Abstract group 4608.pk: $C_2^5.D_6^2$

• Label: 4608.pk
• Order: \( 2^{9} \cdot 3^{2} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$C_2^5.D_6^2$
Order:	\(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$(C_6\times A_4).C_2^6.C_2^3$, of order \(36864\)\(\medspace = 2^{12} \cdot 3^{2} \)
Composition factors:	$C_2$ x 9, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	12
Elements	1	511	26	1536	1190	1344	4608
Conjugacy classes	1	35	3	28	48	23	138
Divisions	1	35	3	28	39	20	126
Autjugacy classes	1	23	3	18	27	15	87

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	36	16	30	20	8	10	0	2	138
Irr. rational chars.	16	28	16	22	20	10	10	2	2	126

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{4}=d^{2}=e^{12}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,2)(4,6)(5,7)(8,14)(9,11)(10,12)(13,15), (1,2), (1,3)(4,7)(8,10,14,12) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^5$ . $D_6^2$	$(C_2^6:D_6)$ . $S_3$	$(C_2^4:D_6)$ . $S_4$	$C_2^6$ . $(S_3\times D_6)$	all 166

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 241 normal subgroups (137 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 $C_2$
Commutator:	a subgroup isomorphic to $C_2^3:C_6^2$
Frattini:	a subgroup isomorphic to $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6:D_4$

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 18 (order at least 256) are shown, as well as all normal subgroups of any index.

Order 4608: $C_2^5.D_6^2$

Order 2304: $C_2^6:S_3^2$ x 2, $(C_2^2\times C_6\times A_4).C_2^3$ x 2, $C_2^6:S_3^2$ x 2, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^4.D_6^2$, $A_4\times C_2^4:D_6$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$

Order 1536: $C_2^4:\GL(2,\mathbb{Z}/4)$, $S_3\times C_2^5:D_4$

Order 1152: $D_6\times \GL(2,\mathbb{Z}/4)$ x 5, $D_6:\GL(2,\mathbb{Z}/4)$ x 4, $D_6:\GL(2,\mathbb{Z}/4)$ x 4, $D_6.\GL(2,\mathbb{Z}/4)$ x 4, $\GL(2,\mathbb{Z}/4):D_6$ x 3, $D_4\times A_4\times D_6$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $\GL(2,\mathbb{Z}/4):D_6$ x 2, $A_4\times D_6:D_4$ x 2, $A_4\times C_2^2:D_{12}$ x 2, $S_3\times C_2^4:C_{12}$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times S_4):C_{12}$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $D_6^2:D_4$, $C_2^3\times A_4\times D_6$, $(C_6\times D_4):S_4$, $C_3\times \GL(2,\mathbb{Z}/4):C_2^2$, $\GL(2,\mathbb{Z}/4):D_6$, $\GL(2,\mathbb{Z}/4):D_6$, $A_4\times C_2^4:S_3$, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$, $C_2^5:C_6^2$

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ x 9, $(C_2^2\times D_{12}):D_4$ x 4, $S_3\times C_2^4:D_4$ x 4, $C_2^6:D_6$ x 2, $C_2^4:(C_2\times D_{12})$ x 2, $C_2^4:(C_2\times D_{12})$ x 2, $C_2^6:D_6$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2^4:(C_4\times D_6)$ x 2, $C_2^4:(C_4\times D_6)$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $C_2^5:D_{12}$, $(C_2^3\times C_{12}):D_4$, $C_2^6:D_6$, $C_3\times C_2^5:D_4$, $C_2^5:C_4\times S_3$, $\GL(2,\mathbb{Z}/4):C_2^3$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $C_2^7:C_6$, $D_4^2:D_6$, $C_2^6:D_6$

