Abstract group 1728.8372: $C_{48}.D_{18}$

• Label: 1728.8372
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{4} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{5} \cdot 3 \)
• $\card{Z(G)}$: \( 2^{3} \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \)
• Perm deg.: $44$
• Trans deg.: $288$
• Rank: $3$

Group information

Description:	$C_{48}.D_{18}$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(144\)\(\medspace = 2^{4} \cdot 3^{2} \)
Automorphism group:	$C_{18}.(C_2^4\times C_{12}).C_2$, of order \(6912\)\(\medspace = 2^{8} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	3	4	6	8	9	12	16	18	24	36	48	72	144
Elements	1	39	8	40	96	80	18	104	160	54	208	72	416	144	288	1728
Conjugacy classes	1	4	5	5	14	10	9	19	20	18	38	27	64	54	72	360
Divisions	1	4	3	4	8	4	2	8	4	4	8	4	8	4	4	70
Autjugacy classes	1	3	3	3	7	3	2	7	3	4	7	4	5	4	2	58

Dimension	1	2	4	6	8	12	16	24	32	48	96
Irr. complex chars.	96	216	48	0	0	0	0	0	0	0	0	360
Irr. rational chars.	8	16	14	4	8	6	4	4	2	3	1	70

Minimal presentations

Permutation degree:	$44$
Transitive degree:	$288$
Rank:	$3$
Inequivalent generating triples:	$209664$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	4	8	96
Arbitrary	4	6	24

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{18}=c^{48}=[a,c]=1, b^{a}=b^{17}c^{24}, c^{b}=c^{25} \rangle$
Permutation group:	Degree $44$ $\langle(1,2)(3,6)(4,9)(5,10)(7,11)(8,14)(12,17)(13,19)(15,21)(16,23)(18,24)(20,25) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $(C_{16}.D_{18})$
Semidirect product:	$(D_9\times C_{48})$ $\,\rtimes\,$ $C_2$ (2)	$(C_{16}\times D_9)$ $\,\rtimes\,$ $C_6$ (2)	$(C_{144}:C_6)$ $\,\rtimes\,$ $C_2$	$(C_{18}:C_{48})$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{36}$ . $C_{24}$	$C_{48}$ . $D_{18}$ (2)	$C_{48}$ . $(C_6\times S_3)$ (2)	$(C_9:Q_8)$ . $C_{24}$	all 84

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

Homology

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

Subgroups

There are 946 subgroups in 278 conjugacy classes, 126 normal (78 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{24}$	$G/Z \simeq$ $C_2\times D_{18}$
Commutator:	$G' \simeq$ $C_{18}$	$G/G' \simeq$ $C_2^2\times C_{24}$
Frattini:	$\Phi \simeq$ $C_{24}$	$G/\Phi \simeq$ $C_6\times D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{144}:C_6$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_{48}.D_{18}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_8\times D_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $\OD_{32}:C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times C_9$

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 1728: $C_{48}.D_{18}$

Order 864: $C_{144}:C_6$ x 2, $D_9\times C_{48}$ x 2, $C_{144}:C_6$, $C_{18}:C_{48}$, $C_{24}.D_{18}$

Order 576: $C_{16}.D_{18}$, $D_{12}.C_{24}$

Order 432: $C_3\times C_{144}$ x 2, $C_9:C_{48}$ x 2, $C_{72}:C_6$ x 2, $D_9\times C_{24}$ x 2, $D_{36}:C_6$, $C_6\times C_{72}$, $C_{36}.C_{12}$

Order 288: $C_9\times \OD_{32}$ x 2, $C_{144}:C_2$ x 2, $C_{16}\times D_9$ x 2, $C_{48}:C_6$ x 2, $S_3\times C_{48}$ x 2, $C_{24}.D_6$, $C_3^2\times \OD_{32}$, $C_6:C_{48}$, $C_{18}:C_{16}$, $C_8.D_{18}$

Order 216: $D_{18}:C_6$ x 2, $C_{12}\times D_9$ x 2, $C_3\times C_{72}$ x 2, $C_9:C_{24}$ x 2, $C_6\times C_{36}$, $C_3\times D_{36}$, $C_{36}.C_6$

