Abstract group 768.1084932: $C_2^4:C_{48}$

• Label: 768.1084932
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{6} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{6} \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \cdot 3 \)
• Perm deg.: $24$
• Trans deg.: $192$
• Rank: $3$

Group information

Description:	$C_2^4:C_{48}$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_2.C_2^6.D_6^2$, of order \(18432\)\(\medspace = 2^{11} \cdot 3^{2} \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$2$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	31	8	32	56	64	64	128	128	256	768
Conjugacy classes	1	15	2	16	14	32	16	64	32	64	256
Divisions	1	15	1	8	7	8	4	8	4	4	60
Autjugacy classes	1	5	1	4	2	4	2	2	2	1	24

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	192	0	64	0	0	0	0	0	0	256
Irr. rational chars.	8	12	8	8	4	8	4	4	4	60

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$192$
Rank:	$3$
Inequivalent generating triples:	$7280$

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:	$\langle a, b, c, d, e \mid a^{6}=b^{2}=c^{2}=d^{2}=e^{16}=[a,b]=[a,e]=[b,c]=[b,d]=[b,e]=[c,d]=[c,e]=[d,e]=1, c^{a}=cd, d^{a}=c \rangle$
Permutation group:	Degree $24$ $\langle(19,20,22)(21,23,24), (1,2,4,7,5,8,11,14,3,6,9,12,10,13,15,16), (20,23) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 17 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 16 & 7 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 13 & 24 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 7 & 14 \\ 26 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 1 & 14 \\ 14 & 15 \end{array}\right), \left(\begin{array}{rr} 2 & 21 \\ 21 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 10 \\ 2 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 2 \\ 6 & 1 \end{array}\right), \left(\begin{array}{rr} 27 & 6 \\ 18 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 15 & 14 \\ 14 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/28\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $C_{16}$ $\, \times\, $ $A_4$
Semidirect product:	$C_2^4$ $\,\rtimes\,$ $C_{48}$	$(C_2^4\times C_{16})$ $\,\rtimes\,$ $C_3$	$(C_2^3\times C_{16})$ $\,\rtimes\,$ $C_6$	$C_2^3$ $\,\rtimes\,$ $(C_2\times C_{48})$	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $C_{24}$	$(C_2^3\times A_4)$ . $C_8$	$(C_8\times A_4)$ . $C_2^3$	$C_2^3$ . $(C_8\times A_4)$	all 39

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

Homology

Abelianization:	$C_{2}^{2} \times C_{48} \simeq C_{2}^{2} \times C_{16} \times C_{3}$
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 1540 subgroups in 595 conjugacy classes, 147 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2\times C_{16}$	$G/Z \simeq$ $A_4$
Commutator:	$G' \simeq$ $C_2^2$	$G/G' \simeq$ $C_2^2\times C_{48}$
Frattini:	$\Phi \simeq$ $C_8$	$G/\Phi \simeq$ $C_2^3\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^4\times C_{16}$	$G/\operatorname{Fit} \simeq$ $C_3$
Radical:	$R \simeq$ $C_2^4:C_{48}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^5$	$G/\operatorname{soc} \simeq$ $C_{24}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4\times C_{16}$
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: $C_2^4:C_{48}$

Order 384: $C_2^3:C_{48}$ x 6, $C_2^4:C_{24}$

Order 256: $C_2^4\times C_{16}$

Order 192: $C_2^3:C_{24}$ x 6, $A_4\times C_{16}$ x 4, $C_2^2\times C_{48}$, $C_2^4:C_{12}$

Order 128: $C_2^3\times C_{16}$ x 14, $C_2^4\times C_8$

Order 96: $C_2\times C_{48}$ x 6, $C_2^3:C_{12}$ x 6, $C_8\times A_4$ x 4, $C_2^3\times A_4$, $C_2^2\times C_{24}$

Order 64: $C_2^2\times C_{16}$ x 50, $C_2^3\times C_8$ x 14, $C_2^4\times C_4$

Order 48: $C_2^2\times A_4$ x 7, $C_2\times C_{24}$ x 6, $C_4\times A_4$ x 4, $C_{48}$ x 4, $C_2^2\times C_{12}$

