Abstract group 1152.155815: $C_{24}:(C_2\times S_4)$

• Label: 1152.155815
• Order: \( 2^{7} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $15$
• Trans deg.: $72$
• Rank: $3$

Group information

Description:	$C_{24}:(C_2\times S_4)$
Order:	\(1152\)\(\medspace = 2^{7} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^5.D_6^2$, of order \(4608\)\(\medspace = 2^{9} \cdot 3^{2} \)
Composition factors:	$C_2$ x 7, $C_3$ 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	139	26	212	230	160	256	128	1152
Conjugacy classes	1	8	3	9	8	4	8	10	51
Divisions	1	8	3	9	8	4	7	4	44
Autjugacy classes	1	8	3	9	8	4	7	4	44

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	8	10	8	15	6	0	4	0	0	51
Irr. rational chars.	8	10	8	5	6	3	2	1	1	44

Minimal presentations

Permutation degree:	$15$
Transitive degree:	$72$
Rank:	$3$
Inequivalent generating triples:	$80640$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	24	24
Arbitrary	7	9	9

Constructions

Presentation:	${\langle a, b, c, d, e \mid a^{2}=b^{2}=c^{24}=d^{2}=e^{6}=[a,b]=[a,d]=[b,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(2,5)(3,8)(6,7)(10,11), (2,6)(3,8)(5,7)(13,15), (1,2,3,5,4,7,8,6), (1,3,4,8) \!\cdots\! \rangle$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_8:S_4)$ $\,\rtimes\,$ $S_3$	$C_8$ $\,\rtimes\,$ $(S_3\times S_4)$	$(C_4:S_4)$ $\,\rtimes\,$ $D_6$	$(C_8\times A_4)$ $\,\rtimes\,$ $D_6$	all 27
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6:S_4)$ . $D_4$	$(C_6.S_4)$ . $D_4$	$(C_4:S_4)$ . $D_6$	$C_6$ . $(D_4\times S_4)$	all 20

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

Homology

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

Subgroups

There are 6310 subgroups in 616 conjugacy classes, 49 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{12}:S_4$
Commutator:	$G' \simeq$ $C_{12}\times A_4$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $D_6\times S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^2\times C_{24}$	$G/\operatorname{Fit} \simeq$ $D_6$
Radical:	$R \simeq$ $C_{24}:(C_2\times S_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 1152: $C_{24}:(C_2\times S_4)$

Order 576: $D_{12}:S_4$, $C_{12}.(C_2\times S_4)$, $A_4\times C_{24}:C_2$, $C_{24}:S_4$, $C_{24}:S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$

Order 384: $\SD_{16}:S_4$, $C_6.D_4^2$

Order 288: $C_{12}:S_4$ x 2, $A_4\times D_{12}$, $A_4\times C_3:Q_8$, $C_{12}:S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $A_4\times C_{24}$, $C_{24}:D_6$, $C_{12}.S_4$, $D_6\times S_4$

Order 192: $C_8:S_4$ x 2, $Q_8:S_4$, $D_4:S_4$, $C_8:S_4$, $C_{24}:D_4$, $C_{24}:D_4$, $C_{24}:D_4$, $(C_2\times C_6):\SD_{16}$, $D_{12}:D_4$, $C_8:(C_4\times S_3)$, $C_8:D_{12}$, $D_{12}.D_4$, $D_4:D_{12}$, $D_6:\SD_{16}$, $C_{12}.(C_2\times D_4)$, $Q_8:S_4$, $D_4\times S_4$, $D_8:D_6$, $C_{24}:C_2^3$, $D_{12}:D_4$, $C_{12}.(C_2\times D_4)$, $A_4\times \SD_{16}$

Order 144: $C_6\times S_4$ x 2, $S_3\times S_4$ x 2, $C_{24}:S_3$, $C_3\times D_{24}$, $C_{24}:C_6$, $C_{12}.D_6$, $D_{12}:S_3$, $A_4\times D_6$, $C_6:S_4$, $C_{12}\times A_4$, $C_{12}:D_6$, $D_{12}:S_3$, $A_4\times C_3:C_4$, $C_6.S_4$