Order 192: $\OD_{32}:C_6$, $\OD_{32}:S_3$

Order 144: $C_{144}$ x 4, $C_{24}:C_6$ x 2, $S_3\times C_{24}$ x 2, $C_{72}:C_2$ x 2, $C_8\times D_9$ x 2, $C_3\times C_{48}$ x 2, $C_3:C_{48}$ x 2, $C_2\times C_{72}$ x 2, $C_9:C_{16}$ x 2, $C_{12}.C_{12}$, $D_{36}:C_2$, $D_{12}:C_6$, $C_6\times C_{24}$, $C_9:\OD_{16}$

Order 108: $C_9:C_{12}$ x 2, $C_3\times D_{18}$ x 2, $C_3\times C_{36}$ x 2, $C_6\times C_{18}$

Order 96: $C_3\times \OD_{32}$ x 5, $C_2\times C_{48}$ x 3, $C_{48}:C_2$ x 2, $S_3\times C_{16}$ x 2, $C_6:C_{16}$, $\OD_{16}:C_6$, $C_8.D_6$

Order 72: $C_{72}$ x 4, $C_2\times C_{36}$ x 2, $C_9:D_4$ x 2, $C_4\times D_9$ x 2, $C_6\wr C_2$ x 2, $S_3\times C_{12}$ x 2, $C_3\times C_{24}$ x 2, $C_3:C_{24}$ x 2, $C_9:C_8$ x 2, $D_{36}$, $C_9:Q_8$, $C_6\times C_{12}$, $C_3\times D_{12}$, $C_3^2:Q_8$

Order 64: $\OD_{32}:C_2$

Order 54: $C_3\times C_{18}$ x 2, $C_3\times D_9$ x 2

Order 48: $C_{48}$ x 8, $C_2\times C_{24}$ x 5, $C_3\times \OD_{16}$ x 3, $C_{24}:C_2$ x 2, $S_3\times C_8$ x 2, $C_3:C_{16}$ x 2, $D_4:C_6$, $D_{12}:C_2$, $C_3:\OD_{16}$

Order 36: $C_{36}$ x 4, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_2\times C_{18}$ x 2, $D_{18}$ x 2, $C_6\times S_3$ x 2, $C_9:C_4$ x 2, $C_6^2$

Order 32: $\OD_{32}$ x 3, $C_2\times C_{16}$ x 3, $\OD_{16}:C_2$

Order 27: $C_3\times C_9$

Order 24: $C_{24}$ x 8, $C_2\times C_{12}$ x 5, $C_3\times D_4$ x 3, $C_3:D_4$ x 2, $C_4\times S_3$ x 2, $C_3:C_8$ x 2, $D_{12}$, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_{18}$ x 4, $C_3\times C_6$ x 2, $C_3\times S_3$ x 2, $D_9$ x 2

Order 16: $C_{16}$ x 4, $\OD_{16}$ x 3, $C_2\times C_8$ x 3, $D_4:C_2$

Order 12: $C_{12}$ x 8, $C_2\times C_6$ x 5, $D_6$ x 2, $C_3:C_4$ x 2

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_8$ x 4, $D_4$ x 3, $C_2\times C_4$ x 3, $Q_8$

Order 6: $C_6$ x 8, $S_3$ x 2

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $C_{48}.D_{18}$

Order 864: $C_{144}:C_6$, $C_{18}:C_{48}$, $C_{144}:C_6$, $D_9\times C_{48}$, $C_{24}.D_{18}$

Order 576: $C_{16}.D_{18}$, $D_{12}.C_{24}$

Order 432: $C_9:C_{48}$ x 2, $D_{36}:C_6$, $C_3\times C_{144}$, $C_6\times C_{72}$, $C_{36}.C_{12}$, $C_{72}:C_6$, $D_9\times C_{24}$

Order 288: $C_9\times \OD_{32}$ x 2, $C_{24}.D_6$, $C_{144}:C_2$, $C_{16}\times D_9$, $C_3^2\times \OD_{32}$, $C_6:C_{48}$, $C_{48}:C_6$, $S_3\times C_{48}$, $C_{18}:C_{16}$, $C_8.D_{18}$