Order 576: $S_3\times \GL(2,\mathbb{Z}/4)$ x 22, $C_2^2\times A_4\times D_6$ x 13, $S_3\times D_4\times A_4$ x 8, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_6:\GL(2,\mathbb{Z}/4)$ x 5, $C_6\times \GL(2,\mathbb{Z}/4)$ x 5, $C_6:\GL(2,\mathbb{Z}/4)$ x 5, $C_6:\GL(2,\mathbb{Z}/4)$ x 5, $C_2^4.S_3^2$ x 5, $C_6:D_4\times A_4$ x 4, $\GL(2,\mathbb{Z}/4):S_3$ x 4, $C_2^4:S_3^2$ x 4, $\GL(2,\mathbb{Z}/4):S_3$ x 4, $C_2^4.S_3^2$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^2:D_6^2$ x 3, $A_4\times D_6:C_4$ x 3, $C_2^4:C_6^2$ x 2, $C_2\times A_4\times D_{12}$ x 2, $C_4\times A_4\times D_6$ x 2, $(C_3\times D_4):S_4$ x 2, $D_4\times C_3:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_6$ x 2, $\GL(2,\mathbb{Z}/4):C_6$ x 2, $\GL(2,\mathbb{Z}/4):S_3$ x 2, $D_{12}:S_4$ x 2, $S_4\times D_{12}$ x 2, $C_4\times S_3\times S_4$ x 2, $(C_4\times S_3):S_4$ x 2, $D_{12}:S_4$ x 2, $D_{12}:S_4$ x 2, $C_2^4.C_6^2$ x 2, $C_2^4:S_3^2$ x 2, $A_4\times C_6.D_4$ x 2, $C_2^4:S_3^2$ x 2, $(C_2\times D_6):D_{12}$ x 2, $D_6^2:C_4$ x 2, $C_2^4:C_6^2$, $D_6^2:C_2^2$, $(C_2\times C_{12}):S_4$, $\GL(2,\mathbb{Z}/4):C_6$, $S_3\times C_4:S_4$, $(C_4\times S_3):S_4$, $(C_4\times S_3):S_4$, $S_3\times A_4:Q_8$, $C_6^2.C_2^4$, $(C_6\times C_{12}):D_4$, $C_3\times C_2^4:D_6$, $(C_2\times C_{12}):D_{12}$, $(C_6\times C_{12}):D_4$, $(C_6\times D_{12}):C_4$

Order 512: $C_2^6:D_4$

Order 384: $C_2^5:D_6$ x 23, $C_2\wr C_2^2\times S_3$ x 16, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 14, $\GL(2,\mathbb{Z}/4):C_2^2$ x 11, $C_2^4:D_{12}$ x 10, $C_2^5.D_6$ x 10, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 10, $C_2^5.D_6$ x 10, $C_2^5.D_6$ x 8, $C_2^4:D_{12}$ x 8, $C_2^3:C_4\times D_6$ x 8, $C_2^6:C_6$ x 7, $C_2^5:D_6$ x 7, $C_2^4:D_{12}$ x 6, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^2\times \GL(2,\mathbb{Z}/4)$ x 5, $C_3\times C_2^4:D_4$ x 4, $(C_2^2\times C_{12}):D_4$ x 4, $C_2^4:(C_4\times S_3)$ x 4, $(C_2^2\times D_{12}):C_4$ x 4, $C_2^4:D_{12}$ x 4, $C_2^5.D_6$ x 4, $D_4:(S_3\times D_4)$ x 4, $S_3\times D_4:D_4$ x 4, $D_4^2:S_3$ x 4, $S_3\times D_4^2$ x 4, $C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times D_{12}):C_4$ x 4, $S_3\times C_2^2.C_4^2$ x 4, $(C_2^2\times C_4):D_{12}$ x 4, $(C_2^2\times C_4):D_{12}$ x 4, $C_2^3:(S_3\times D_4)$ x 3, $S_3\times C_2^3:D_4$ x 3, $C_2^5:D_6$ x 3, $C_2^4:(C_2\times C_{12})$ x 2, $C_2^4:(C_2\times C_{12})$ x 2, $C_2^5.D_6$ x 2, $C_2^5.D_6$ x 2, $(C_2^2\times C_{12}):D_4$ x 2, $C_2^4.D_{12}$ x 2, $C_2^4.D_{12}$ x 2, $C_2^4.D_{12}$ x 2, $C_2^4.D_{12}$ x 2, $C_{12}:C_2^5$ x 2, $C_2^6:C_6$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $\GL(2,\mathbb{Z}/4):C_2^2$ x 2, $C_2^5:D_6$ x 2, $C_2^4:D_{12}$ x 2, $C_2^5.D_6$ x 2, $C_2^5:C_{12}$ x 2, $C_2\wr C_2^2\times C_6$ x 2, $C_2^6:C_6$ x 2, $C_2^6:S_3$ x 2, $(C_6\times D_4):D_4$ x 2, $C_2^5:C_{12}$, $C_2^5.D_6$, $D_6\times C_2^5$, $A_4\times C_2^5$, $D_{12}:C_2^4$, $\GL(2,\mathbb{Z}/4):C_2^2$, $D_4^2:C_6$, $C_2^6:C_6$, $C_2^5:D_6$, $(C_6\times D_4):D_4$, $C_2^5:D_6$, $(C_2\times D_{12}):D_4$, $C_4^2:C_2^2\times S_3$, $(C_2\times D_{12}):D_4$, $C_4^2:C_2^2\times S_3$, $(C_2\times D_4):D_{12}$, $C_4^2:C_2^2\times S_3$

