Abstract group 768.56452: $C_{24}.(C_4\times C_8)$

• Label: 768.56452
• Order: \( 2^{8} \cdot 3 \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \)
• $\card{Z(G)}$: \( 2^{4} \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{9} \)
• Perm deg.: $39$
• Trans deg.: $192$
• Rank: $2$

Group information

Description:	$C_{24}.(C_4\times C_8)$
Order:	\(768\)\(\medspace = 2^{8} \cdot 3 \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_3:((C_2^4\times C_8).C_2^6)$, of order \(24576\)\(\medspace = 2^{13} \cdot 3 \)
Composition factors:	$C_2$ x 8, $C_3$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	3	4	6	8	12	16	24	48
Elements	1	7	2	24	14	32	48	448	64	128	768
Conjugacy classes	1	5	1	14	7	20	24	48	32	64	216
Divisions	1	5	1	8	5	8	8	8	8	4	56
Autjugacy classes	1	4	1	5	4	6	5	3	6	2	37

Dimension	1	2	4	8	16	32
Irr. complex chars.	32	184	0	0	0	0	216
Irr. rational chars.	4	14	20	8	6	4	56

Minimal presentations

Permutation degree:	$39$
Transitive degree:	$192$
Rank:	$2$
Inequivalent generating pairs:	$6$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	6	20

Constructions

Presentation:	$\langle a, b, c \mid b^{48}=c^{2}=[a,c]=[b,c]=1, a^{8}=b^{24}, b^{a}=b^{23}c \rangle$
Permutation group:	Degree $39$ $\langle(1,2,5,12,8,13,23,11,9,14,4,22,7,25,6,24)(3,15,17,28,20,30,32,29,21,31,16,10,19,26,18,27) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 24 \\ 8 & 25 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 0 & 17 \end{array}\right), \left(\begin{array}{rr} 1 & 16 \\ 16 & 17 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 28 & 11 \\ 25 & 3 \end{array}\right), \left(\begin{array}{rr} 7 & 14 \\ 26 & 13 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 24 & 21 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 30 & 31 \\ 1 & 2 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/32\Z)$
	$\left\langle \left(\begin{array}{rr} 0 & 9 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 35 & 17 \\ 51 & 52 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 33 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 33 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 51 & 23 \\ 6 & 17 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/68\Z)$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not isomorphic to a non-trivial semidirect product
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{24}$ . $(Q_8:C_4)$ (2)	$C_8$ . $(C_6.Q_{16})$ (2)	$(C_4\times C_{24})$ . $D_4$ (2)	$(C_2\times C_{24})$ . $D_8$	all 79

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

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

Subgroups

There are 436 subgroups in 210 conjugacy classes, 119 normal (73 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2\times C_8$	$G/Z \simeq$ $D_{24}$
Commutator:	$G' \simeq$ $C_{24}$	$G/G' \simeq$ $C_4\times C_8$
Frattini:	$\Phi \simeq$ $C_2\times C_4\times C_8$	$G/\Phi \simeq$ $D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_4\times C_{48}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{24}.(C_4\times C_8)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_8:C_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_8.(C_4\times C_8)$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 768: $C_{24}.(C_4\times C_8)$

Order 384: $C_2\times C_{24}.C_8$ x 2, $C_2\times C_4\times C_{48}$

Order 256: $C_8.(C_4\times C_8)$

Order 192: $C_{24}.C_8$ x 6, $C_2^2\times C_{48}$ x 2, $C_6:\OD_{32}$ x 2, $C_4\times C_{48}$ x 2, $C_2\times C_4\times C_{24}$

Order 128: $C_2^3.\OD_{16}$ x 2, $C_2\times C_4\times C_{16}$

Order 96: $C_2\times C_{48}$ x 8, $C_3:\OD_{32}$ x 6, $C_4\times C_{24}$ x 4, $C_6:C_{16}$ x 2, $C_2^2\times C_{24}$ x 2, $C_2\times C_4\times C_{12}$

Order 64: $C_8.C_8$ x 6, $C_4\times C_{16}$ x 2, $C_2\times \OD_{32}$ x 2, $C_2^2\times C_{16}$ x 2, $C_2\times C_4\times C_8$