Order 216: $C_3\times C_{72}$ x 2, $C_6\times C_{36}$, $D_{18}:C_6$, $C_3\times D_{36}$, $C_{12}\times D_9$, $C_{36}.C_6$, $C_9:C_{24}$

Order 192: $\OD_{32}:C_6$, $\OD_{32}:S_3$

Order 144: $C_3:C_{48}$ x 2, $C_2\times C_{72}$ x 2, $C_{144}$ x 2, $C_9:C_{16}$ x 2, $C_{12}.C_{12}$, $C_{24}:C_6$, $S_3\times C_{24}$, $C_{72}:C_2$, $C_8\times D_9$, $D_{36}:C_2$, $C_3\times C_{48}$, $D_{12}:C_6$, $C_6\times C_{24}$, $C_9:\OD_{16}$

Order 108: $C_3\times C_{36}$ x 2, $C_9:C_{12}$, $C_6\times C_{18}$, $C_3\times D_{18}$

Order 96: $C_3\times \OD_{32}$ x 4, $C_2\times C_{48}$ x 2, $C_{48}:C_2$, $S_3\times C_{16}$, $C_6:C_{16}$, $\OD_{16}:C_6$, $C_8.D_6$

Order 72: $C_{72}$ x 4, $C_2\times C_{36}$ x 2, $C_3\times C_{24}$ x 2, $C_9:D_4$, $D_{36}$, $C_4\times D_9$, $C_9:Q_8$, $C_6\times C_{12}$, $C_6\wr C_2$, $C_3\times D_{12}$, $S_3\times C_{12}$, $C_3^2:Q_8$, $C_3:C_{24}$, $C_9:C_8$

Order 64: $\OD_{32}:C_2$

Order 54: $C_3\times C_{18}$ x 2, $C_3\times D_9$

Order 48: $C_{48}$ x 5, $C_2\times C_{24}$ x 4, $C_3\times \OD_{16}$ x 2, $C_3:C_{16}$ x 2, $C_{24}:C_2$, $D_4:C_6$, $S_3\times C_8$, $D_{12}:C_2$, $C_3:\OD_{16}$

Order 36: $C_{36}$ x 4, $C_3\times C_{12}$ x 2, $C_2\times C_{18}$ x 2, $C_3:C_{12}$, $D_{18}$, $C_6^2$, $C_6\times S_3$, $C_9:C_4$

Order 32: $\OD_{32}$ x 2, $C_2\times C_{16}$ x 2, $\OD_{16}:C_2$

Order 27: $C_3\times C_9$

Order 24: $C_{24}$ x 7, $C_2\times C_{12}$ x 4, $C_3\times D_4$ x 2, $C_3:D_4$, $D_{12}$, $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$, $C_3:C_8$

Order 18: $C_{18}$ x 4, $C_3\times C_6$ x 2, $C_3\times S_3$, $D_9$

Order 16: $C_{16}$ x 3, $\OD_{16}$ x 2, $C_2\times C_8$ x 2, $D_4:C_2$

Order 12: $C_{12}$ x 7, $C_2\times C_6$ x 4, $D_6$, $C_3:C_4$

Order 9: $C_9$ x 2, $C_3^2$

Order 8: $C_8$ x 3, $D_4$ x 2, $C_2\times C_4$ x 2, $Q_8$

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

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

Order 3: $C_3$ x 3

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $C_{48}.D_{18}$ ($C_1$)

Order 864: $C_{144}:C_6$ ($C_2$) x 2, $D_9\times C_{48}$ ($C_2$) x 2, $C_{144}:C_6$ ($C_2$), $C_{18}:C_{48}$ ($C_2$), $C_{24}.D_{18}$ ($C_2$)

Order 576: $C_{16}.D_{18}$ ($C_3$)