Order 288: $C_2\times A_4\times D_6$ x 52, $D_6\times S_4$ x 12, $C_3:\GL(2,\mathbb{Z}/4)$ x 10, $C_3\times \GL(2,\mathbb{Z}/4)$ x 10, $D_6:S_4$ x 10, $D_6:S_4$ x 10, $D_6.S_4$ x 10, $A_4\times C_3:D_4$ x 8, $C_2^3:C_6^2$ x 7, $D_6^2:C_2$ x 5, $C_2\times C_6.S_4$ x 5, $C_2\times A_4:C_{12}$ x 5, $C_2^3.C_6^2$ x 4, $A_4\times D_{12}$ x 4, $C_4\times S_3\times A_4$ x 4, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 4, $C_2^3.S_3^2$ x 4, $C_6^2.D_4$ x 4, $C_2^3:S_3^2$ x 3, $C_6^2:D_4$ x 3, $C_2\times C_6:S_4$ x 3, $C_2\times C_6\times S_4$ x 3, $C_6^2:C_2^3$ x 2, $C_2^3.C_6^2$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $D_4\times S_3^2$ x 2, $C_6:C_4\times A_4$ x 2, $C_{12}:S_4$ x 2, $C_{12}\times S_4$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_2^3.S_3^2$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_6^2.C_2^3$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.C_2^3$, $D_{12}:D_6$, $D_{12}:D_6$, $C_{12}:S_4$, $C_{12}.S_4$, $C_{12}:S_4$, $C_{12}.S_4$, $C_6^2:D_4$, $C_6^2.D_4$, $C_6^2.D_4$, $C_6^2:C_2^3$, $C_6^2.C_2^3$

Order 256: $C_2^5:D_4$ x 16, $C_2^5:D_4$ x 4, $C_2^5:D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_4^2:C_2^4$, $C_2^5:D_4$, $C_2^6:C_4$

Order 192: $D_6\times C_2^4$ x 3, $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $D_6:C_2^4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:C_{12}$, $C_2^5\times C_6$, $A_4\times C_2^4$, $C_{12}:C_2^4$, $C_2^3\times D_{12}$, $C_{12}:C_2^4$, $C_2^5:C_6$, $C_2^4:C_{12}$, $C_2^4.D_6$, $C_2^4:D_6$

Order 144: $A_4\times D_6$ x 4, $C_2^2:C_6^2$ x 3

Order 128: $C_2^4:D_4$

Order 96: $C_2^3\times D_6$ x 8, $C_2^4\times C_6$ x 2, $C_2^3\times A_4$ x 2, $C_2^2\times S_4$ x 2, $C_2^2.S_4$ x 2, $C_2^2:D_{12}$, $D_6.D_4$, $C_2^3\times C_{12}$, $C_6.C_2^4$, $D_4\times D_6$, $C_2^3:C_{12}$, $C_2^4:C_6$, $C_2^4:S_3$, $D_6:D_4$

Order 72: $S_3\times A_4$ x 2, $C_6\times A_4$

Order 64: $C_2^6$, $D_4\times C_2^3$, $C_2^4:C_4$

Order 48: $C_2^2\times D_6$ x 7, $C_2^3\times C_6$ x 4, $C_2^2\times A_4$ x 3, $C_6:D_4$ x 2, $C_6\times D_4$, $C_2\times D_{12}$, $C_4\times D_6$, $C_2^2:C_{12}$, $C_6.D_4$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$ x 2, $C_2^3\times C_4$, $C_2^2\wr C_2$

Order 24: $C_2\times D_6$ x 9, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$, $C_6:C_4$, $C_2\times A_4$

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

Order 12: $C_2\times C_6$ x 4, $D_6$ x 4, $A_4$

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

Order 6: $S_3$ x 2, $C_6$

Order 4: $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 4608: $C_2^5.D_6^2$ ($C_1$)

Order 2304: $C_2^6:S_3^2$ ($C_2$) x 2, $(C_2^2\times C_6\times A_4).C_2^3$ ($C_2$) x 2, $C_2^6:S_3^2$ ($C_2$) x 2, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $S_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^4.D_6^2$ ($C_2$), $A_4\times C_2^4:D_6$ ($C_2$), $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ ($C_2$)

