Abstract group 18432.bx: $A_4^2.C_2^5:C_4$

• Label: 18432.bx
• Order: \( 2^{11} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$A_4^2.C_2^5:C_4$
Order:	\(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$S_4\times S_5$, of order \(294912\)\(\medspace = 2^{15} \cdot 3^{2} \)
Composition factors:	$C_2$ x 11, $C_3$ 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	8	12
Elements	1	1039	80	6384	2480	6912	1536	18432
Conjugacy classes	1	31	2	36	31	12	9	122
Divisions	1	31	2	34	31	6	9	114
Autjugacy classes	1	22	2	23	16	3	6	73

Dimension	1	2	4	6	8	9	12	18	24	36
Irr. complex chars.	16	4	22	16	12	16	20	4	6	6	122
Irr. rational chars.	8	8	22	16	12	8	20	8	6	6	114

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid a^{4}=b^{6}=c^{12}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,2,4,5)(7,8)(9,10)(11,13)(12,14)(15,16), (1,3,5,8)(2,6)(4,7)(9,12,11,15)(10,14,13,16), (2,5)(6,7)(9,11)(10,13)(12,16,14,15)\rangle$
Transitive group:	24T12277	24T12306			more information
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_4^2:C_2^4)$ . $D_4$	$A_4^2$ . $(C_2^5:C_4)$	$C_2^7$ . $(C_6^2:C_4)$	$C_2^7$ . $(C_6^2:C_4)$	all 48

Elements of the group are displayed as permutations of degree 16.

Homology

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

Subgroups

There are 64 normal subgroups (54 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 $C_2^3\times A_4^2$
Frattini:	a subgroup isomorphic to $C_2^3$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2^4.C_2$

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 36 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 18432: $A_4^2.C_2^5:C_4$

Order 9216: $A_4^2:C_2\wr C_4$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2.C_2^5.C_2$, $A_4^2:\OD_{16}:C_2^2$, $A_4^2:C_2^4:C_4$

Order 4608: $A_4^2.C_2^4.C_2$ x 7, $(C_2^3\times A_4^2).C_4$ x 3, $A_4^2:C_2^2\times D_4$ x 2, $(C_2^3\times A_4^2):C_4$ x 2, $C_2^6.C_6^2.C_2$, $C_2^3\times A_4^2.C_2^2$, $C_2^3:(A_4^2:C_4)$, $A_4^2:(C_2\times \OD_{16})$, $A_4^2:C_2^3:C_4$

Order 3072: $A_4.C_2^6.C_2^2$, $C_2^8.D_6$

Order 2304: $A_4^2:C_2^4$ x 11, $D_4\times \PSOPlus(4,3)$ x 6, $A_4^2.C_2^3.C_2$ x 5, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times A_4^2):C_4$ x 3, $C_2^5.C_6^2.C_2$ x 2, $C_2\times A_4^2:D_4$ x 2, $C_4\times A_4^2:C_2^2$ x 2, $C_2^6:C_6^2$ x 2, $A_4^2:\OD_{16}$ x 2, $C_2^3\times C_2\times A_4^2$, $(A_4\times C_2^4):D_6$, $A_4^2:C_4\times C_2^2$, $A_4^2:\OD_{16}$, $A_4^2:(C_2\times C_8)$

Order 2048: $C_2^6.C_2^4.C_2$

Order 1536: $C_2^7:D_6$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:D_{12}$ x 3, $C_2^7.D_6$ x 3, $C_2^7:D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^5.C_2^3.S_3$ x 3, $\GL(2,\mathbb{Z}/4)\times C_2^2:C_4$ x 3, $C_2^6:S_4$, $C_2^4\times \GL(2,\mathbb{Z}/4)$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^7:D_6$, $C_2^8.S_3$, $C_2^6.C_2.D_6$, $C_2^6.C_2.D_6$, $A_4:C_4\times C_2^2\wr C_2$, $C_2^8:C_6$, $C_2^2\times C_2^3.(C_6\times D_4)$