Order 432: $C_3\times C_{144}$ ($C_2^2$) x 2, $C_9:C_{48}$ ($C_2^2$) x 2, $C_{72}:C_6$ ($C_4$) x 2, $D_9\times C_{24}$ ($C_2^2$) x 2, $D_{36}:C_6$ ($C_4$), $C_6\times C_{72}$ ($C_2^2$), $C_{36}.C_{12}$ ($C_4$)

Order 288: $C_{144}:C_2$ ($C_6$) x 2, $C_{16}\times D_9$ ($C_6$) x 2, $C_9\times \OD_{32}$ ($C_6$), $C_3^2\times \OD_{32}$ ($S_3$), $C_{18}:C_{16}$ ($C_6$), $C_8.D_{18}$ ($C_6$)

Order 216: $D_{18}:C_6$ ($C_8$) x 2, $C_{12}\times D_9$ ($C_2\times C_4$) x 2, $C_9:C_{24}$ ($C_2\times C_4$) x 2, $C_6\times C_{36}$ ($C_2\times C_4$), $C_3\times D_{36}$ ($C_8$), $C_{36}.C_6$ ($C_8$), $C_3\times C_{72}$ ($C_2^3$), $C_3\times C_{72}$ ($C_2\times C_4$)

Order 144: $C_{72}:C_2$ ($C_{12}$) x 2, $C_8\times D_9$ ($C_2\times C_6$) x 2, $C_3\times C_{48}$ ($D_6$) x 2, $C_{144}$ ($C_2\times C_6$) x 2, $C_9:C_{16}$ ($C_2\times C_6$) x 2, $D_{36}:C_2$ ($C_{12}$), $C_2\times C_{72}$ ($C_2\times C_6$), $C_6\times C_{24}$ ($D_6$), $C_9:\OD_{16}$ ($C_{12}$)

Order 108: $C_9:C_{12}$ ($C_2\times C_8$) x 2, $C_3\times D_{18}$ ($C_2\times C_8$) x 2, $C_6\times C_{18}$ ($C_2\times C_8$), $C_3\times C_{36}$ ($C_2\times C_8$), $C_3\times C_{36}$ ($C_2^2\times C_4$)

Order 96: $C_3\times \OD_{32}$ ($C_3\times S_3$), $C_3\times \OD_{32}$ ($D_9$)

Order 72: $C_9:D_4$ ($C_{24}$) x 2, $C_4\times D_9$ ($C_2\times C_{12}$) x 2, $C_9:C_8$ ($C_2\times C_{12}$) x 2, $C_2\times C_{36}$ ($C_2\times C_{12}$), $D_{36}$ ($C_{24}$), $C_9:Q_8$ ($C_{24}$), $C_6\times C_{12}$ ($C_4\times S_3$), $C_{72}$ ($C_2\times C_{12}$), $C_{72}$ ($C_2^2\times C_6$), $C_3\times C_{24}$ ($C_4\times S_3$), $C_3\times C_{24}$ ($C_2\times D_6$)

Order 54: $C_3\times C_{18}$ ($C_2^2\times C_8$)

Order 48: $C_{48}$ ($D_{18}$) x 2, $C_{48}$ ($C_6\times S_3$) x 2, $C_2\times C_{24}$ ($D_{18}$), $C_2\times C_{24}$ ($C_6\times S_3$)

Order 36: $D_{18}$ ($C_2\times C_{24}$) x 2, $C_9:C_4$ ($C_2\times C_{24}$) x 2, $C_3\times C_{12}$ ($S_3\times C_8$), $C_3\times C_{12}$ ($C_4\times D_6$), $C_2\times C_{18}$ ($C_2\times C_{24}$), $C_{36}$ ($C_2^2\times C_{12}$), $C_{36}$ ($C_2\times C_{24}$), $C_6^2$ ($S_3\times C_8$)

Order 32: $\OD_{32}$ ($C_3\times D_9$)

Order 27: $C_3\times C_9$ ($\OD_{32}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_4\times D_9$), $C_2\times C_{12}$ ($S_3\times C_{12}$), $C_{24}$ ($C_4\times D_9$), $C_{24}$ ($C_6\times D_6$), $C_{24}$ ($S_3\times C_{12}$), $C_{24}$ ($C_2\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_8\times D_6$), $C_{18}$ ($C_2^2\times C_{24}$)