Order 32: $C_2^2\times C_8$ x 50, $C_2\times C_{16}$ x 44, $C_2^3\times C_4$ x 14, $C_2^5$

Order 24: $C_2\times A_4$ x 7, $C_2\times C_{12}$ x 6, $C_{24}$ x 4, $C_2^2\times C_6$

Order 16: $C_2^2\times C_4$ x 50, $C_2\times C_8$ x 44, $C_2^4$ x 15, $C_{16}$ x 8

Order 12: $C_2\times C_6$ x 7, $C_{12}$ x 4, $A_4$

Order 8: $C_2^3$ x 57, $C_2\times C_4$ x 44, $C_8$ x 8

Order 6: $C_6$ x 7

Order 4: $C_2^2$ x 57, $C_4$ x 8

Order 3: $C_3$

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_2^4:C_{48}$

Order 384: $C_2^3:C_{48}$, $C_2^4:C_{24}$

Order 256: $C_2^4\times C_{16}$

Order 192: $C_2^3:C_{24}$ x 2, $C_2^2\times C_{48}$, $A_4\times C_{16}$, $C_2^4:C_{12}$

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

Order 96: $C_8\times A_4$ x 2, $C_2^3:C_{12}$ x 2, $C_2\times C_{48}$, $C_2^3\times A_4$, $C_2^2\times C_{24}$

Order 64: $C_2^2\times C_{16}$ x 9, $C_2^3\times C_8$ x 6, $C_2^4\times C_4$

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

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

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

Order 16: $C_2^2\times C_4$ x 16, $C_2\times C_8$ x 13, $C_2^4$ x 5, $C_{16}$ x 2

Order 12: $C_2\times C_6$ x 2, $C_{12}$ x 2, $A_4$

Order 8: $C_2\times C_4$ x 13, $C_2^3$ x 12, $C_8$ x 4

Order 6: $C_6$ x 2

Order 4: $C_2^2$ x 12, $C_4$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_2^4:C_{48}$ ($C_1$)

Order 384: $C_2^3:C_{48}$ ($C_2$) x 6, $C_2^4:C_{24}$ ($C_2$)

Order 256: $C_2^4\times C_{16}$ ($C_3$)

Order 192: $A_4\times C_{16}$ ($C_2^2$) 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: $C_2^3\times C_{16}$ ($C_6$) x 6, $C_2^4\times C_8$ ($C_6$)

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: $C_2^2\times C_{16}$ ($C_2\times C_6$) x 4, $C_2^3\times C_8$ ($C_2\times C_6$) x 3, $C_2^3\times C_8$ ($C_{12}$) x 3, $C_2^4\times C_4$ ($C_{12}$), $C_2^2\times C_{16}$ ($A_4$)

Order 48: $C_2^2\times A_4$ ($C_{16}$) x 4, $C_2^2\times A_4$ ($C_2\times C_8$) 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: $C_2\times C_{16}$ ($C_2\times A_4$) x 6, $C_2^3\times C_4$ ($C_2\times C_{12}$) x 3, $C_2^3\times C_4$ ($C_{24}$) x 3, $C_2^2\times C_8$ ($C_2\times C_{12}$) x 3, $C_2^5$ ($C_{24}$), $C_2^2\times C_8$ ($C_2^2\times C_6$), $C_2^2\times C_8$ ($C_2\times A_4$)

Order 24: $C_2\times A_4$ ($C_2\times C_{16}$) x 6, $C_2\times A_4$ ($C_2^2\times C_8$)

Order 16: $C_2^4$ ($C_{48}$) x 4, $C_{16}$ ($C_2^2\times A_4$) x 4, $C_2\times C_8$ ($C_2^2\times A_4$) x 3, $C_2\times C_8$ ($C_4\times A_4$) x 3, $C_2^4$ ($C_2\times C_{24}$) x 3, $C_2^2\times C_4$ ($C_2\times C_{24}$) x 3, $C_2^2\times C_4$ ($C_2^2\times C_{12}$), $C_2^2\times C_4$ ($C_4\times A_4$)