Order 128: $D_4^2:C_2$

Order 96: $C_8:D_6$ x 4, $C_8:D_6$ x 4, $C_4:S_4$ x 3, $C_4\times S_4$ x 3, $C_8\times A_4$ x 2, $A_4:C_8$ x 2, $D_4:C_{12}$ x 2, $D_4\times D_6$ x 2, $C_4.D_{12}$ x 2, $C_{12}:D_4$ x 2, $D_{12}:C_4$ x 2, $C_6:\SD_{16}$ x 2, $C_6:D_8$ x 2, $C_3:(C_4\times Q_8)$, $C_2^3.D_6$, $D_6:D_4$, $C_2^2:D_{12}$, $C_2^3.D_6$, $C_8:C_{12}$, $C_{12}.D_4$, $D_6:C_8$, $C_2^2\times S_4$, $C_{24}:C_4$, $D_4:D_6$, $C_{12}.D_4$, $C_2^2\times D_{12}$, $C_6.C_2^4$, $Q_8\times A_4$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_6\times D_8$, $C_2^2\times C_{24}$, $C_{12}:D_4$, $C_2^3.D_6$, $D_6:D_4$, $C_{12}.D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_8:D_6$, $C_8:D_6$, $D_6:Q_8$, $C_4:D_{12}$, $D_{12}:C_4$

Order 72: $C_3\times D_{12}$ x 3, $C_3\times S_4$ x 2, $C_6\times A_4$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3:D_{12}$, $D_6:S_3$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_8:C_2^3$, $C_2^2\times \SD_{16}$, $Q_8:D_4$, $D_4^2$, $C_8:D_4$, $C_8:D_4$, $C_8:D_4$, $D_4.D_4$, $Q_8:D_4$, $C_2^2:\SD_{16}$, $D_4:D_4$, $Q_8:D_4$, $D_4:D_4$, $\SD_{16}:C_4$, $C_8:D_4$

Order 48: $S_3\times D_4$ x 8, $C_2\times S_4$ x 6, $C_{24}:C_2$ x 5, $C_6:D_4$ x 5, $C_{24}:C_2$ x 4, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_3:D_8$ x 4, $D_6:C_4$ x 4, $D_4:S_3$ x 3, $C_6:Q_8$ x 3, $C_4\times A_4$ x 3, $C_6.D_4$ x 3, $C_3:\SD_{16}$ x 3, $C_6:C_8$ x 2, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2, $C_4\times D_6$ x 2, $A_4:C_4$ x 2, $C_3\times D_8$ x 2, $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $D_{24}$, $C_2^2\times A_4$, $C_2^2\times C_{12}$, $D_{12}:C_2$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 3, $S_3^2$ x 2, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$, $C_6:S_3$, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 6, $C_4:D_4$ x 5, $D_4:C_4$ x 4, $D_8:C_2$ x 4, $C_2^2\times D_4$ x 3, $C_2\times D_8$ x 3, $C_4\times D_4$ x 3, $C_2^2:C_8$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $Q_8:C_4$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_8:C_4$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4:D_4$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 24: $C_2\times D_6$ x 11, $C_3:D_4$ x 10, $D_{12}$ x 9, $C_3\times D_4$ x 9, $C_6:C_4$ x 7, $C_4\times S_3$ x 7, $C_2\times C_{12}$ x 4, $C_3:Q_8$ x 4, $C_{24}$ x 4, $S_4$ x 4, $C_3:C_8$ x 4, $C_2^2\times C_6$ x 3, $C_2\times A_4$ x 3, $C_3\times Q_8$

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2\times D_4$ x 20, $C_2^2:C_4$ x 9, $\SD_{16}$ x 8, $D_8$ x 5, $C_4:C_4$ x 5, $C_2^2\times C_4$ x 5, $C_2\times C_8$ x 4, $D_4:C_2$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $C_4^2$ x 2, $C_2^4$ x 2

Order 12: $D_6$ x 19, $C_2\times C_6$ x 10, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 23, $C_2\times C_4$ x 15, $C_2^3$ x 13, $Q_8$ x 4, $C_8$ x 4

Order 6: $C_6$ x 8, $S_3$ x 7

Order 4: $C_2^2$ x 22, $C_4$ x 9

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1152: $C_{24}:(C_2\times S_4)$

Order 576: $D_{12}:S_4$, $C_{12}.(C_2\times S_4)$, $A_4\times C_{24}:C_2$, $C_{24}:S_4$, $C_{24}:S_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $D_{12}:S_4$