Order 16: $C_{16}$ ($C_3\times D_{18}$) x 2, $C_2\times C_8$ ($C_3\times D_{18}$)

Order 12: $C_2\times C_6$ ($S_3\times C_{24}$), $C_2\times C_6$ ($C_8\times D_9$), $C_{12}$ ($S_3\times C_{24}$), $C_{12}$ ($C_8\times D_9$), $C_{12}$ ($C_4\times D_{18}$), $C_{12}$ ($C_{12}\times D_6$)

Order 9: $C_3^2$ ($\OD_{32}:S_3$), $C_9$ ($\OD_{32}:C_6$)

Order 8: $C_2\times C_4$ ($C_{12}\times D_9$), $C_8$ ($C_{12}\times D_9$), $C_8$ ($C_6\times D_{18}$)

Order 6: $C_6$ ($D_6\times C_{24}$), $C_6$ ($C_8\times D_{18}$)

Order 4: $C_2^2$ ($D_9\times C_{24}$), $C_4$ ($C_{12}\times D_{18}$), $C_4$ ($D_9\times C_{24}$)

Order 3: $C_3$ ($C_{16}.D_{18}$), $C_3$ ($D_{12}.C_{24}$)

Order 2: $C_2$ ($C_{24}\times D_{18}$)

Order 1: $C_1$ ($C_{48}.D_{18}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $C_{48}.D_{18}$ ($C_1$)

Order 864: $C_{144}:C_6$ ($C_2$), $C_{18}:C_{48}$ ($C_2$), $C_{144}:C_6$ ($C_2$), $D_9\times C_{48}$ ($C_2$), $C_{24}.D_{18}$ ($C_2$)

Order 576: $C_{16}.D_{18}$ ($C_3$)

Order 432: $C_9:C_{48}$ ($C_2^2$) x 2, $D_{36}:C_6$ ($C_4$), $C_3\times C_{144}$ ($C_2^2$), $C_6\times C_{72}$ ($C_2^2$), $C_{36}.C_{12}$ ($C_4$), $C_{72}:C_6$ ($C_4$), $D_9\times C_{24}$ ($C_2^2$)

Order 288: $C_9\times \OD_{32}$ ($C_6$), $C_{144}:C_2$ ($C_6$), $C_{16}\times D_9$ ($C_6$), $C_3^2\times \OD_{32}$ ($S_3$), $C_{18}:C_{16}$ ($C_6$), $C_8.D_{18}$ ($C_6$)

Order 216: $C_6\times C_{36}$ ($C_2\times C_4$), $D_{18}:C_6$ ($C_8$), $C_3\times D_{36}$ ($C_8$), $C_{12}\times D_9$ ($C_2\times C_4$), $C_{36}.C_6$ ($C_8$), $C_3\times C_{72}$ ($C_2^3$), $C_3\times C_{72}$ ($C_2\times C_4$), $C_9:C_{24}$ ($C_2\times C_4$)

Order 144: $C_9:C_{16}$ ($C_2\times C_6$) x 2, $C_{72}:C_2$ ($C_{12}$), $C_8\times D_9$ ($C_2\times C_6$), $D_{36}:C_2$ ($C_{12}$), $C_3\times C_{48}$ ($D_6$), $C_2\times C_{72}$ ($C_2\times C_6$), $C_{144}$ ($C_2\times C_6$), $C_6\times C_{24}$ ($D_6$), $C_9:\OD_{16}$ ($C_{12}$)

Order 108: $C_9:C_{12}$ ($C_2\times C_8$), $C_6\times C_{18}$ ($C_2\times C_8$), $C_3\times D_{18}$ ($C_2\times C_8$), $C_3\times C_{36}$ ($C_2\times C_8$), $C_3\times C_{36}$ ($C_2^2\times C_4$)

Order 96: $C_3\times \OD_{32}$ ($C_3\times S_3$), $C_3\times \OD_{32}$ ($D_9$)

