Abstract group 288.398: $A_4:C_{24}$

• Label: 288.398
• Order: \( 2^{5} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{2} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$A_4:C_{24}$
Order:	\(288\)\(\medspace = 2^{5} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3\times S_4$, of order \(192\)\(\medspace = 2^{6} \cdot 3 \)
Composition factors:	$C_2$ x 5, $C_3$ x 2
Derived length:	$3$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	7	26	8	38	48	64	96	288
Conjugacy classes	1	3	5	4	9	8	14	16	60
Divisions	1	3	3	2	5	2	4	2	22
Autjugacy classes	1	3	3	2	5	2	4	2	22

Dimension	1	2	3	4	6	8	12	24
Irr. complex chars.	24	12	24	0	0	0	0	0	60
Irr. rational chars.	2	5	2	5	3	2	2	1	22

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	$144$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	24
Arbitrary	3	5	9

Constructions

Presentation:	$\langle a, b, c, d \mid a^{8}=b^{3}=c^{2}=d^{6}=[a,d]=[c,d]=1, b^{a}=b^{2}, c^{a}=cd^{3}, c^{b}=d^{3}, d^{b}=cd \rangle$
Permutation group:	Degree $15$ $\langle(2,3)(5,6,7,9,8,10,11,12), (13,14,15), (5,7,8,11)(6,9,10,12), (5,8)(6,10)(7,11)(9,12), (2,3,4), (1,2)(3,4), (1,3)(2,4)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 8 & 17 \\ 19 & 12 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 7 & 10 \\ 10 & 7 \end{array}\right), \left(\begin{array}{rr} 17 & 0 \\ 10 & 17 \end{array}\right), \left(\begin{array}{rr} 3 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 11 & 15 \\ 5 & 16 \end{array}\right), \left(\begin{array}{rr} 13 & 4 \\ 8 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/20\Z)$
Direct product:	$C_3$ $\, \times\, $ $(A_4:C_8)$
Semidirect product:	$A_4$ $\,\rtimes\,$ $C_{24}$	$(C_3\times A_4)$ $\,\rtimes\,$ $C_8$	$C_2^2$ $\,\rtimes\,$ $(C_3:C_{24})$	$(C_2\times C_6)$ $\,\rtimes\,$ $(C_3:C_8)$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{12}$ . $S_4$	$C_4$ . $(C_3\times S_4)$	$(C_6\times A_4)$ . $C_4$	$(C_4\times A_4)$ . $C_6$	all 12

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

Homology

Abelianization:	$C_{24} \simeq C_{8} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 166 subgroups in 58 conjugacy classes, 20 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{12}$	$G/Z \simeq$ $S_4$
Commutator:	$G' \simeq$ $A_4$	$G/G' \simeq$ $C_{24}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_3\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{12}$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $A_4:C_{24}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $C_3:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 288: $A_4:C_{24}$

Order 144: $C_{12}\times A_4$

Order 96: $A_4:C_8$, $C_2^2:C_{24}$

Order 72: $C_6\times A_4$, $C_3:C_{24}$

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

Order 36: $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_2\times A_4$ x 2, $C_2^2\times C_6$, $C_3:C_8$

Order 18: $C_3\times C_6$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

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

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 288: $A_4:C_{24}$

Order 144: $C_{12}\times A_4$

Order 96: $A_4:C_8$, $C_2^2:C_{24}$

Order 72: $C_6\times A_4$, $C_3:C_{24}$

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

Order 36: $C_3\times C_{12}$, $C_3\times A_4$

Order 32: $C_2^2:C_8$

Order 24: $C_2\times C_{12}$ x 2, $C_{24}$ x 2, $C_2\times A_4$ x 2, $C_2^2\times C_6$, $C_3:C_8$

Order 18: $C_3\times C_6$

Order 16: $C_2\times C_8$ x 2, $C_2^2\times C_4$

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

Order 9: $C_3^2$

Order 8: $C_2\times C_4$ x 2, $C_8$ x 2, $C_2^3$

Order 6: $C_6$ x 5

Order 4: $C_2^2$ x 3, $C_4$ x 2

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 288: $A_4:C_{24}$ ($C_1$)

Order 144: $C_{12}\times A_4$ ($C_2$)

Order 96: $A_4:C_8$ ($C_3$)

Order 72: $C_6\times A_4$ ($C_4$)

Order 48: $C_2^2\times C_{12}$ ($S_3$), $C_4\times A_4$ ($C_6$)

Order 36: $C_3\times A_4$ ($C_8$)

Order 24: $C_2^2\times C_6$ ($C_3:C_4$), $C_2\times A_4$ ($C_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times S_3$)

Order 12: $C_2\times C_6$ ($C_3:C_8$), $A_4$ ($C_{24}$), $C_{12}$ ($S_4$)

Order 8: $C_2^3$ ($C_3:C_{12}$)

Order 6: $C_6$ ($A_4:C_4$)

Order 4: $C_2^2$ ($C_3:C_{24}$), $C_4$ ($C_3\times S_4$)

Order 3: $C_3$ ($A_4:C_8$)

Order 2: $C_2$ ($A_4:C_{12}$)

Order 1: $C_1$ ($A_4:C_{24}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 288: $A_4:C_{24}$ ($C_1$)

Order 144: $C_{12}\times A_4$ ($C_2$)

Order 96: $A_4:C_8$ ($C_3$)

Order 72: $C_6\times A_4$ ($C_4$)

Order 48: $C_2^2\times C_{12}$ ($S_3$), $C_4\times A_4$ ($C_6$)

Order 36: $C_3\times A_4$ ($C_8$)

Order 24: $C_2^2\times C_6$ ($C_3:C_4$), $C_2\times A_4$ ($C_{12}$)

Order 16: $C_2^2\times C_4$ ($C_3\times S_3$)

Order 12: $C_2\times C_6$ ($C_3:C_8$), $A_4$ ($C_{24}$), $C_{12}$ ($S_4$)

Order 8: $C_2^3$ ($C_3:C_{12}$)

Order 6: $C_6$ ($A_4:C_4$)

Order 4: $C_2^2$ ($C_3:C_{24}$), $C_4$ ($C_3\times S_4$)

Order 3: $C_3$ ($A_4:C_8$)

Order 2: $C_2$ ($A_4:C_{12}$)

Order 1: $C_1$ ($A_4:C_{24}$)

Series

Derived series

$A_4:C_{24}$

$\rhd$

$A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$A_4:C_{24}$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$A_4:C_{24}$

$\rhd$

$A_4$

Upper central series

$C_1$

$\lhd$

$C_{12}$

Supergroups

This group is a maximal subgroup of 24 larger groups in the database.

This group is a maximal quotient of 13 larger groups in the database.

Character theory

Complex character table

See the $60 \times 60$ 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.