Order 384: $\SD_{16}:S_4$, $C_6.D_4^2$

Order 288: $C_{12}:S_4$ x 2, $A_4\times D_{12}$, $A_4\times C_3:Q_8$, $C_{12}:S_4$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $A_4\times C_{24}$, $C_{24}:D_6$, $C_{12}.S_4$, $D_6\times S_4$

Order 192: $C_8:S_4$ x 2, $Q_8:S_4$, $D_4:S_4$, $C_8:S_4$, $C_{24}:D_4$, $C_{24}:D_4$, $C_{24}:D_4$, $(C_2\times C_6):\SD_{16}$, $D_{12}:D_4$, $C_8:(C_4\times S_3)$, $C_8:D_{12}$, $D_{12}.D_4$, $D_4:D_{12}$, $D_6:\SD_{16}$, $C_{12}.(C_2\times D_4)$, $Q_8:S_4$, $D_4\times S_4$, $D_8:D_6$, $C_{24}:C_2^3$, $D_{12}:D_4$, $C_{12}.(C_2\times D_4)$, $A_4\times \SD_{16}$

Order 144: $C_6\times S_4$ x 2, $C_{24}:S_3$, $C_3\times D_{24}$, $C_{24}:C_6$, $C_{12}.D_6$, $D_{12}:S_3$, $A_4\times D_6$, $C_6:S_4$, $S_3\times S_4$, $C_{12}\times A_4$, $C_{12}:D_6$, $D_{12}:S_3$, $A_4\times C_3:C_4$, $C_6.S_4$

Order 128: $D_4^2:C_2$

Order 96: $C_8:D_6$ x 4, $C_8:D_6$ x 4, $C_4:S_4$ x 3, $C_4\times S_4$ x 3, $C_8\times A_4$ x 2, $A_4:C_8$ x 2, $D_4:C_{12}$ x 2, $D_4\times D_6$ x 2, $C_4.D_{12}$ x 2, $C_{12}:D_4$ x 2, $D_{12}:C_4$ x 2, $C_6:\SD_{16}$ x 2, $C_6:D_8$ x 2, $C_3:(C_4\times Q_8)$, $C_2^3.D_6$, $D_6:D_4$, $C_2^2:D_{12}$, $C_2^3.D_6$, $C_8:C_{12}$, $C_{12}.D_4$, $D_6:C_8$, $C_2^2\times S_4$, $C_{24}:C_4$, $D_4:D_6$, $C_{12}.D_4$, $C_2^2\times D_{12}$, $C_6.C_2^4$, $Q_8\times A_4$, $D_4\times A_4$, $\GL(2,\mathbb{Z}/4)$, $C_6\times D_8$, $C_2^2\times C_{24}$, $C_{12}:D_4$, $C_2^3.D_6$, $D_6:D_4$, $C_{12}.D_4$, $C_2^3.D_6$, $C_{12}:D_4$, $C_8:D_6$, $C_8:D_6$, $D_6:Q_8$, $C_4:D_{12}$, $D_{12}:C_4$

Order 72: $C_3\times D_{12}$ x 3, $C_3\times S_4$ x 2, $C_6\times A_4$, $S_3\times D_6$, $S_3\times A_4$, $C_3:S_4$, $C_{12}:S_3$, $C_3^2:Q_8$, $C_3:D_{12}$, $D_6:S_3$, $C_6.D_6$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_8:C_2^3$, $C_2^2\times \SD_{16}$, $Q_8:D_4$, $D_4^2$, $C_8:D_4$, $C_8:D_4$, $C_8:D_4$, $D_4.D_4$, $Q_8:D_4$, $C_2^2:\SD_{16}$, $D_4:D_4$, $Q_8:D_4$, $D_4:D_4$, $\SD_{16}:C_4$, $C_8:D_4$