Order 72: $C_2\times C_{36}$ ($C_2\times C_{12}$), $C_9:D_4$ ($C_{24}$), $D_{36}$ ($C_{24}$), $C_4\times D_9$ ($C_2\times C_{12}$), $C_9:Q_8$ ($C_{24}$), $C_6\times C_{12}$ ($C_4\times S_3$), $C_{72}$ ($C_2\times C_{12}$), $C_{72}$ ($C_2^2\times C_6$), $C_3\times C_{24}$ ($C_4\times S_3$), $C_3\times C_{24}$ ($C_2\times D_6$), $C_9:C_8$ ($C_2\times C_{12}$)

Order 54: $C_3\times C_{18}$ ($C_2^2\times C_8$)

Order 48: $C_2\times C_{24}$ ($D_{18}$), $C_2\times C_{24}$ ($C_6\times S_3$), $C_{48}$ ($D_{18}$), $C_{48}$ ($C_6\times S_3$)

Order 36: $C_3\times C_{12}$ ($S_3\times C_8$), $C_3\times C_{12}$ ($C_4\times D_6$), $C_2\times C_{18}$ ($C_2\times C_{24}$), $D_{18}$ ($C_2\times C_{24}$), $C_{36}$ ($C_2^2\times C_{12}$), $C_{36}$ ($C_2\times C_{24}$), $C_6^2$ ($S_3\times C_8$), $C_9:C_4$ ($C_2\times C_{24}$)

Order 32: $\OD_{32}$ ($C_3\times D_9$)

Order 27: $C_3\times C_9$ ($\OD_{32}:C_2$)

Order 24: $C_2\times C_{12}$ ($C_4\times D_9$), $C_2\times C_{12}$ ($S_3\times C_{12}$), $C_{24}$ ($C_4\times D_9$), $C_{24}$ ($C_6\times D_6$), $C_{24}$ ($S_3\times C_{12}$), $C_{24}$ ($C_2\times D_{18}$)

Order 18: $C_3\times C_6$ ($C_8\times D_6$), $C_{18}$ ($C_2^2\times C_{24}$)

Order 16: $C_2\times C_8$ ($C_3\times D_{18}$), $C_{16}$ ($C_3\times D_{18}$)

Order 12: $C_2\times C_6$ ($S_3\times C_{24}$), $C_2\times C_6$ ($C_8\times D_9$), $C_{12}$ ($S_3\times C_{24}$), $C_{12}$ ($C_8\times D_9$), $C_{12}$ ($C_4\times D_{18}$), $C_{12}$ ($C_{12}\times D_6$)

Order 9: $C_3^2$ ($\OD_{32}:S_3$), $C_9$ ($\OD_{32}:C_6$)

Order 8: $C_2\times C_4$ ($C_{12}\times D_9$), $C_8$ ($C_{12}\times D_9$), $C_8$ ($C_6\times D_{18}$)

Order 6: $C_6$ ($D_6\times C_{24}$), $C_6$ ($C_8\times D_{18}$)

Order 4: $C_2^2$ ($D_9\times C_{24}$), $C_4$ ($C_{12}\times D_{18}$), $C_4$ ($D_9\times C_{24}$)

Order 3: $C_3$ ($C_{16}.D_{18}$), $C_3$ ($D_{12}.C_{24}$)

Order 2: $C_2$ ($C_{24}\times D_{18}$)

Order 1: $C_1$ ($C_{48}.D_{18}$)

Series

Derived series

$C_{48}.D_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_1$

Chief series

$C_{48}.D_{18}$

$\rhd$

$D_9\times C_{48}$

$\rhd$

$C_3\times C_{144}$

$\rhd$

$C_3\times C_{48}$

$\rhd$

$C_{48}$

$\rhd$

$C_{24}$

$\rhd$

$C_{12}$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_{48}.D_{18}$

$\rhd$

$C_{18}$

$\rhd$

$C_9$

Upper central series

$C_1$

$\lhd$

$C_{24}$

$\lhd$

$C_3\times \OD_{32}$

Character theory

Complex character table

See the $360 \times 360$ 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 $70 \times 70$ rational character table.