Abstract group 1728.47489: $(Q_8\times C_3^2):S_4$

• Label: 1728.47489
• Order: \( 2^{6} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3 \)
• Perm deg.: $14$
• Trans deg.: $48$
• Rank: $3$

Group information

Description:	$(Q_8\times C_3^2):S_4$
Order:	\(1728\)\(\medspace = 2^{6} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2\times C_2^4:\He_3.C_2^4$, of order \(13824\)\(\medspace = 2^{9} \cdot 3^{3} \)
Composition factors:	$C_2$ x 6, $C_3$ x 3
Derived length:	$4$

This group is nonabelian and monomial (hence solvable).

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	91	296	84	584	144	240	288	1728
Conjugacy classes	1	4	14	4	29	2	14	4	72
Divisions	1	4	9	4	17	2	8	2	47
Autjugacy classes	1	4	5	2	12	1	4	1	30

Dimension	1	2	3	4	6	8	12	16
Irr. complex chars.	6	12	18	6	18	12	0	0	72
Irr. rational chars.	2	6	6	6	10	6	7	4	47

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$48$
Rank:	$3$
Inequivalent generating triples:	$152880$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	16
Arbitrary	6	8	8

Constructions

Presentation:	${\langle a, b, c, d, e, f \mid a^{2}=b^{3}=d^{2}=e^{4}=f^{6}=[a,d]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $14$ $\langle(1,2)(3,7)(5,8)(9,10)(11,13)(12,14), (9,11,12)(10,13,14), (3,4,7)(5,6,8) \!\cdots\! \rangle$
Direct product:	$C_3$ $\, \times\, $ $((C_3\times Q_8):S_4)$
Semidirect product:	$(Q_8\times C_3^2)$ $\,\rtimes\,$ $S_4$ (2)	$C_3^2$ $\,\rtimes\,$ $(Q_8:S_4)$	$Q_8$ $\,\rtimes\,$ $(C_3^2:S_4)$ (2)	$(D_4:C_6^2)$ $\,\rtimes\,$ $S_3$	all 14
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times C_6^2)$ . $S_4$	$C_2$ . $(C_6^2:S_4)$	$C_2^3$ . $(C_3^2:S_4)$	$C_6$ . $((C_2\times C_6):S_4)$	all 8

Elements of the group are displayed as permutations of degree 14.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$2$

Subgroups

There are 5270 subgroups in 496 conjugacy classes, 34 normal (20 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $(C_2\times C_6):S_4$
Commutator:	$G' \simeq$ $C_3\times Q_8:A_4$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_6^2:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $(Q_8\times C_3^2):S_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1728: $(Q_8\times C_3^2):S_4$

Order 864: $C_3^2\times Q_8:A_4$

Order 576: $C_3\times Q_8:S_4$ x 3, $(C_3\times Q_8):S_4$, $(C_6\times D_{12}):C_2^2$

Order 432: $C_3^2:\GL(2,3)$ x 2, $C_6^2:D_6$

Order 288: $C_3\times Q_8:A_4$ x 8, $C_6^2.D_4$ x 2, $C_6^2.D_4$ x 2, $C_6^2.C_2^3$, $C_3\times C_{12}.D_4$, $D_4:C_6^2$

Order 216: $C_6^2:C_6$ x 2, $C_3^2:S_4$ x 2, $C_3^2\times \SL(2,3)$ x 2

Order 192: $Q_8:S_4$ x 3, $C_3\times D_4:D_4$, $(C_3\times D_4):D_4$

Order 144: $C_3\times \GL(2,3)$ x 6, $C_6\times S_4$ x 3, $C_{12}.D_6$ x 2, $D_{12}:C_6$ x 2, $C_{12}.C_{12}$ x 2, $C_4.C_6^2$ x 2, $C_4:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_3:\GL(2,3)$ x 2, $C_{12}:C_{12}$, $C_6:S_4$, $C_6\times D_{12}$

Order 108: $C_3^2\times A_4$ x 2, $C_3^2:D_6$

Order 96: $Q_8:A_4$ x 6, $C_6.C_2^4$ x 3, $D_4:C_{12}$ x 2, $C_{12}.D_4$ x 2, $D_8:C_6$ x 2, $D_4:D_6$ x 2, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_{12}:D_4$, $C_{12}:D_4$

Order 72: $C_6\times A_4$ x 20, $C_3\times \SL(2,3)$ x 16, $C_3\times S_4$ x 6, $D_4\times C_3^2$ x 4, $C_6\wr C_2$ x 4, $C_2\times C_6^2$ x 3, $C_3:S_4$ x 2, $Q_8\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $C_3\times D_{12}$ x 2, $C_3:C_{24}$ x 2, $C_6\times D_6$, $C_6:C_{12}$

