Abstract group $(C_3:S_3)^3.D_6$

• Label: 69984.ck
• Order: \( 2^{5} \cdot 3^{7} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{7} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$(C_3:S_3)^3.D_6$
Order:	\(69984\)\(\medspace = 2^{5} \cdot 3^{7} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	$C_3^6.(\SD_{16}\times S_4)$, of order \(279936\)\(\medspace = 2^{7} \cdot 3^{7} \) (generators)
Outer automorphisms:	$C_2^2$, of order \(4\)\(\medspace = 2^{2} \)
Composition factors:	$C_2$ x 5, $C_3$ x 7
Derived length:	$4$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	9	12
Elements	1	999	1376	25272	9936	5184	27216	69984
Conjugacy classes	1	3	14	6	10	2	10	46
Divisions	1	3	14	6	10	1	5	40
Autjugacy classes	1	3	9	4	6	1	3	27

Dimension	1	2	3	4	6	24	32	48	64	96
Irr. complex chars.	4	5	4	0	1	20	3	4	0	5	46
Irr. rational chars.	4	3	4	1	1	12	1	8	1	5	40

Minimal Presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$2$
Inequivalent generating pairs:	$3276$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid b^{12}=c^{2}=d^{6}=e^{3}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,24,12,8,23,16,5,22,15,7,26,14)(2,21,17,9,27,10,4,25,13,3,19,11)(6,20,18) \!\cdots\! \rangle$
Transitive group:	27T1247	36T17777	36T17805	36T17941	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^6.A_4)$ $\,\rtimes\,$ $Q_8$	$C_3^6$ $\,\rtimes\,$ $(A_4:Q_8)$	$(C_3^5:D_6)$ $\,\rtimes\,$ $(C_3:Q_8)$		more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^6.C_4)$ . $S_4$	$((C_3:S_3)^3)$ . $D_6$	$(C_3^5:S_3.S_4)$ . $C_2$	$(C_3^5:S_3)$ . $(C_2\times S_4)$	all 7

Elements of the group are displayed as permutations of degree 27.

Homology

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

Subgroups

There are 2596344 subgroups in 4028 conjugacy classes, 13 normal (11 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_3:S_3)^3.D_6$
Commutator:	$G' \simeq$ $C_3^6.(C_2\times A_4)$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $(C_3:S_3)^3.D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $A_4:Q_8$
Radical:	$R \simeq$ $(C_3:S_3)^3.D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $A_4:Q_8$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:Q_8$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2{\rm wrC}_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 69984: $(C_3:S_3)^3.D_6$

Order 34992: $C_3^5:S_3.S_4$ x 2, $C_3^6.(C_4\times A_4)$

Order 23328: $C_3^4.C_6^2:Q_8$

Order 17496: $C_3^5:S_3.D_6$, $C_3^6.(C_2\times A_4)$

Order 11664: $C_3^5:C_6.D_4$ x 2, $C_3^5:C_{12}:C_4$ x 2, $C_3^6.(C_2^2\times C_4)$, $C_3^6.(C_2\times Q_8)$, $C_3^6.C_4:C_4$

Order 8748: $C_3^5:S_3.S_3$ x 2, $C_3^6.A_4$, $C_3^6.C_{12}$

Order 5832: $C_3^6.(C_2\times C_4)$ x 4, $C_3^6.Q_8$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $(C_3:S_3)^3$

Order 4374: $C_3^6.C_6$

Order 2916: $C_3^6.C_4$ x 4, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4:C_6^2$

Order 2592: $C_3^2:S_3^2:Q_8$

Order 2187: $C_3^2{\rm wrC}_3$

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

Order 1458: $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $(C_3^2\times S_3^2):C_4$ x 4, $C_3^4:(C_2\times Q_8)$ x 2, $C_3^2\wr C_2.Q_8$ x 2, $C_3^2\wr C_2.D_4$ x 2, $(C_3^3\times C_6).D_4$ x 2, $C_3^3:C_{12}:C_4$ x 2, $C_2\times C_3^4:Q_8$, $C_3^4:(C_2^2\times C_4)$, $(C_3^3\times C_6).D_4$