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ ($C_2^2$) x 2, $D_6\times \GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $D_6:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $D_6:\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $D_6.\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ ($C_2^2$) x 2, $(C_2^2\times S_4):C_{12}$ ($C_2^2$) x 2, $C_2^5.S_3^2$ ($C_2^2$) x 2, $C_2^5.S_3^2$ ($C_2^2$) x 2, $C_2^5.S_3^2$ ($C_2^2$) x 2, $C_2^3\times A_4\times D_6$ ($C_2^2$), $D_4\times A_4\times D_6$ ($C_2^2$), $(C_6\times D_4):S_4$ ($C_2^2$), $C_3\times \GL(2,\mathbb{Z}/4):C_2^2$ ($C_2^2$), $A_4\times C_2^4:S_3$ ($C_2^2$), $A_4\times D_6:D_4$ ($C_2^2$), $A_4\times C_2^2:D_{12}$ ($C_2^2$), $S_3\times C_2^4:C_{12}$ ($C_2^2$), $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^2$), $C_2^5:C_6^2$ ($C_2^2$)

Order 768: $C_2^3:\GL(2,\mathbb{Z}/4)$ ($S_3$), $C_2^6:D_6$ ($S_3$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2^3$) x 2, $C_2^2:D_6^2$ ($D_4$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $C_6\times \GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($D_4$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($D_4$) x 2, $C_2^4.S_3^2$ ($D_4$) x 2, $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $C_2^4:C_6^2$ ($C_2^3$), $C_2^2\times A_4\times D_6$ ($D_4$), $C_2^4:C_6^2$ ($C_2^3$), $C_6:D_4\times A_4$ ($C_2^3$), $C_6:D_4\times A_4$ ($D_4$), $C_2\times A_4\times D_{12}$ ($D_4$), $C_4\times A_4\times D_6$ ($D_4$), $C_2^4.C_6^2$ ($C_2^3$), $A_4\times C_6.D_4$ ($C_2^3$)

Order 384: $C_2^2.\GL(2,\mathbb{Z}/4)$ ($D_6$) x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ ($D_6$) x 2, $C_2^2.\GL(2,\mathbb{Z}/4)$ ($D_6$), $D_6\times C_2^5$ ($D_6$), $C_{12}:C_2^5$ ($D_6$), $\GL(2,\mathbb{Z}/4):C_2^2$ ($D_6$), $C_2^6:C_6$ ($D_6$), $C_2^5:D_6$ ($D_6$), $C_2^5:D_6$ ($D_6$), $C_2^4:D_{12}$ ($D_6$), $C_2^5.D_6$ ($D_6$), $C_2^6:C_6$ ($D_6$)

Order 288: $C_2\times A_4\times D_6$ ($C_2\times D_4$) x 7, $C_2\times C_6.S_4$ ($C_2\times D_4$) x 2, $C_2\times A_4:C_{12}$ ($C_2\times D_4$) x 2, $C_2\times C_6:S_4$ ($C_2\times D_4$) x 2, $C_2\times C_6\times S_4$ ($C_2\times D_4$) x 2, $C_2^3.C_6^2$ ($C_2\times D_4$), $C_6:C_4\times A_4$ ($C_2\times D_4$), $C_2^3:C_6^2$ ($C_2^4$), $C_2^3:C_6^2$ ($C_2\times D_4$)

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 2, $D_6\times C_2^4$ ($C_2\times D_6$) x 2, $C_2\times \GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 2, $C_2^4:C_{12}$ ($C_2\times D_6$), $C_2^5\times C_6$ ($C_2\times D_6$), $D_6\times C_2^4$ ($C_3:D_4$), $A_4\times C_2^4$ ($C_2\times D_6$), $C_{12}:C_2^4$ ($C_2\times D_6$), $D_6:C_2^4$ ($C_3:D_4$), $D_6:C_2^4$ ($C_2\times D_6$), $C_2^3\times D_{12}$ ($C_3:D_4$), $C_{12}:C_2^4$ ($C_3:D_4$), $C_2^5:C_6$ ($C_2\times D_6$), $C_2^4:C_{12}$ ($C_2\times D_6$), $C_2^4.D_6$ ($C_2\times D_6$), $C_2^4:D_6$ ($S_4$)

Order 144: $A_4\times D_6$ ($C_2^2\wr C_2$) x 4, $C_2^2:C_6^2$ ($C_2^2\times D_4$) x 3

Order 128: $C_2^4:D_4$ ($S_3^2$)

