Abstract group 768.1088504: $C_2\times A_4:\OD_{32}$

• Label: 768.1088504
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $22$
• Trans deg.: $96$
• Rank: $3$

Group information

Description:	$C_2\times A_4:\OD_{32}$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$(C_2^3\times A_4).C_2^6$, of order \(6144\)\(\medspace = 2^{11} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	16	24
Elements	1	31	8	32	56	64	64	384	128	768
Conjugacy classes	1	11	1	12	7	24	8	32	16	112
Divisions	1	11	1	8	5	8	4	8	4	50
Autjugacy classes	1	7	1	6	3	6	3	2	3	32

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	32	40	32	0	8	0	0	0	0	112
Irr. rational chars.	8	12	8	8	4	2	4	2	2	50

Minimal presentations

Permutation degree:	$22$
Transitive degree:	$96$
Rank:	$3$
Inequivalent generating triples:	$20160$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	5	7	11

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{16}=c^{6}=d^{2}=e^{2}=[a,c]=[a,d]=[a,e]= \!\cdots\! \rangle}$
Permutation group:	Degree $22$ $\langle(1,2)(3,5)(4,6)(7,8,10,13,11,14,17,20,9,12,15,18,16,19,21,22), (1,2)(3,4) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 23 & 13 \\ 4 & 9 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 29 & 16 \\ 16 & 13 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 25 & 34 \\ 34 & 25 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 34 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 35 & 17 \\ 51 & 52 \end{array}\right), \left(\begin{array}{rr} 0 & 9 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 33 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 59 & 0 \\ 34 & 59 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/68\Z)$
Direct product:	$C_2$ $\, \times\, $ $(A_4:\OD_{32})$
Semidirect product:	$(C_2\times A_4)$ $\,\rtimes\,$ $\OD_{32}$ (2)	$A_4$ $\,\rtimes\,$ $(C_2\times \OD_{32})$	$(A_4:C_{16})$ $\,\rtimes\,$ $C_2^2$ (4)	$(C_2\times A_4:C_{16})$ $\,\rtimes\,$ $C_2$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2^4\times C_8)$ . $S_3$	$(C_2^2\times C_8)$ . $S_4$	$C_8$ . $(C_2^2\times S_4)$	$C_8$ . $(C_2^2.S_4)$ (3)	all 39

Elements of the group are displayed as matrices in $\GL_{2}(\Z/{32}\Z)$.

Homology

Abelianization:	$C_{2}^{2} \times C_{8} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 1626 subgroups in 489 conjugacy classes, 101 normal (43 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $C_2\times S_4$
Commutator:	$G' \simeq$ $C_2\times A_4$	$G/G' \simeq$ $C_2^2\times C_8$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^2\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_8$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2\times A_4:\OD_{32}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^4$	$G/\operatorname{soc} \simeq$ $C_6:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^3:\OD_{32}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 768: $C_2\times A_4:\OD_{32}$

Order 384: $A_4:\OD_{32}$ x 4, $C_2\times A_4:C_{16}$ x 2, $C_2^4:C_{24}$

Order 256: $C_2^3:\OD_{32}$

Order 192: $C_2^3:C_{24}$ x 6, $A_4:C_{16}$ x 4, $C_6:\OD_{32}$, $C_2^4:C_{12}$

Order 128: $C_2^2:\OD_{32}$ x 8, $C_2^3:C_{16}$ x 4, $C_2^2\times \OD_{32}$ x 2, $C_2^4\times C_8$

Order 96: $C_2^3:C_{12}$ x 6, $C_8\times A_4$ x 4, $C_3:\OD_{32}$ x 4, $C_6:C_{16}$ x 2, $C_2^3\times A_4$, $C_2^2\times C_{24}$

Order 64: $C_2^2:C_{16}$ x 16, $C_2\times \OD_{32}$ x 16, $C_2^3\times C_8$ x 14, $C_2^2\times C_{16}$ x 4, $C_2^4\times C_4$

Order 48: $C_2\times C_{24}$ x 6, $C_2^2\times A_4$ x 5, $C_4\times A_4$ x 4, $C_3:C_{16}$ x 4, $C_2^2\times C_{12}$

Order 32: $C_2^2\times C_8$ x 43, $\OD_{32}$ x 16, $C_2\times C_{16}$ x 16, $C_2^3\times C_4$ x 14, $C_2^5$

Order 24: $C_2\times C_{12}$ x 6, $C_2\times A_4$ x 5, $C_{24}$ x 4, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 43, $C_2\times C_8$ x 38, $C_2^4$ x 11, $C_{16}$ x 8