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

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

Order 24: $C_2\times C_{12}$ x 12, $C_{24}$ x 8, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 12, $C_{16}$ x 8, $C_4^2$ x 3, $C_2^2\times C_4$ x 2

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5

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

Order 6: $C_6$ x 5

Order 4: $C_4$ x 8, $C_2^2$ x 5

Order 3: $C_3$

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 768: $C_{24}.(C_4\times C_8)$

Order 384: $C_2\times C_4\times C_{48}$, $C_2\times C_{24}.C_8$

Order 256: $C_8.(C_4\times C_8)$

Order 192: $C_2^2\times C_{48}$ x 2, $C_{24}.C_8$ x 2, $C_2\times C_4\times C_{24}$, $C_6:\OD_{32}$, $C_4\times C_{48}$

Order 128: $C_2^3.\OD_{16}$, $C_2\times C_4\times C_{16}$

Order 96: $C_2\times C_{48}$ x 6, $C_4\times C_{24}$ x 2, $C_3:\OD_{32}$ x 2, $C_2^2\times C_{24}$ x 2, $C_6:C_{16}$, $C_2\times C_4\times C_{12}$

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

Order 48: $C_2\times C_{24}$ x 10, $C_2^2\times C_{12}$ x 2, $C_4\times C_{12}$ x 2, $C_{48}$ x 2, $C_3:C_{16}$

Order 32: $C_2\times C_{16}$ x 7, $C_2^2\times C_8$ x 2, $C_4\times C_8$ x 2, $\OD_{32}$ x 2, $C_2\times C_4^2$

Order 24: $C_2\times C_{12}$ x 8, $C_{24}$ x 6, $C_2^2\times C_6$

Order 16: $C_2\times C_8$ x 10, $C_{16}$ x 3, $C_4^2$ x 2, $C_2^2\times C_4$ x 2

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

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

Order 6: $C_6$ x 4

Order 4: $C_4$ x 5, $C_2^2$ x 4

Order 3: $C_3$

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 768: $C_{24}.(C_4\times C_8)$ ($C_1$)

Order 384: $C_2\times C_{24}.C_8$ ($C_2$) x 2, $C_2\times C_4\times C_{48}$ ($C_2$)

Order 192: $C_{24}.C_8$ ($C_4$) x 4, $C_2^2\times C_{48}$ ($C_4$) x 2, $C_2\times C_4\times C_{24}$ ($C_2^2$)

Order 128: $C_2\times C_4\times C_{16}$ ($S_3$)

Order 96: $C_2\times C_{48}$ ($C_8$) x 4, $C_4\times C_{24}$ ($D_4$) x 2, $C_4\times C_{24}$ ($C_2\times C_4$) x 2, $C_2^2\times C_{24}$ ($D_4$), $C_2^2\times C_{24}$ ($C_2\times C_4$), $C_2\times C_4\times C_{12}$ ($Q_8$)

Order 64: $C_2^2\times C_{16}$ ($C_3:C_4$) x 2, $C_2\times C_4\times C_8$ ($D_6$)

Order 48: $C_2\times C_{24}$ ($C_2^2:C_4$) x 3, $C_2\times C_{24}$ ($\SD_{16}$) x 2, $C_2\times C_{24}$ ($\OD_{16}$) x 2, $C_2\times C_{24}$ ($C_2\times C_8$) x 2, $C_4\times C_{12}$ ($C_4:C_4$) x 2, $C_2^2\times C_{12}$ ($C_4:C_4$), $C_2\times C_{24}$ ($Q_{16}$), $C_2\times C_{24}$ ($D_8$), $C_2\times C_{24}$ ($C_4^2$)

Order 32: $C_2\times C_{16}$ ($C_3:C_8$) x 4, $C_4\times C_8$ ($C_3:D_4$) x 2, $C_4\times C_8$ ($C_4\times S_3$) x 2, $C_2^2\times C_8$ ($C_6:C_4$), $C_2^2\times C_8$ ($D_{12}$), $C_2\times C_4^2$ ($C_3:Q_8$)