Order 972: $C_3^3:S_3^2$ x 8, $C_3^4:C_{12}$ x 4, $C_3^4:C_{12}$ x 4, $C_3^3:C_6^2$ x 2, $C_3^4:C_3:C_4$ x 2, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_{12}$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 648: $C_3^4:(C_2\times C_4)$ x 19, $C_3^4:Q_8$ x 18, $C_3^3:(C_4\times S_3)$ x 12, $C_3:S_3^3$ x 6, $C_2\times C_3^4:C_4$ x 4, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:C_2^3$, $S_3\wr C_3$

Order 486: $C_3^4:S_3$ x 13, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 8, $C_3^4:C_6$ x 3, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:C_6$

Order 324: $C_3^4:C_4$ x 94, $C_3^2:S_3^2$ x 30, $C_3^2:S_3^2$ x 24, $C_3\wr C_4$ x 16, $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 6, $C_3^2\times S_3^2$ x 6, $C_3^4:C_4$ x 2, $C_3^3:C_{12}$ x 2, $C_3^3:C_{12}$ x 2, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^3:A_4$

Order 288: $C_6^2:Q_8$

Order 243: $C_3^5$ x 13, $C_3^4:C_3$ x 3, $C_3^3:C_9$ x 3, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $S_3^3$ x 6, $C_3^2:C_4\times S_3$ x 4, $C_6.S_3^2$ x 4, $C_6:S_3^2$ x 2, $C_3^3:Q_8$

Order 162: $C_3^3:S_3$ x 286, $C_3^2\wr C_2$ x 56, $C_3^3:C_6$ x 24, $S_3\times C_3^3$ x 24, $C_3^3:C_6$ x 9, $C_3^3\times C_6$, $C_3^3:C_6$

Order 144: $S_3^2:C_4$ x 4, $C_2\times \PSU(3,2)$ x 3, $C_2.\PSU(3,2)$ x 3, $C_6^2:C_4$ x 2, $(C_3\times C_{12}):C_4$ x 2, $C_2.\SOPlus(4,2)$ x 2, $C_6^2:C_4$

Order 108: $C_3:S_3^2$ x 86, $C_3\times S_3^2$ x 16, $C_3^2:D_6$ x 8, $C_3:S_3^2$ x 6, $C_3^3:C_4$ x 6, $C_3^2:C_{12}$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 2, $C_3^2\times D_6$ x 2

Order 96: $A_4:Q_8$

Order 81: $C_3^4$ x 285, $C_3\wr C_3$ x 9, $C_3^2:C_9$ x 3, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_2\times C_3^2:C_4$ x 25, $\PSU(3,2)$ x 18, $C_6.D_6$ x 12, $S_3\times D_6$ x 6, $C_{12}:S_3$ x 2, $C_6:D_6$

Order 54: $C_3^2:S_3$ x 762, $C_3^2:C_6$ x 83, $S_3\times C_3^2$ x 82, $C_3^2\times C_6$ x 8, $C_3^2:C_6$ x 3

Order 48: $A_4:C_4$ x 2, $C_4\times A_4$

Order 36: $C_3^2:C_4$ x 96, $S_3^2$ x 36, $C_6:S_3$ x 31, $C_3:C_{12}$ x 16, $C_6\times S_3$ x 8, $C_3\times C_{12}$ x 2, $C_3^2:C_4$ x 2, $C_6^2$

Order 32: $C_2^2:Q_8$

Order 27: $C_3^3$ x 755, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3\times C_9$