Order 1152: $A_4^2:C_2^3$ x 45, $C_2^3\times A_4^2$ x 6, $A_4:\GL(2,\mathbb{Z}/4)$ x 6, $C_2\times A_4^2:C_4$ x 4, $C_4\times \PSOPlus(4,3)$ x 4, $C_6:S_4\times D_4$ x 3, $D_4\times A_4^2$ x 3, $A_4^2:D_4$ x 3, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times A_4):D_{12}$ x 3, $(C_2^3\times C_{12}):D_6$ x 3, $C_2^5.C_6^2$ x 2, $C_2\times A_4^2:C_4$ x 2, $A_4^2:C_8$ x 2, $C_6^2:D_4:C_4$, $C_2^3\times C_6:S_4$, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^5:C_6^2$

Order 1024: $C_2^6.C_2^3.C_2$ x 6, $C_2^5.C_2^4.C_2$ x 5, $C_2^2\wr C_4$ x 2, $C_2^7.C_2^3$, $C_2^5.C_2^3.C_2^2$

Order 768: $C_2^3\times \GL(2,\mathbb{Z}/4)$ x 37, $D_4\times \GL(2,\mathbb{Z}/4)$ x 24, $C_2^5:S_4$ x 19, $C_2^6:D_6$ x 12, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 12, $A_4.(C_2.C_4:D_4)$ x 12, $C_2^6:D_6$ x 11, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 10, $C_2^7:C_6$ x 8, $A_4.(C_2^5.C_2)$ x 7, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $(C_2^3\times S_4).C_2^2$ x 6, $A_4.(C_2.C_4:D_4)$ x 6, $A_4.(C_2^3.D_4)$ x 6, $(C_2^3\times A_4).C_2^3$ x 6, $A_4.(C_2^3.D_4)$ x 6, $A_4.(C_2^3.D_4)$ x 6, $A_4.C_2^5.C_2$ x 6, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $A_4.(C_2^3.D_4)$ x 3, $(C_2^3\times A_4).C_2^3$ x 3, $(C_2^3\times A_4).C_2^3$ x 3, $A_4:C_4\times C_2^2:C_4$ x 3, $C_2^7:C_6$ x 3, $C_2^7:C_6$ x 3, $C_2^6:C_{12}$ x 3, $C_2^7.C_6$ x 3, $C_2^6:D_6$, $S_4\times C_2^5$, $C_2^4\times A_4:C_4$, $C_2^8.C_3$, $C_2^8.C_3$

Order 576: $A_4^2:C_2^2$ x 39, $C_2^2\times C_6:S_4$ x 19, $C_2^2\times A_4^2$ x 13, $D_4\times C_3:S_4$ x 12, $C_6:\GL(2,\mathbb{Z}/4)$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:C_6^2$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2^4.C_6^2$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $A_4^2:C_4$ x 2, $C_4\times A_4^2$ x 2, $A_4^2:C_4$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_3^2:C_2\wr C_4$ x 2, $C_2^4:C_6^2$, $C_6^2.C_2^4$, $C_6^2.(C_2^2\times C_4)$, $C_6^2.(C_2^2\times C_4)$

Order 512: $C_2^7.C_2^2$ x 10, $C_2^7.C_4$ x 8, $C_2^3.C_2^5.C_2$ x 6, $C_2^7:C_4$ x 6, $C_2^7.C_2^2$ x 5, $C_2^4.C_2^5$ x 5, $C_2^3:D_4^2$ x 4, $C_2^6.C_2^3$ x 4, $C_2^6:D_4$ x 4, $C_2^3.C_2^4.C_2^2$ x 4, $C_2^4.C_2^5$ x 4, $C_2^5.C_2^3.C_2$ x 4, $C_2^2.C_2^6.C_2$ x 2, $C_2^5.(C_2\times D_4)$ x 2, $C_2^6.D_4$ x 2, $C_2^3.(C_2^3\times C_4).C_2$ x 2, $(C_2^3\times D_4).C_2^3$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $(C_2^2\times D_4).C_2^4$ x 2, $C_2^7:C_4$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.C_2^5.C_2$ x 2, $C_2^6:D_4$ x 2, $C_2^2.C_2^6.C_2$, $C_2^6.C_2^3$, $C_2^4\wr C_2$