Order 24: $C_2\times C_{12}$ ($C_4:C_8$) x 2, $C_{24}$ ($D_4:C_4$) x 2, $C_{24}$ ($C_2^2:C_8$) x 2, $C_{24}$ ($Q_8:C_4$) x 2, $C_2\times C_{12}$ ($C_2.C_4^2$), $C_2\times C_{12}$ ($C_8.C_4$), $C_2\times C_{12}$ ($C_8:C_4$), $C_2\times C_{12}$ ($C_8:C_4$), $C_{24}$ ($C_8:C_4$), $C_{24}$ ($C_4\times C_8$), $C_2^2\times C_6$ ($C_8.C_4$)

Order 16: $C_2\times C_8$ ($C_6:C_8$) x 2, $C_2\times C_8$ ($C_{24}:C_2$) x 2, $C_2\times C_8$ ($D_6:C_4$) x 2, $C_2\times C_8$ ($C_3:\OD_{16}$) x 2, $C_4^2$ ($C_6.D_4$) x 2, $C_2\times C_8$ ($C_3:Q_{16}$), $C_2\times C_8$ ($D_{24}$), $C_2\times C_8$ ($C_6.D_4$), $C_2\times C_8$ ($C_{12}:C_4$), $C_2^2\times C_4$ ($C_{12}:C_4$)

Order 12: $C_2\times C_6$ ($C_4.C_4^2$), $C_2\times C_6$ ($C_8:C_8$), $C_2\times C_6$ ($C_8:C_8$), $C_{12}$ ($C_4.C_4^2$), $C_{12}$ ($C_2^2.C_4^2$), $C_{12}$ ($C_8:C_8$), $C_{12}$ ($C_8:C_8$)

Order 8: $C_2\times C_4$ ($C_{12}:C_8$) x 2, $C_8$ ($C_{12}.D_4$) x 2, $C_8$ ($D_{12}:C_4$) x 2, $C_8$ ($C_6.Q_{16}$) x 2, $C_2^3$ ($C_{24}.C_4$), $C_2\times C_4$ ($C_6.C_4^2$), $C_2\times C_4$ ($C_{24}.C_4$), $C_2\times C_4$ ($C_{24}:C_4$), $C_2\times C_4$ ($C_{24}:C_4$), $C_8$ ($C_{12}:C_8$), $C_8$ ($C_6.C_4^2$)

Order 6: $C_6$ ($C_{16}.C_8$) x 2, $C_6$ ($C_4^2.D_4$)

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

Order 3: $C_3$ ($C_8.(C_4\times C_8)$)

Order 2: $C_2$ ($C_{48}.C_8$) x 2, $C_2$ ($(C_2\times C_{24}):C_8$)

Order 1: $C_1$ ($C_{24}.(C_4\times C_8)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 768: $C_{24}.(C_4\times C_8)$ ($C_1$)

Order 384: $C_2\times C_4\times C_{48}$ ($C_2$), $C_2\times C_{24}.C_8$ ($C_2$)

Order 192: $C_2^2\times C_{48}$ ($C_4$) x 2, $C_2\times C_4\times C_{24}$ ($C_2^2$), $C_{24}.C_8$ ($C_4$)

Order 128: $C_2\times C_4\times C_{16}$ ($S_3$)

Order 96: $C_2\times C_{48}$ ($C_8$) x 2, $C_4\times C_{24}$ ($D_4$), $C_4\times C_{24}$ ($C_2\times C_4$), $C_2^2\times C_{24}$ ($D_4$), $C_2^2\times C_{24}$ ($C_2\times C_4$), $C_2\times C_4\times C_{12}$ ($Q_8$)

Order 64: $C_2^2\times C_{16}$ ($C_3:C_4$) x 2, $C_2\times C_4\times C_8$ ($D_6$)

Order 48: $C_2\times C_{24}$ ($\OD_{16}$) x 2, $C_2\times C_{24}$ ($C_2\times C_8$) x 2, $C_2\times C_{24}$ ($C_2^2:C_4$) x 2, $C_2^2\times C_{12}$ ($C_4:C_4$), $C_2\times C_{24}$ ($Q_{16}$), $C_2\times C_{24}$ ($\SD_{16}$), $C_2\times C_{24}$ ($D_8$), $C_2\times C_{24}$ ($C_4^2$), $C_4\times C_{12}$ ($C_4:C_4$)

