($C_2\times A_4\times D_6$) x 2, $C_4^2$ ($C_2\times A_4\times D_6$), $C_2^4$ ($C_6^2:C_2^3$)

Order 12: $D_6$ ($C_2^6:C_6$) x 12, $C_2\times C_6$ ($C_2^6:C_6$) x 8, $C_{12}$ ($C_2^6:C_6$) x 4, $C_3:C_4$ ($C_2^6:C_6$) x 4, $D_6$ ($A_4\times D_4:C_2^2$) x 2, $C_2\times C_6$ ($A_4\times C_2^5$), $C_2\times C_6$ ($C_6\times D_4^2$), $A_4$ ($S_3\times D_4^2$)

Order 8: $C_2^3$ ($C_2^2\times A_4\times D_6$) x 8, $D_4$ ($S_3\times D_4\times A_4$) x 8, $C_2\times C_4$ ($C_2^2\times A_4\times D_6$) x 7, $C_2^3$ ($C_6^2:C_2^4$) x 2, $C_2^3$ ($C_6^2.C_2^4$)

Order 6: $S_3$ ($A_4\times D_4^2$) x 4, $C_6$ ($C_2^7:C_6$) x 2, $C_6$ ($D_4:C_2^3\times A_4$)

Order 4: $C_2^2$ ($D_4\times A_4\times D_6$) x 8, $C_4$ ($D_4\times A_4\times D_6$) x 4, $C_2^2$ ($C_2^3\times A_4\times D_6$), $C_2^2$ ($C_{12}^2:C_2^3$)

Order 3: $C_3$ ($C_2\times A_4\times D_4^2$)

Order 2: $C_2$ ($C_{12}.(A_4\times C_2^4)$) x 2, $C_2$ ($A_4\times D_{12}:C_2^3$)

Order 1: $C_1$ ($A_4.(S_3\times D_4^2)$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $300 \times 300$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $225 \times 225$ rational character table (warning: may be slow to load).