Order 64: $D_4:D_4$

Order 54: $C_3^2:C_6$ x 2, $C_3^2\times C_6$

Order 48: $C_6\times D_4$ x 10, $D_4:C_6$ x 6, $\GL(2,3)$ x 6, $C_2\times S_4$ x 3, $C_6:D_4$ x 2, $C_3\times \SD_{16}$ x 2, $C_3\times D_8$ x 2, $C_3\times \OD_{16}$ x 2, $Q_8:S_3$ x 2, $C_3:D_8$ x 2, $C_3:\OD_{16}$ x 2, $C_2\times D_{12}$, $C_4\times C_{12}$, $C_{12}:C_4$

Order 36: $C_3\times A_4$ x 20, $C_6\times S_3$ x 6, $C_6^2$ x 5, $C_3\times C_{12}$ x 2, $C_3:C_{12}$ x 2, $C_6:S_3$

Order 32: $D_8:C_2$ x 2, $C_4\wr C_2$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_4:D_4$

Order 27: $C_3^3$

Order 24: $C_3\times D_4$ x 20, $C_2\times A_4$ x 15, $\SL(2,3)$ x 12, $C_2^2\times C_6$ x 11, $C_2\times C_{12}$ x 8, $S_4$ x 6, $C_3\times Q_8$ x 6, $C_3:D_4$ x 4, $D_{12}$ x 2, $C_{24}$ x 2, $C_3:C_8$ x 2, $C_6:C_4$, $C_2\times D_6$

Order 18: $C_3\times C_6$ x 11, $C_3\times S_3$ x 7, $C_3:S_3$ x 2

Order 16: $C_2\times D_4$ x 5, $\SD_{16}$ x 2, $D_8$ x 2, $\OD_{16}$ x 2, $D_4:C_2$ x 2, $C_4^2$

Order 12: $C_2\times C_6$ x 20, $A_4$ x 15, $C_{12}$ x 8, $D_6$ x 6, $C_3:C_4$ x 2

Order 9: $C_3^2$ x 9

Order 8: $D_4$ x 10, $C_2^3$ x 4, $C_2\times C_4$ x 3, $Q_8$ x 2, $C_8$ x 2

Order 6: $C_6$ x 17, $S_3$ x 7

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1728: $(Q_8\times C_3^2):S_4$

Order 864: $C_3^2\times Q_8:A_4$

Order 576: $(C_3\times Q_8):S_4$, $C_3\times Q_8:S_4$, $(C_6\times D_{12}):C_2^2$

Order 432: $C_6^2:D_6$, $C_3^2:\GL(2,3)$

Order 288: $C_3\times Q_8:A_4$ x 4, $C_6^2.D_4$, $C_6^2.C_2^3$, $C_6^2.D_4$, $C_3\times C_{12}.D_4$, $D_4:C_6^2$

Order 216: $C_6^2:C_6$ x 2, $C_3^2:S_4$, $C_3^2\times \SL(2,3)$

Order 192: $C_3\times D_4:D_4$, $(C_3\times D_4):D_4$, $Q_8:S_4$

Order 144: $C_4:C_6^2$ x 2, $C_6^2:C_2^2$ x 2, $C_{12}.D_6$, $D_{12}:C_6$, $C_{12}:C_{12}$, $C_{12}.C_{12}$, $C_6:S_4$, $C_6\times S_4$, $C_4.C_6^2$, $C_6\times D_{12}$, $C_3:\GL(2,3)$, $C_3\times \GL(2,3)$

Order 108: $C_3^2\times A_4$ x 2, $C_3^2:D_6$

Order 96: $C_6.C_2^4$ x 3, $Q_8:A_4$ x 2, $D_4:C_{12}$, $\OD_{16}:C_6$, $C_{12}.D_4$, $C_{12}.D_4$, $D_8:C_6$, $C_{12}:D_4$, $D_4:D_6$, $C_{12}:D_4$

Order 72: $C_6\times A_4$ x 8, $C_3\times \SL(2,3)$ x 4, $C_2\times C_6^2$ x 3, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $C_6\wr C_2$ x 2, $C_6\times D_6$, $C_3:S_4$, $C_3\times S_4$, $Q_8\times C_3^2$, $C_6:C_{12}$, $C_3\times D_{12}$, $C_3:C_{24}$

Order 64: $D_4:D_4$

Order 54: $C_3^2\times C_6$, $C_3^2:C_6$

Order 48: $C_6\times D_4$ x 9, $D_4:C_6$ x 3, $C_6:D_4$ x 2, $C_2\times S_4$, $C_2\times D_{12}$, $\GL(2,3)$, $C_3\times \SD_{16}$, $C_3\times D_8$, $C_3\times \OD_{16}$, $C_4\times C_{12}$, $Q_8:S_3$, $C_3:D_8$, $C_{12}:C_4$, $C_3:\OD_{16}$

Order 36: $C_3\times A_4$ x 8, $C_6^2$ x 5, $C_6\times S_3$ x 4, $C_3\times C_{12}$, $C_3:C_{12}$, $C_6:S_3$