Order 48: $S_3\times D_4$ x 6, $C_{24}:C_2$ x 5, $C_2\times S_4$ x 5, $C_6:D_4$ x 5, $C_{24}:C_2$ x 4, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_3:D_8$ x 4, $D_6:C_4$ x 4, $D_4:S_3$ x 3, $C_6:Q_8$ x 3, $C_4\times A_4$ x 3, $C_6.D_4$ x 3, $C_3:\SD_{16}$ x 3, $C_6:C_8$ x 2, $C_2^2\times D_6$ x 2, $C_6.C_2^3$ x 2, $C_4\times D_6$ x 2, $A_4:C_4$ x 2, $C_3\times D_8$ x 2, $C_2\times C_{24}$ x 2, $C_4:C_{12}$ x 2, $C_2^2:C_{12}$ x 2, $C_6.D_4$ x 2, $C_{12}:C_4$ x 2, $D_{24}$, $C_2^2\times A_4$, $C_2^2\times C_{12}$, $D_{12}:C_2$, $C_3\times \SD_{16}$, $Q_8:S_3$, $C_{12}:C_4$

Order 36: $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_3:C_{12}$, $C_6:S_3$, $C_3\times A_4$, $S_3^2$

Order 32: $C_2\times \SD_{16}$ x 6, $C_4:D_4$ x 5, $D_4:C_4$ x 4, $D_8:C_2$ x 4, $C_2^2\times D_4$ x 3, $C_2\times D_8$ x 3, $C_4\times D_4$ x 3, $C_2^2:C_8$ x 2, $C_4^2:C_2$ x 2, $C_2^2:Q_8$ x 2, $C_2^2\wr C_2$ x 2, $Q_8:C_4$ x 2, $D_4:C_2^2$, $C_2^2\times Q_8$, $C_8:C_4$, $C_2\times \OD_{16}$, $C_2^2\times C_8$, $C_4:D_4$, $C_4\times Q_8$, $C_8:C_4$, $C_4:C_8$

Order 24: $C_2\times D_6$ x 10, $C_3:D_4$ x 9, $C_3\times D_4$ x 9, $D_{12}$ x 8, $C_6:C_4$ x 7, $C_4\times S_3$ x 6, $C_2\times C_{12}$ x 4, $C_3:Q_8$ x 4, $C_{24}$ x 4, $S_4$ x 4, $C_3:C_8$ x 4, $C_2^2\times C_6$ x 3, $C_2\times A_4$ x 3, $C_3\times Q_8$

Order 18: $C_3\times S_3$ x 3, $C_3\times C_6$, $C_3:S_3$

Order 16: $C_2\times D_4$ x 18, $C_2^2:C_4$ x 9, $\SD_{16}$ x 8, $D_8$ x 5, $C_4:C_4$ x 5, $C_2^2\times C_4$ x 5, $C_2\times C_8$ x 4, $D_4:C_2$ x 3, $C_2\times Q_8$ x 3, $\OD_{16}$ x 2, $C_4^2$ x 2, $C_2^4$ x 2

Order 12: $D_6$ x 16, $C_2\times C_6$ x 10, $C_{12}$ x 7, $C_3:C_4$ x 7, $A_4$ x 2

Order 9: $C_3^2$

Order 8: $D_4$ x 21, $C_2\times C_4$ x 14, $C_2^3$ x 12, $Q_8$ x 4, $C_8$ x 4

Order 6: $C_6$ x 8, $S_3$ x 7

Order 4: $C_2^2$ x 20, $C_4$ x 9

Order 3: $C_3$ x 3

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1152: $C_{24}:(C_2\times S_4)$ ($C_1$)

Order 576: $D_{12}:S_4$ ($C_2$), $C_{12}.(C_2\times S_4)$ ($C_2$), $A_4\times C_{24}:C_2$ ($C_2$), $C_{24}:S_4$ ($C_2$), $C_{24}:S_4$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $D_{12}:S_4$ ($C_2$)

Order 288: $C_{12}:S_4$ ($C_2^2$) x 2, $A_4\times D_{12}$ ($C_2^2$), $A_4\times C_3:Q_8$ ($C_2^2$), $C_{12}:S_4$ ($C_2^2$), $A_4\times C_{24}$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$)

Order 192: $C_8:S_4$ ($S_3$), $C_{24}:C_2^3$ ($S_3$)

Order 144: $C_6:S_4$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $C_6.S_4$ ($D_4$)

Order 96: $C_4:S_4$ ($D_6$) x 2, $C_8\times A_4$ ($D_6$), $C_2^2\times D_{12}$ ($D_6$), $C_6.C_2^4$ ($D_6$), $C_2^2\times C_{24}$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2\times D_4$)