Order 24: $C_4\times S_3$ x 4, $C_3:Q_8$, $C_2\times D_6$, $C_2\times A_4$

Order 18: $C_3:S_3$ x 314, $C_3\times S_3$ x 49, $C_3\times C_6$ x 29

Order 16: $C_4:C_4$ x 3, $C_2^2:C_4$ x 2, $C_2\times Q_8$, $C_2^2\times C_4$

Order 12: $D_6$ x 12, $C_3:C_4$ x 6, $C_{12}$ x 5, $C_2\times C_6$, $A_4$

Order 9: $C_3^2$ x 286, $C_9$

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 2, $C_2^3$

Order 6: $S_3$ x 22, $C_6$ x 10

Order 4: $C_4$ x 6, $C_2^2$ x 3

Order 3: $C_3$ x 14

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 69984: $(C_3:S_3)^3.D_6$

Order 34992: $C_3^5:S_3.S_4$, $C_3^6.(C_4\times A_4)$

Order 23328: $C_3^4.C_6^2:Q_8$

Order 17496: $C_3^5:S_3.D_6$, $C_3^6.(C_2\times A_4)$

Order 11664: $C_3^5:C_6.D_4$, $C_3^6.(C_2^2\times C_4)$, $C_3^5:C_{12}:C_4$, $C_3^6.(C_2\times Q_8)$, $C_3^6.C_4:C_4$

Order 8748: $C_3^6.A_4$, $C_3^5:S_3.S_3$, $C_3^6.C_{12}$

Order 5832: $C_3^6.(C_2\times C_4)$ x 3, $C_3^6.Q_8$ x 2, $(C_3:S_3)^3$, $C_3^6:(C_2\times C_4)$

Order 4374: $C_3^6.C_6$

Order 2916: $C_3^6.C_4$ x 3, $C_3^5.D_6$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:C_{12}$

Order 2592: $C_3^2:S_3^2:Q_8$

Order 2187: $C_3^2{\rm wrC}_3$

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

Order 1458: $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^4:(C_2\times Q_8)$ x 2, $C_2\times C_3^4:Q_8$, $C_3^4:(C_2^2\times C_4)$, $C_3^2\wr C_2.Q_8$, $C_3^2\wr C_2.D_4$, $(C_3^2\times S_3^2):C_4$, $(C_3^3\times C_6).D_4$, $(C_3^3\times C_6).D_4$, $C_3^3:C_{12}:C_4$

Order 972: $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:C_6^2$, $C_3^4:C_{12}$, $C_3^4:C_{12}$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 648: $C_3^4:Q_8$ x 12, $C_3^4:(C_2\times C_4)$ x 11, $C_3^3:(C_4\times S_3)$ x 4, $C_2\times C_3^4:C_4$ x 3, $C_3:S_3^3$ x 3, $C_3^4:C_2^3$, $C_3^3:(C_4\times S_3)$, $S_3\wr C_3$, $C_3^3:(C_4\times S_3)$

Order 486: $C_3^4:S_3$ x 8, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:C_6$

Order 324: $C_3^4:C_4$ x 42, $C_3^2:S_3^2$ x 17, $C_3^2:S_3^2$ x 10, $C_3\wr C_4$ x 4, $C_3^2:S_3^2$ x 3, $C_3^2:S_3^2$ x 3, $C_3^2\times S_3^2$ x 3, $C_3^3:D_6$, $C_3^2:C_6^2$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:A_4$, $C_3^3:C_{12}$

Order 288: $C_6^2:Q_8$

Order 243: $C_3^5$ x 8, $C_3^4:C_3$ x 2, $C_3^3:C_9$ x 2, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $S_3^3$ x 4, $C_6:S_3^2$, $C_3^3:Q_8$, $C_3^2:C_4\times S_3$, $C_6.S_3^2$

Order 162: $C_3^3:S_3$ x 113, $C_3^2\wr C_2$ x 30, $C_3^3:C_6$ x 10, $S_3\times C_3^3$ x 10, $C_3^3:C_6$ x 4, $C_3^3\times C_6$, $C_3^3:C_6$

Order 144: $C_2\times \PSU(3,2)$ x 3, $C_2.\PSU(3,2)$ x 2, $C_6^2:C_4$, $C_6^2:C_4$, $(C_3\times C_{12}):C_4$, $C_2.\SOPlus(4,2)$, $S_3^2:C_4$