Order 288: $\PSOPlus(4,3)$ x 2, $C_2\times A_4^2$

Order 256: $C_2^8$, $D_4\times C_2^5$, $C_2^6:C_4$

Order 144: $A_4^2$

Order 128: $C_2^7$ x 2, $C_2^5\times C_4$

Order 64: $C_2^6$

Order 32: $C_2^5$, $C_2^2\wr C_2$

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

Order 8: $C_2^3$ x 2, $C_2\times C_4$

Order 4: $C_2^2$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 18432: $A_4^2.C_2^5:C_4$ ($C_1$)

Order 9216: $A_4^2:C_2\wr C_4$ ($C_2$) x 2, $A_4^2:C_2\wr C_4$ ($C_2$) x 2, $A_4^2.C_2^5.C_2$ ($C_2$), $A_4^2:\OD_{16}:C_2^2$ ($C_2$), $A_4^2:C_2^4:C_4$ ($C_2$)

Order 4608: $A_4^2.C_2^4.C_2$ ($C_2^2$) x 2, $A_4^2.C_2^4.C_2$ ($C_4$) x 2, $(C_2^3\times A_4^2).C_4$ ($C_2^2$) x 2, $(C_2^3\times A_4^2):C_4$ ($C_2^2$) x 2, $C_2^6.C_6^2.C_2$ ($C_4$), $C_2^3\times A_4^2.C_2^2$ ($C_4$), $A_4^2:C_2^2\times D_4$ ($C_2^2$)

Order 2304: $A_4^2:C_2^4$ ($C_2\times C_4$) x 2, $C_2^5.C_6^2.C_2$ ($C_2\times C_4$), $C_2^3\times C_2\times A_4^2$ ($C_2\times C_4$), $A_4^2.C_2^3.C_2$ ($C_2\times C_4$), $C_2\times A_4^2:D_4$ ($D_4$), $C_4\times A_4^2:C_2^2$ ($D_4$), $A_4^2:C_2^4$ ($D_4$), $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ ($D_4$), $C_2^6:C_6^2$ ($C_2\times C_4$)

Order 1152: $A_4^2:C_2^3$ ($C_2^2:C_4$) x 2, $C_2^3\times A_4^2$ ($C_2^2:C_4$), $C_2^3\times A_4^2$ ($C_2^2\times C_4$), $A_4^2:C_2^3$ ($C_2\times D_4$), $C_2^5.C_6^2$ ($C_2^2:C_4$), $C_2\times A_4^2:C_4$ ($C_2\times D_4$)

Order 576: $A_4^2:C_2^2$ ($C_2^3:C_4$) x 2, $C_2^2\times A_4^2$ ($C_2^3:C_4$)

Order 512: $C_2^6:D_4$ ($C_3^2:C_4$)

Order 288: $\PSOPlus(4,3)$ ($C_2\wr C_4$) x 2, $C_2\times A_4^2$ ($C_2^4:C_4$)

Order 256: $C_2^8$ ($C_2\times C_3^2:C_4$), $D_4\times C_2^5$ ($C_2\times C_3^2:C_4$), $C_2^6:C_4$ ($C_2\times C_3^2:C_4$)

Order 144: $A_4^2$ ($C_2^5:C_4$)

Order 128: $C_2^7$ ($C_6^2:C_4$), $C_2^7$ ($C_6^2:C_4$), $C_2^5\times C_4$ ($C_6^2:C_4$)

Order 64: $C_2^6$ ($C_2\times C_6^2:C_4$)

Order 32: $C_2^5$ ($C_6^2.(C_2^2\times C_4)$), $C_2^2\wr C_2$ ($A_4^2:C_4$)

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

Order 8: $C_2^3$ ($A_4^2:C_4\times C_2^2$), $C_2^3$ ($(C_2^2\times A_4^2):C_4$), $C_2\times C_4$ ($(C_2^2\times A_4^2):C_4$)

Order 4: $C_2^2$ ($C_2^3:(A_4^2:C_4)$)

Order 2: $C_2$ ($A_4^2:C_2^4:C_4$)

Order 1: $C_1$ ($A_4^2.C_2^5:C_4$)

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 9 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $122 \times 122$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $114 \times 114$ rational character table.