Abstract group 384.18221: $C_2^5:A_4$

• Label: 384.18221
• Order: \( 2^{7} \cdot 3 \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $12$
• Trans deg.: $24$
• Rank: $3$

Group information

Description:	$C_2^5:A_4$
Order:	\(384\)\(\medspace = 2^{7} \cdot 3 \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.C_2^6.C_3.S_3$, of order \(73728\)\(\medspace = 2^{13} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$
Derived length:	$2$

This group is nonabelian, monomial (hence solvable), and metabelian.

Group statistics

Order	1	2	3	4	6
Elements	1	79	128	48	128	384
Conjugacy classes	1	15	2	4	2	24
Divisions	1	15	1	4	1	22
Autjugacy classes	1	5	1	1	1	9

Dimension	1	2	3	6
Irr. complex chars.	6	0	10	8	24
Irr. rational chars.	2	2	10	8	22

Minimal presentations

Permutation degree:	$12$
Transitive degree:	$24$
Rank:	$3$
Inequivalent generating triples:	$455$

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:	${\langle a, b, c, d, e, f, g \mid a^{6}=b^{2}=c^{2}=d^{2}=e^{2}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $12$ $\langle(1,2)(3,8)(4,6)(5,7), (3,5,4)(6,8,7)(10,12,11), (1,3)(2,6)(4,5)(7,8), (1,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 1 & 0 & 1 & 0 & 0 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 \end{array}\right) \right\rangle \subseteq \GL_{5}(\F_{2})$
Transitive group:	24T1023	24T1030	24T1031	24T1034	all 5
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_2^5$ $\,\rtimes\,$ $A_4$	$C_2^6$ $\,\rtimes\,$ $C_6$	$(C_2^4:A_4)$ $\,\rtimes\,$ $C_2$	$C_2^4$ $\,\rtimes\,$ $(C_2\times A_4)$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^4$ . $(C_2\times A_4)$	$C_2^2$ . $(C_2^3:A_4)$			more information

Elements of the group are displayed as permutations of degree 12.

Homology

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

Subgroups

There are 4726 subgroups in 824 conjugacy classes, 21 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^5:A_4$
Commutator:	$G' \simeq$ $C_2^6$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $C_2^3:A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4:D_4$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^5:A_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_2\times A_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 384: $C_2^5:A_4$

Order 192: $C_2^4:A_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^3:A_4$ x 4, $C_2^3:A_4$

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

Order 48: $C_2^2:A_4$ x 13

Order 32: $C_2^2\wr C_2$ x 24, $C_2^2\times D_4$ x 20, $C_2^5$ x 14, $C_2^3:C_4$ x 12, $C_2^3\times C_4$

Order 24: $C_2\times A_4$ x 5

Order 16: $C_2^4$ x 133, $C_2\times D_4$ x 64, $C_2^2:C_4$ x 16, $C_2^2\times C_4$ x 10

Order 12: $A_4$ x 13

Order 8: $C_2^3$ x 271, $D_4$ x 32, $C_2\times C_4$ x 16

Order 6: $C_6$

Order 4: $C_2^2$ x 139, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 384: $C_2^5:A_4$

Order 192: $C_2^4:A_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^3:A_4$, $C_2^3:A_4$

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

Order 48: $C_2^2:A_4$ x 3

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

Order 24: $C_2\times A_4$ x 2

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

Order 12: $A_4$ x 3

Order 8: $C_2^3$ x 17, $D_4$ x 2, $C_2\times C_4$ x 2

Order 6: $C_6$

Order 4: $C_2^2$ x 14, $C_4$

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 384: $C_2^5:A_4$ ($C_1$)

Order 192: $C_2^4:A_4$ ($C_2$)

Order 128: $C_2^4:D_4$ ($C_3$)

Order 64: $C_2^6$ ($C_6$)

Order 32: $C_2^2\wr C_2$ ($A_4$) x 4, $C_2^5$ ($A_4$)

Order 16: $C_2^4$ ($C_2\times A_4$) x 5

Order 8: $C_2^3$ ($C_2^2:A_4$)

Order 4: $C_2^2$ ($C_2^3:A_4$) x 4, $C_2^2$ ($C_2^3:A_4$)

Order 1: $C_1$ ($C_2^5:A_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 384: $C_2^5:A_4$ ($C_1$)

Order 192: $C_2^4:A_4$ ($C_2$)

Order 128: $C_2^4:D_4$ ($C_3$)

Order 64: $C_2^6$ ($C_6$)

Order 32: $C_2^5$ ($A_4$), $C_2^2\wr C_2$ ($A_4$)

Order 16: $C_2^4$ ($C_2\times A_4$) x 2

Order 8: $C_2^3$ ($C_2^2:A_4$)

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

Order 1: $C_1$ ($C_2^5:A_4$)

Series

Derived series

$C_2^5:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_2^5:A_4$

$\rhd$

$C_2^4:A_4$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2^5:A_4$

$\rhd$

$C_2^6$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 15 larger groups in the database.

This group is a maximal quotient of 11 larger groups in the database.

Character theory

Complex character table

See the $24 \times 24$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $22 \times 22$ rational character table.