Order 96: $C_2^3\times D_6$ ($S_3\times D_4$) x 4, $C_2^3\times D_6$ ($C_6:D_4$) x 3, $C_2^2\times S_4$ ($S_3\times D_4$) x 2, $C_2^2.S_4$ ($S_3\times D_4$) x 2, $C_2^2:D_{12}$ ($C_2\times S_4$), $D_6.D_4$ ($C_2\times S_4$), $C_2^4\times C_6$ ($C_2^2\times D_6$), $C_2^4\times C_6$ ($C_6:D_4$), $C_2^3\times D_6$ ($C_2\times S_4$), $C_2^3\times A_4$ ($C_2^2\times D_6$), $C_2^3\times A_4$ ($S_3\times D_4$), $C_2^3\times C_{12}$ ($C_6:D_4$), $C_6.C_2^4$ ($C_6:D_4$), $D_4\times D_6$ ($C_2\times S_4$), $C_2^3:C_{12}$ ($S_3\times D_4$), $C_2^4:C_6$ ($C_2\times S_4$), $C_2^4:S_3$ ($C_2\times S_4$), $D_6:D_4$ ($C_2\times S_4$)

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

Order 64: $C_2^6$ ($S_3\times D_6$), $D_4\times C_2^3$ ($S_3\times D_6$), $C_2^4:C_4$ ($S_3\times D_6$)

Order 48: $C_2^2\times D_6$ ($D_6:D_4$) x 4, $C_2^2\times A_4$ ($D_4\times D_6$) x 3, $C_2^3\times C_6$ ($D_4\times D_6$) x 2, $C_2^2\times D_6$ ($C_2^2\times S_4$) x 2, $C_2^3\times C_6$ ($C_2^2\times S_4$), $C_2^3\times C_6$ ($C_2^3:D_6$), $C_2^2\times D_6$ ($\GL(2,\mathbb{Z}/4)$), $C_6\times D_4$ ($C_2^2\times S_4$), $C_6:D_4$ ($C_2^2\times S_4$), $C_6:D_4$ ($\GL(2,\mathbb{Z}/4)$), $C_2\times D_{12}$ ($\GL(2,\mathbb{Z}/4)$), $C_4\times D_6$ ($\GL(2,\mathbb{Z}/4)$), $C_2^2:C_{12}$ ($C_2^2\times S_4$), $C_6.D_4$ ($C_2^2\times S_4$)

Order 36: $C_3\times A_4$ ($C_2^4:D_4$)

Order 32: $C_2^5$ ($D_6^2$), $C_2^5$ ($D_6:D_6$), $C_2^3\times C_4$ ($D_6:D_6$), $C_2^2\wr C_2$ ($S_3\times S_4$)

Order 24: $C_2\times D_6$ ($D_4\times S_4$) x 4, $C_2\times D_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$) x 3, $C_2\times D_6$ ($C_2^4:D_6$) x 2, $C_2\times C_{12}$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_6:C_4$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_2^2\times C_6$ ($C_2^3\times S_4$), $C_2^2\times C_6$ ($C_2\times \GL(2,\mathbb{Z}/4)$), $C_2^2\times C_6$ ($C_2^4:D_6$), $C_2\times A_4$ ($C_2^4:D_6$)

Order 16: $C_2^4$ ($D_4\times S_3^2$) x 2, $C_2^2:C_4$ ($D_6\times S_4$), $C_2^4$ ($D_6^2:C_2$), $C_2^4$ ($D_6\times S_4$), $C_2\times D_4$ ($D_6\times S_4$)

Order 12: $D_6$ ($C_2^2:\GL(2,\mathbb{Z}/4)$) x 4, $C_2\times C_6$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_2\times C_6$ ($C_2^2\times \GL(2,\mathbb{Z}/4)$), $C_2\times C_6$ ($C_2^5:D_6$), $A_4$ ($C_2\wr C_2^2\times S_3$)

Order 8: $C_2^3$ ($C_2^2:D_6^2$), $C_2^3$ ($S_3\times \GL(2,\mathbb{Z}/4)$), $C_2^3$ ($D_6^2:C_2^2$), $C_2\times C_4$ ($S_3\times \GL(2,\mathbb{Z}/4)$)

Order 6: $S_3$ ($C_2^3:\GL(2,\mathbb{Z}/4)$) x 2, $C_6$ ($C_2^3:\GL(2,\mathbb{Z}/4)$)

Order 4: $C_2^2$ ($\GL(2,\mathbb{Z}/4):D_6$) x 2, $C_2^2$ ($D_6^2:D_4$), $C_2^2$ ($D_6\times \GL(2,\mathbb{Z}/4)$)

Order 3: $C_3$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 2: $C_2$ ($C_2^4.D_6^2$)

Order 1: $C_1$ ($C_2^5.D_6^2$)

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 4 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $138 \times 138$ 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 $126 \times 126$ rational character table.