Abstract group 3888.ee: $C_3^4:(S_3\times Q_8)$

• Label: 3888.ee
• Order: \( 2^{4} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{6} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \cdot 3 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $3$

Group information

Description:	$C_3^4:(S_3\times Q_8)$
Order:	\(3888\)\(\medspace = 2^{4} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^4:(S_3\times \GL(2,3))$, of order \(23328\)\(\medspace = 2^{5} \cdot 3^{6} \)
Composition factors:	$C_2$ x 4, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	135	242	1944	594	972	3888
Conjugacy classes	1	3	5	6	3	3	21
Divisions	1	3	5	6	3	3	21
Autjugacy classes	1	3	5	2	3	1	15

Dimension	1	2	4	8	16	24	48
Irr. complex chars.	8	6	1	2	1	2	1	21
Irr. rational chars.	8	6	1	2	1	2	1	21

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$3$
Inequivalent generating triples:	$725760$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{4}=c^{3}=d^{6}=e^{3}=f^{3}=g^{3}=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,24,3,21)(2,23,5,25)(4,20,9,19)(6,26,7,22)(8,27)(10,18,13,14)(11,17,12,15) \!\cdots\! \rangle$
Transitive group:	27T529	36T4823	36T4903		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^4:Q_8)$ $\,\rtimes\,$ $S_3$	$(C_3^4:S_3)$ $\,\rtimes\,$ $Q_8$	$(C_3^4:S_3)$ $\,\rtimes\,$ $Q_8$	$C_3^4$ $\,\rtimes\,$ $(S_3\times Q_8)$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^4:C_4)$ . $D_6$ (3)	$(C_3^4:D_6)$ . $C_2^2$	$(C_3^4:C_6)$ . $C_2^3$	$(C_3^4:C_{12})$ . $C_2^2$ (3)	all 8
Aut. group:	$\Aut(C_4^3.C_2\wr S_3)$

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2} \times C_{6}$
Commutator length:	$1$

Subgroups

There are 12229 subgroups in 242 conjugacy classes, 29 normal (14 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^4:(S_3\times Q_8)$
Commutator:	$G' \simeq$ $C_3^4:C_6$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $S_3\times \PSU(3,2)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^4:C_3$	$G/\operatorname{Fit} \simeq$ $C_2\times Q_8$
Radical:	$R \simeq$ $C_3^4:(S_3\times Q_8)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^2$	$G/\operatorname{soc} \simeq$ $S_3\times \PSU(3,2)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

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

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 3888: $C_3^4:(S_3\times Q_8)$

Order 1944: $C_3^3:\PSU(3,2)$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3^3:\PSU(3,2)$

Order 1296: $C_3^4:(C_2\times Q_8)$

Order 972: $C_3^4:C_3:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^4:D_6$

Order 648: $C_3^4:Q_8$ x 4, $C_3^4:(C_2\times C_4)$ x 3

Order 486: $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $S_3\times \PSU(3,2)$

Order 324: $C_3^4:C_4$ x 6, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^4:C_3$

Order 216: $C_3^3:Q_8$ x 3, $C_3^2:C_4\times S_3$ x 3, $C_3^3:Q_8$

Order 162: $C_3^2\wr C_2$ x 2, $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$

Order 144: $C_2\times \PSU(3,2)$ x 2

Order 108: $C_3:S_3^2$ x 3, $C_3^3:C_4$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 2, $C_3^4$

Order 72: $\PSU(3,2)$ x 11, $C_2\times C_3^2:C_4$ x 6

Order 54: $C_3^2:S_3$ x 4, $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 3, $C_3^2:C_6$ x 3, $C_3^2:S_3$ x 2

Order 48: $S_3\times Q_8$

Order 36: $C_3^2:C_4$ x 24, $S_3^2$ x 5, $C_6:S_3$ x 2

Order 27: $C_3^3$ x 5, $\He_3$ x 4

Order 24: $C_4\times S_3$ x 3, $C_3:Q_8$ x 3, $C_3\times Q_8$

Order 18: $C_3:S_3$ x 17, $C_3\times S_3$ x 14, $C_3\times C_6$ x 2

Order 16: $C_2\times Q_8$

Order 12: $D_6$ x 3, $C_{12}$ x 3, $C_3:C_4$ x 3

Order 9: $C_3^2$ x 19

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

Order 6: $S_3$ x 8, $C_6$ x 3

Order 4: $C_4$ x 6, $C_2^2$

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3888: $C_3^4:(S_3\times Q_8)$

Order 1944: $C_3^3:\PSU(3,2)$, $C_3^3:\PSU(3,2)$, $C_3^4:(C_4\times S_3)$

Order 1296: $C_3^4:(C_2\times Q_8)$

Order 972: $C_3^4:D_6$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 648: $C_3^4:Q_8$ x 2, $C_3^4:(C_2\times C_4)$