Order 12: $A_4$ ($C_2^2\times C_{16}$)

Order 8: $C_2^3$ ($C_2\times C_{48}$) x 6, $C_2\times C_4$ ($C_8\times A_4$) x 3, $C_2\times C_4$ ($C_2^3:C_{12}$) x 3, $C_8$ ($C_2^3:C_{12}$) x 3, $C_2^3$ ($C_8\times A_4$), $C_2^3$ ($C_2^2\times C_{24}$), $C_8$ ($C_2^3\times A_4$)

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

Order 2: $C_2$ ($C_2^3:C_{48}$) x 6, $C_2$ ($C_2^4:C_{24}$)

Order 1: $C_1$ ($C_2^4:C_{48}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_2^4:C_{48}$ ($C_1$)

Order 384: $C_2^3:C_{48}$ ($C_2$), $C_2^4:C_{24}$ ($C_2$)

Order 256: $C_2^4\times C_{16}$ ($C_3$)

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

Order 128: $C_2^4\times C_8$ ($C_6$), $C_2^3\times C_{16}$ ($C_6$)

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

Order 64: $C_2^4\times C_4$ ($C_{12}$), $C_2^3\times C_8$ ($C_2\times C_6$), $C_2^3\times C_8$ ($C_{12}$), $C_2^2\times C_{16}$ ($C_2\times C_6$), $C_2^2\times C_{16}$ ($A_4$)

Order 48: $C_2^2\times A_4$ ($C_2\times C_8$), $C_2^2\times A_4$ ($C_{16}$), $C_4\times A_4$ ($C_2\times C_8$), $C_4\times A_4$ ($C_2^2\times C_4$)

Order 32: $C_2^5$ ($C_{24}$), $C_2^3\times C_4$ ($C_2\times C_{12}$), $C_2^3\times C_4$ ($C_{24}$), $C_2^2\times C_8$ ($C_2\times C_{12}$), $C_2^2\times C_8$ ($C_2^2\times C_6$), $C_2^2\times C_8$ ($C_2\times A_4$), $C_2\times C_{16}$ ($C_2\times A_4$)

Order 24: $C_2\times A_4$ ($C_2^2\times C_8$), $C_2\times A_4$ ($C_2\times C_{16}$)

Order 16: $C_2\times C_8$ ($C_2^2\times A_4$), $C_2\times C_8$ ($C_4\times A_4$), $C_2^4$ ($C_2\times C_{24}$), $C_2^4$ ($C_{48}$), $C_2^2\times C_4$ ($C_2^2\times C_{12}$), $C_2^2\times C_4$ ($C_4\times A_4$), $C_2^2\times C_4$ ($C_2\times C_{24}$), $C_{16}$ ($C_2^2\times A_4$)

Order 12: $A_4$ ($C_2^2\times C_{16}$)

Order 8: $C_2^3$ ($C_8\times A_4$), $C_2^3$ ($C_2\times C_{48}$), $C_2^3$ ($C_2^2\times C_{24}$), $C_2\times C_4$ ($C_8\times A_4$), $C_2\times C_4$ ($C_2^3:C_{12}$), $C_8$ ($C_2^3\times A_4$), $C_8$ ($C_2^3:C_{12}$)

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

Order 2: $C_2$ ($C_2^3:C_{48}$), $C_2$ ($C_2^4:C_{24}$)

Order 1: $C_1$ ($C_2^4:C_{48}$)

Series

Derived series

$C_2^4:C_{48}$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_2^4:C_{48}$

$\rhd$

$C_2^3:C_{48}$

$\rhd$

$C_2^3\times C_{16}$

$\rhd$

$C_2^2\times C_{16}$

$\rhd$

$C_{16}$

$\rhd$

$C_8$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^4:C_{48}$

$\rhd$

$C_2^2$

Upper central series

$C_1$

$\lhd$

$C_2^2\times C_{16}$

Supergroups

This group is a maximal subgroup of 10 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 $256 \times 256$ 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 $60 \times 60$ rational character table.