Order 108: $C_3:S_3^2$ x 36, $C_3\times S_3^2$ x 7, $C_3^2:D_6$ x 4, $C_3:S_3^2$ x 4, $C_3^3:C_4$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$, $C_3^2\times D_6$, $C_3^2:C_{12}$, $C_3^2:C_{12}$

Order 96: $A_4:Q_8$

Order 81: $C_3^4$ x 112, $C_3\wr C_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $C_2\times C_3^2:C_4$ x 15, $\PSU(3,2)$ x 12, $C_6.D_6$ x 4, $S_3\times D_6$ x 3, $C_6:D_6$, $C_{12}:S_3$

Order 54: $C_3^2:S_3$ x 250, $C_3^2:C_6$ x 35, $S_3\times C_3^2$ x 34, $C_3^2\times C_6$ x 4, $C_3^2:C_6$ x 2

Order 48: $C_4\times A_4$, $A_4:C_4$

Order 36: $C_3^2:C_4$ x 43, $C_6:S_3$ x 18, $S_3^2$ x 16, $C_3:C_{12}$ x 4, $C_6\times S_3$ x 3, $C_3\times C_{12}$, $C_3^2:C_4$, $C_6^2$

Order 32: $C_2^2:Q_8$

Order 27: $C_3^3$ x 247, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_4\times S_3$, $C_3:Q_8$, $C_2\times D_6$, $C_2\times A_4$

Order 18: $C_3:S_3$ x 128, $C_3\times S_3$ x 21, $C_3\times C_6$ x 16

Order 16: $C_4:C_4$ x 2, $C_2^2:C_4$, $C_2\times Q_8$, $C_2^2\times C_4$

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

Order 9: $C_3^2$ x 113, $C_9$

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

Order 6: $S_3$ x 13, $C_6$ x 6

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

Order 3: $C_3$ x 9

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 69984: $(C_3:S_3)^3.D_6$ ($C_1$)

Order 34992: $C_3^5:S_3.S_4$ ($C_2$) x 2, $C_3^6.(C_4\times A_4)$ ($C_2$)

Order 17496: $C_3^6.(C_2\times A_4)$ ($C_2^2$)

Order 11664: $C_3^6.(C_2^2\times C_4)$ ($S_3$)

Order 8748: $C_3^6.A_4$ ($Q_8$)

Order 5832: $(C_3:S_3)^3$ ($D_6$)

Order 2916: $C_3^6.C_4$ ($S_4$), $C_3^5:D_6$ ($C_3:Q_8$)

Order 1458: $C_3^5:S_3$ ($C_2\times S_4$)

Order 729: $C_3^6$ ($A_4:Q_8$)

Order 1: $C_1$ ($(C_3:S_3)^3.D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 69984: $(C_3:S_3)^3.D_6$ ($C_1$)

Order 34992: $C_3^5:S_3.S_4$ ($C_2$), $C_3^6.(C_4\times A_4)$ ($C_2$)

Order 17496: $C_3^6.(C_2\times A_4)$ ($C_2^2$)

Order 11664: $C_3^6.(C_2^2\times C_4)$ ($S_3$)

Order 8748: $C_3^6.A_4$ ($Q_8$)

Order 5832: $(C_3:S_3)^3$ ($D_6$)

Order 2916: $C_3^6.C_4$ ($S_4$), $C_3^5:D_6$ ($C_3:Q_8$)

Order 1458: $C_3^5:S_3$ ($C_2\times S_4$)

Order 729: $C_3^6$ ($A_4:Q_8$)

Order 1: $C_1$ ($(C_3:S_3)^3.D_6$)

Series

Derived series

$(C_3:S_3)^3.D_6$

$\rhd$

$C_3^6.(C_2\times A_4)$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$(C_3:S_3)^3.D_6$

$\rhd$

$C_3^6.(C_4\times A_4)$

$\rhd$

$C_3^6.(C_2\times A_4)$

$\rhd$

$C_3^6.A_4$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$(C_3:S_3)^3.D_6$

$\rhd$

$C_3^6.(C_2\times A_4)$

$\rhd$

$C_3^6.A_4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $46 \times 46$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $40 \times 40$ rational character table.