Order 486: $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $S_3\times \PSU(3,2)$

Order 324: $C_3^4:C_4$ x 2, $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^4:C_3$

Order 216: $C_3^3:Q_8$, $C_3^3:Q_8$, $C_3^2:C_4\times S_3$

Order 162: $C_3^2\wr C_2$ x 2, $C_3^3:S_3$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$

Order 144: $C_2\times \PSU(3,2)$ x 2

Order 108: $C_3:S_3^2$ x 3, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:D_6$

Order 81: $C_3\times \He_3$ x 2, $C_3^4$

Order 72: $\PSU(3,2)$ x 5, $C_2\times C_3^2:C_4$ x 2

Order 54: $C_3^2:S_3$ x 4, $S_3\times C_3^2$ x 4, $C_3^2:C_6$ x 3, $C_3^2:C_6$ x 3, $C_3^2:S_3$ x 2

Order 48: $S_3\times Q_8$

Order 36: $C_3^2:C_4$ x 7, $S_3^2$ x 3, $C_6:S_3$ x 2

Order 27: $C_3^3$ x 5, $\He_3$ x 4

Order 24: $C_4\times S_3$, $C_3:Q_8$, $C_3\times Q_8$

Order 18: $C_3:S_3$ x 11, $C_3\times S_3$ x 8, $C_3\times C_6$ x 2

Order 16: $C_2\times Q_8$

Order 12: $D_6$ x 3, $C_{12}$, $C_3:C_4$

Order 9: $C_3^2$ x 11

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

Order 6: $S_3$ x 8, $C_6$ x 3

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

Order 3: $C_3$ x 5

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3888: $C_3^4:(S_3\times Q_8)$ ($C_1$)

Order 1944: $C_3^3:\PSU(3,2)$ ($C_2$) x 3, $C_3^4:(C_4\times S_3)$ ($C_2$) x 3, $C_3^3:\PSU(3,2)$ ($C_2$)

Order 972: $C_3^4:C_3:C_4$ ($C_2^2$) x 3, $C_3^4:C_{12}$ ($C_2^2$) x 3, $C_3^4:D_6$ ($C_2^2$)

Order 648: $C_3^4:Q_8$ ($S_3$)

Order 486: $C_3^4:S_3$ ($Q_8$), $C_3^4:C_6$ ($C_2^3$), $C_3^4:S_3$ ($Q_8$)

Order 324: $C_3^4:C_4$ ($D_6$) x 3

Order 243: $C_3^4:C_3$ ($C_2\times Q_8$)

Order 162: $C_3^3:S_3$ ($C_2\times D_6$)

Order 81: $C_3^4$ ($S_3\times Q_8$)

Order 54: $C_3^2:S_3$ ($\PSU(3,2)$)

Order 27: $C_3^3$ ($C_2\times \PSU(3,2)$)

Order 9: $C_3^2$ ($S_3\times \PSU(3,2)$)

Order 1: $C_1$ ($C_3^4:(S_3\times Q_8)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3888: $C_3^4:(S_3\times Q_8)$ ($C_1$)

Order 1944: $C_3^3:\PSU(3,2)$ ($C_2$), $C_3^3:\PSU(3,2)$ ($C_2$), $C_3^4:(C_4\times S_3)$ ($C_2$)

Order 972: $C_3^4:D_6$ ($C_2^2$), $C_3^4:C_3:C_4$ ($C_2^2$), $C_3^4:C_{12}$ ($C_2^2$)

Order 648: $C_3^4:Q_8$ ($S_3$)

Order 486: $C_3^4:S_3$ ($Q_8$), $C_3^4:C_6$ ($C_2^3$), $C_3^4:S_3$ ($Q_8$)

Order 324: $C_3^4:C_4$ ($D_6$)

Order 243: $C_3^4:C_3$ ($C_2\times Q_8$)

Order 162: $C_3^3:S_3$ ($C_2\times D_6$)

Order 81: $C_3^4$ ($S_3\times Q_8$)

Order 54: $C_3^2:S_3$ ($\PSU(3,2)$)

Order 27: $C_3^3$ ($C_2\times \PSU(3,2)$)

Order 9: $C_3^2$ ($S_3\times \PSU(3,2)$)

Order 1: $C_1$ ($C_3^4:(S_3\times Q_8)$)

Series

Derived series

$C_3^4:(S_3\times Q_8)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

Chief series

$C_3^4:(S_3\times Q_8)$

$\rhd$

$C_3^3:\PSU(3,2)$

$\rhd$

$C_3^4:C_{12}$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^4:(S_3\times Q_8)$

$\rhd$

$C_3^4:C_6$

$\rhd$

$C_3^4:C_3$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $21 \times 21$ rational character table. Alternatively, you may search for characters of this group with desired properties.