Order 48: $C_{24}:C_2$ ($S_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($D_8:C_2$)

Order 32: $C_2^2\times C_8$ ($S_3^2$)

Order 24: $D_{12}$ ($C_2\times S_4$), $C_3:Q_8$ ($C_2\times S_4$), $C_{24}$ ($C_2\times S_4$), $C_2^2\times C_6$ ($S_3\times D_4$), $C_2\times A_4$ ($S_3\times D_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_8:D_6$), $A_4$ ($C_8:D_6$), $C_{12}$ ($C_2^2\times S_4$)

Order 8: $C_2^3$ ($C_{12}:D_6$), $C_8$ ($S_3\times S_4$)

Order 6: $C_6$ ($D_4\times S_4$)

Order 4: $C_2^2$ ($C_{24}:D_6$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($\SD_{16}:S_4$)

Order 2: $C_2$ ($D_{12}:S_4$)

Order 1: $C_1$ ($C_{24}:(C_2\times S_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1152: $C_{24}:(C_2\times S_4)$ ($C_1$)

Order 576: $D_{12}:S_4$ ($C_2$), $C_{12}.(C_2\times S_4)$ ($C_2$), $A_4\times C_{24}:C_2$ ($C_2$), $C_{24}:S_4$ ($C_2$), $C_{24}:S_4$ ($C_2$), $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2$), $D_{12}:S_4$ ($C_2$)

Order 288: $C_{12}:S_4$ ($C_2^2$) x 2, $A_4\times D_{12}$ ($C_2^2$), $A_4\times C_3:Q_8$ ($C_2^2$), $C_{12}:S_4$ ($C_2^2$), $A_4\times C_{24}$ ($C_2^2$), $C_{12}.S_4$ ($C_2^2$)

Order 192: $C_8:S_4$ ($S_3$), $C_{24}:C_2^3$ ($S_3$)

Order 144: $C_6:S_4$ ($D_4$), $C_{12}\times A_4$ ($C_2^3$), $C_6.S_4$ ($D_4$)

Order 96: $C_4:S_4$ ($D_6$) x 2, $C_8\times A_4$ ($D_6$), $C_2^2\times D_{12}$ ($D_6$), $C_6.C_2^4$ ($D_6$), $C_2^2\times C_{24}$ ($D_6$)

Order 72: $C_6\times A_4$ ($C_2\times D_4$)

Order 48: $C_{24}:C_2$ ($S_4$), $C_2^2\times C_{12}$ ($C_2\times D_6$), $C_4\times A_4$ ($C_2\times D_6$)

Order 36: $C_3\times A_4$ ($D_8:C_2$)

Order 32: $C_2^2\times C_8$ ($S_3^2$)

Order 24: $D_{12}$ ($C_2\times S_4$), $C_3:Q_8$ ($C_2\times S_4$), $C_{24}$ ($C_2\times S_4$), $C_2^2\times C_6$ ($S_3\times D_4$), $C_2\times A_4$ ($S_3\times D_4$)

Order 16: $C_2^2\times C_4$ ($S_3\times D_6$)

Order 12: $C_2\times C_6$ ($C_8:D_6$), $A_4$ ($C_8:D_6$), $C_{12}$ ($C_2^2\times S_4$)

Order 8: $C_2^3$ ($C_{12}:D_6$), $C_8$ ($S_3\times S_4$)

Order 6: $C_6$ ($D_4\times S_4$)

Order 4: $C_2^2$ ($C_{24}:D_6$), $C_4$ ($D_6\times S_4$)

Order 3: $C_3$ ($\SD_{16}:S_4$)

Order 2: $C_2$ ($D_{12}:S_4$)

Order 1: $C_1$ ($C_{24}:(C_2\times S_4)$)

Series

Derived series

$C_{24}:(C_2\times S_4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Chief series

$C_{24}:(C_2\times S_4)$

$\rhd$

$D_{12}:S_4$

$\rhd$

$A_4\times D_{12}$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{24}:(C_2\times S_4)$

$\rhd$

$C_{12}\times A_4$

$\rhd$

$C_6\times A_4$

$\rhd$

$C_3\times A_4$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_8$

Character theory

Complex character table

See the $51 \times 51$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $44 \times 44$ rational character table.