Order 32: $\OD_{16}:C_2$, $D_4:C_2^2$, $D_8:C_2$, $C_4:D_4$, $C_4\wr C_2$

Order 27: $C_3^3$

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

Order 18: $C_3\times C_6$ x 7, $C_3\times S_3$ x 2, $C_3:S_3$

Order 16: $C_2\times D_4$ x 5, $\SD_{16}$, $D_8$, $\OD_{16}$, $C_4^2$, $D_4:C_2$

Order 12: $C_2\times C_6$ x 18, $D_6$ x 4, $A_4$ x 4, $C_{12}$ x 4, $C_3:C_4$

Order 9: $C_3^2$ x 5

Order 8: $D_4$ x 5, $C_2^3$ x 4, $C_2\times C_4$ x 3, $Q_8$, $C_8$

Order 6: $C_6$ x 12, $S_3$ x 2

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1728: $(Q_8\times C_3^2):S_4$ ($C_1$)

Order 864: $C_3^2\times Q_8:A_4$ ($C_2$)

Order 576: $(C_3\times Q_8):S_4$ ($C_3$)

Order 288: $C_3\times Q_8:A_4$ ($S_3$) x 3, $C_3\times Q_8:A_4$ ($C_6$), $D_4:C_6^2$ ($S_3$)

Order 96: $Q_8:A_4$ ($C_3\times S_3$) x 3, $C_6.C_2^4$ ($C_3:S_3$), $C_6.C_2^4$ ($C_3\times S_3$)

Order 72: $Q_8\times C_3^2$ ($S_4$) x 2, $C_2\times C_6^2$ ($S_4$)

Order 32: $D_4:C_2^2$ ($C_3^2:C_6$)

Order 24: $C_3\times Q_8$ ($C_3:S_4$) x 2, $C_3\times Q_8$ ($C_3\times S_4$) x 2, $C_2^2\times C_6$ ($C_3:S_4$), $C_2^2\times C_6$ ($C_3\times S_4$)

Order 18: $C_3\times C_6$ ($C_2^2:S_4$)

Order 9: $C_3^2$ ($Q_8:S_4$)

Order 8: $Q_8$ ($C_3^2:S_4$) x 2, $C_2^3$ ($C_3^2:S_4$)

Order 6: $C_6$ ($(C_2\times C_6):S_4$), $C_6$ ($C_3\times C_2^2:S_4$)

Order 3: $C_3$ ($(C_3\times Q_8):S_4$), $C_3$ ($C_3\times Q_8:S_4$)

Order 2: $C_2$ ($C_6^2:S_4$)

Order 1: $C_1$ ($(Q_8\times C_3^2):S_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1728: $(Q_8\times C_3^2):S_4$ ($C_1$)

Order 864: $C_3^2\times Q_8:A_4$ ($C_2$)

Order 576: $(C_3\times Q_8):S_4$ ($C_3$)

Order 288: $C_3\times Q_8:A_4$ ($C_6$), $C_3\times Q_8:A_4$ ($S_3$), $D_4:C_6^2$ ($S_3$)

Order 96: $C_6.C_2^4$ ($C_3:S_3$), $C_6.C_2^4$ ($C_3\times S_3$), $Q_8:A_4$ ($C_3\times S_3$)

Order 72: $C_2\times C_6^2$ ($S_4$), $Q_8\times C_3^2$ ($S_4$)

Order 32: $D_4:C_2^2$ ($C_3^2:C_6$)

Order 24: $C_2^2\times C_6$ ($C_3:S_4$), $C_2^2\times C_6$ ($C_3\times S_4$), $C_3\times Q_8$ ($C_3:S_4$), $C_3\times Q_8$ ($C_3\times S_4$)

Order 18: $C_3\times C_6$ ($C_2^2:S_4$)

Order 9: $C_3^2$ ($Q_8:S_4$)

Order 8: $C_2^3$ ($C_3^2:S_4$), $Q_8$ ($C_3^2:S_4$)

Order 6: $C_6$ ($(C_2\times C_6):S_4$), $C_6$ ($C_3\times C_2^2:S_4$)

Order 3: $C_3$ ($(C_3\times Q_8):S_4$), $C_3$ ($C_3\times Q_8:S_4$)

Order 2: $C_2$ ($C_6^2:S_4$)

Order 1: $C_1$ ($(Q_8\times C_3^2):S_4$)

Series

Derived series

$(Q_8\times C_3^2):S_4$

$\rhd$

$C_3\times Q_8:A_4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

$(Q_8\times C_3^2):S_4$

$\rhd$

$C_3^2\times Q_8:A_4$

$\rhd$

$C_3\times Q_8:A_4$

$\rhd$

$C_6.C_2^4$

$\rhd$

$D_4:C_2^2$

$\rhd$

$C_2^3$

$\rhd$

$Q_8$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(Q_8\times C_3^2):S_4$

$\rhd$

$C_3\times Q_8:A_4$

Upper central series

$C_1$

$\lhd$

$C_6$

Supergroups

This group is a maximal subgroup of 9 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 $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $47 \times 47$ rational character table.