Order 32: $C_2\times C_{16}$ ($C_3:C_8$) x 2, $C_2^2\times C_8$ ($C_6:C_4$), $C_2^2\times C_8$ ($D_{12}$), $C_4\times C_8$ ($C_3:D_4$), $C_4\times C_8$ ($C_4\times S_3$), $C_2\times C_4^2$ ($C_3:Q_8$)

Order 24: $C_2\times C_{12}$ ($C_4:C_8$) x 2, $C_{24}$ ($C_2^2:C_8$) x 2, $C_2\times C_{12}$ ($C_2.C_4^2$), $C_2\times C_{12}$ ($C_8.C_4$), $C_2\times C_{12}$ ($C_8:C_4$), $C_2\times C_{12}$ ($C_8:C_4$), $C_{24}$ ($D_4:C_4$), $C_{24}$ ($C_8:C_4$), $C_{24}$ ($C_4\times C_8$), $C_{24}$ ($Q_8:C_4$), $C_2^2\times C_6$ ($C_8.C_4$)

Order 16: $C_2\times C_8$ ($C_6:C_8$) x 2, $C_2\times C_8$ ($C_3:\OD_{16}$) x 2, $C_2\times C_8$ ($C_3:Q_{16}$), $C_2\times C_8$ ($D_{24}$), $C_2\times C_8$ ($C_{24}:C_2$), $C_2\times C_8$ ($C_6.D_4$), $C_2\times C_8$ ($D_6:C_4$), $C_2\times C_8$ ($C_{12}:C_4$), $C_4^2$ ($C_6.D_4$), $C_2^2\times C_4$ ($C_{12}:C_4$)

Order 12: $C_2\times C_6$ ($C_4.C_4^2$), $C_2\times C_6$ ($C_8:C_8$), $C_2\times C_6$ ($C_8:C_8$), $C_{12}$ ($C_4.C_4^2$), $C_{12}$ ($C_2^2.C_4^2$), $C_{12}$ ($C_8:C_8$), $C_{12}$ ($C_8:C_8$)

Order 8: $C_2\times C_4$ ($C_{12}:C_8$) x 2, $C_8$ ($C_{12}.D_4$) x 2, $C_2^3$ ($C_{24}.C_4$), $C_2\times C_4$ ($C_6.C_4^2$), $C_2\times C_4$ ($C_{24}.C_4$), $C_2\times C_4$ ($C_{24}:C_4$), $C_2\times C_4$ ($C_{24}:C_4$), $C_8$ ($C_{12}:C_8$), $C_8$ ($D_{12}:C_4$), $C_8$ ($C_6.Q_{16}$), $C_8$ ($C_6.C_4^2$)

Order 6: $C_6$ ($C_4^2.D_4$), $C_6$ ($C_{16}.C_8$)

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

Order 3: $C_3$ ($C_8.(C_4\times C_8)$)

Order 2: $C_2$ ($C_{48}.C_8$), $C_2$ ($(C_2\times C_{24}):C_8$)

Order 1: $C_1$ ($C_{24}.(C_4\times C_8)$)

Series

Derived series

$C_{24}.(C_4\times C_8)$

$\rhd$

$C_{24}$

$\rhd$

$C_1$

Chief series

$C_{24}.(C_4\times C_8)$

$\rhd$

$C_2\times C_{24}.C_8$

$\rhd$

$C_2\times C_4\times C_{24}$

$\rhd$

$C_2^2\times C_{24}$

$\rhd$

$C_2^2\times C_{12}$

$\rhd$

$C_2\times C_{12}$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_{24}.(C_4\times C_8)$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

Upper central series

$C_1$

$\lhd$

$C_2\times C_8$

$\lhd$

$C_2^2\times C_8$

$\lhd$

$C_2\times C_4\times C_8$

$\lhd$

$C_2\times C_4\times C_{16}$

Supergroups

This group is a maximal subgroup of 2 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 $216 \times 216$ 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 $56 \times 56$ rational character table.