Order 12: $C_2\times C_6$ x 5, $C_{12}$ x 4, $A_4$

Order 8: $C_2^3$ x 38, $C_2\times C_4$ x 38, $C_8$ x 8

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 38, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2\times A_4:\OD_{32}$

Order 384: $A_4:\OD_{32}$, $C_2\times A_4:C_{16}$, $C_2^4:C_{24}$

Order 256: $C_2^3:\OD_{32}$

Order 192: $C_2^3:C_{24}$ x 4, $C_6:\OD_{32}$, $A_4:C_{16}$, $C_2^4:C_{12}$

Order 128: $C_2^2:\OD_{32}$ x 2, $C_2^3:C_{16}$ x 2, $C_2^2\times \OD_{32}$ x 2, $C_2^4\times C_8$

Order 96: $C_2^3:C_{12}$ x 4, $C_8\times A_4$ x 3, $C_2^3\times A_4$, $C_3:\OD_{32}$, $C_6:C_{16}$, $C_2^2\times C_{24}$

Order 64: $C_2^3\times C_8$ x 10, $C_2\times \OD_{32}$ x 8, $C_2^2:C_{16}$ x 3, $C_2^2\times C_{16}$ x 2, $C_2^4\times C_4$

Order 48: $C_2\times C_{24}$ x 4, $C_2^2\times A_4$ x 3, $C_4\times A_4$ x 3, $C_2^2\times C_{12}$, $C_3:C_{16}$

Order 32: $C_2^2\times C_8$ x 29, $C_2^3\times C_4$ x 10, $C_2\times C_{16}$ x 6, $\OD_{32}$ x 4, $C_2^5$

Order 24: $C_2\times C_{12}$ x 4, $C_{24}$ x 3, $C_2\times A_4$ x 3, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 29, $C_2\times C_8$ x 25, $C_2^4$ x 7, $C_{16}$ x 2

Order 12: $C_2\times C_6$ x 3, $C_{12}$ x 3, $A_4$

Order 8: $C_2\times C_4$ x 25, $C_2^3$ x 19, $C_8$ x 6

Order 6: $C_6$ x 3

Order 4: $C_2^2$ x 19, $C_4$ x 6

Order 3: $C_3$

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2\times A_4:\OD_{32}$ ($C_1$)

Order 384: $A_4:\OD_{32}$ ($C_2$) x 4, $C_2\times A_4:C_{16}$ ($C_2$) x 2, $C_2^4:C_{24}$ ($C_2$)

Order 192: $A_4:C_{16}$ ($C_2^2$) x 4, $C_2^3:C_{24}$ ($C_2^2$) x 3, $C_2^3:C_{24}$ ($C_4$) x 3, $C_2^4:C_{12}$ ($C_4$)

Order 128: $C_2^4\times C_8$ ($S_3$)

Order 96: $C_8\times A_4$ ($C_2\times C_4$) x 3, $C_2^3:C_{12}$ ($C_2\times C_4$) x 3, $C_2^3:C_{12}$ ($C_8$) x 3, $C_8\times A_4$ ($C_2^3$), $C_2^3\times A_4$ ($C_8$)

Order 64: $C_2^3\times C_8$ ($D_6$) x 3, $C_2^3\times C_8$ ($C_3:C_4$) x 3, $C_2^4\times C_4$ ($C_3:C_4$)

Order 48: $C_2^2\times A_4$ ($C_2\times C_8$) x 3, $C_4\times A_4$ ($C_2\times C_8$) x 3, $C_4\times A_4$ ($C_2^2\times C_4$)

Order 32: $C_2^3\times C_4$ ($C_6:C_4$) x 3, $C_2^3\times C_4$ ($C_3:C_8$) x 3, $C_2^2\times C_8$ ($C_6:C_4$) x 3, $C_2^5$ ($C_3:C_8$), $C_2^2\times C_8$ ($C_2\times D_6$), $C_2^2\times C_8$ ($S_4$)

Order 24: $C_2\times A_4$ ($\OD_{32}$) x 2, $C_2\times A_4$ ($C_2^2\times C_8$)

Order 16: $C_2\times C_8$ ($C_2\times S_4$) x 3, $C_2\times C_8$ ($A_4:C_4$) x 3, $C_2^4$ ($C_6:C_8$) x 3, $C_2^2\times C_4$ ($C_6:C_8$) x 3, $C_2^2\times C_4$ ($C_6.C_2^3$), $C_2^2\times C_4$ ($A_4:C_4$)

Order 12: $A_4$ ($C_2\times \OD_{32}$)

Order 8: $C_2\times C_4$ ($A_4:C_8$) x 3, $C_2\times C_4$ ($C_2^2.S_4$) x 3, $C_8$ ($C_2^2.S_4$) x 3, $C_2^3$ ($C_3:\OD_{32}$) x 2, $C_2^3$ ($A_4:C_8$), $C_2^3$ ($C_{12}.C_2^3$), $C_8$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2\times A_4:C_8$) x 3, $C_4$ ($C_2\times A_4:C_8$) x 3, $C_2^2$ ($C_6:\OD_{32}$), $C_4$ ($C_2^3.S_4$)

Order 2: $C_2$ ($A_4:\OD_{32}$) x 2, $C_2$ ($C_2^2\times A_4:C_8$)

Order 1: $C_1$ ($C_2\times A_4:\OD_{32}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2\times A_4:\OD_{32}$ ($C_1$)

Order 384: $A_4:\OD_{32}$ ($C_2$), $C_2\times A_4:C_{16}$ ($C_2$), $C_2^4:C_{24}$ ($C_2$)

Order 192: $C_2^3:C_{24}$ ($C_2^2$) x 2, $C_2^3:C_{24}$ ($C_4$) x 2, $A_4:C_{16}$ ($C_2^2$), $C_2^4:C_{12}$ ($C_4$)

Order 128: $C_2^4\times C_8$ ($S_3$)

Order 96: $C_8\times A_4$ ($C_2\times C_4$) x 2, $C_2^3:C_{12}$ ($C_2\times C_4$) x 2, $C_2^3:C_{12}$ ($C_8$) x 2, $C_8\times A_4$ ($C_2^3$), $C_2^3\times A_4$ ($C_8$)

Order 64: $C_2^3\times C_8$ ($D_6$) x 2, $C_2^3\times C_8$ ($C_3:C_4$) x 2, $C_2^4\times C_4$ ($C_3:C_4$)

Order 48: $C_2^2\times A_4$ ($C_2\times C_8$) x 2, $C_4\times A_4$ ($C_2\times C_8$) x 2, $C_4\times A_4$ ($C_2^2\times C_4$)

Order 32: $C_2^3\times C_4$ ($C_6:C_4$) x 2, $C_2^3\times C_4$ ($C_3:C_8$) x 2, $C_2^2\times C_8$ ($C_6:C_4$) x 2, $C_2^5$ ($C_3:C_8$), $C_2^2\times C_8$ ($C_2\times D_6$), $C_2^2\times C_8$ ($S_4$)

Order 24: $C_2\times A_4$ ($C_2^2\times C_8$), $C_2\times A_4$ ($\OD_{32}$)

Order 16: $C_2\times C_8$ ($C_2\times S_4$) x 2, $C_2\times C_8$ ($A_4:C_4$) x 2, $C_2^4$ ($C_6:C_8$) x 2, $C_2^2\times C_4$ ($C_6:C_8$) x 2, $C_2^2\times C_4$ ($C_6.C_2^3$), $C_2^2\times C_4$ ($A_4:C_4$)

Order 12: $A_4$ ($C_2\times \OD_{32}$)

Order 8: $C_2\times C_4$ ($A_4:C_8$) x 2, $C_2\times C_4$ ($C_2^2.S_4$) x 2, $C_8$ ($C_2^2.S_4$) x 2, $C_2^3$ ($A_4:C_8$), $C_2^3$ ($C_3:\OD_{32}$), $C_2^3$ ($C_{12}.C_2^3$), $C_8$ ($C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2\times A_4:C_8$) x 2, $C_4$ ($C_2\times A_4:C_8$) x 2, $C_2^2$ ($C_6:\OD_{32}$), $C_4$ ($C_2^3.S_4$)

Order 2: $C_2$ ($A_4:\OD_{32}$), $C_2$ ($C_2^2\times A_4:C_8$)

Order 1: $C_1$ ($C_2\times A_4:\OD_{32}$)

Series

Derived series

$C_2\times A_4:\OD_{32}$

$\rhd$

$C_2\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2\times A_4:\OD_{32}$

$\rhd$

$A_4:\OD_{32}$

$\rhd$

$C_2^3:C_{24}$

$\rhd$

$C_8\times A_4$

$\rhd$

$C_4\times A_4$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$C_2\times A_4:\OD_{32}$

$\rhd$

$C_2\times A_4$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2^2\times C_8$

Supergroups

This group is a maximal subgroup of 6 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $112 \times 112$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $50 \times 50$ rational character table.