Abstract group 2834352.ot: $C_3^7.(C_3^2\wr C_2.D_4)$

• Label: 2834352.ot
• Order: \( 2^{4} \cdot 3^{11} \)
• Exponent: \( 2^{2} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^7.(C_3^2\wr C_2.D_4)$
Order:	\(2834352\)\(\medspace = 2^{4} \cdot 3^{11} \)
Exponent:	\(36\)\(\medspace = 2^{2} \cdot 3^{2} \)
Automorphism group:	Group of order \(102036672\)\(\medspace = 2^{6} \cdot 3^{13} \)
Composition factors:	$C_2$ x 4, $C_3$ x 11
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	18	36
Elements	1	9639	54674	157464	443070	122472	314928	787320	944784	2834352
Conjugacy classes	1	5	106	4	72	53	4	41	12	298
Divisions	1	5	72	2	48	36	2	28	2	196
Autjugacy classes	1	4	27	1	29	19	1	19	1	102

Minimal presentations

Permutation degree:	$36$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{6}=d^{9}=e^{3}=f^{9}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,17,21,12,13,28,31,24,25,6,7,35,2,16,20,10,15,30,32,23,26,4,9,36,3,18,19,11,14,29,33,22,27,5,8,34) \!\cdots\! \rangle$
Transitive group:	36T47667				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_3^6.C_3.S_3^3)$ . $S_3$	$(C_3^6.C_3^3.D_6)$ . $D_6$	$(C_3^6.C_3^5.C_2)$ . $D_4$	$C_3^6$ . $(C_3^3:S_3^2:C_4)$	all 21

Elements of the group are displayed as permutations of degree 36.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 34 normal subgroups (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^7.(C_3^2\wr C_2.D_4)$
Commutator:	$G' \simeq$ $C_3^6.C_3^5.C_2$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_3^6$	$G/\Phi \simeq$ $C_3^3:S_3^2:C_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^7.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2^2:C_4$
Radical:	$R \simeq$ $C_3^7.(C_3^2\wr C_2.D_4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3^5.S_3^2:C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2:C_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^7.C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 1 (order at least 2834352) are shown, as well as all normal subgroups of any index.

Order 2834352: $C_3^7.(C_3^2\wr C_2.D_4)$

Order 1417176: $C_3^6.C_3^4:(C_4\times S_3)$ x 2, $C_3^6.C_3^5.C_2^3$

Order 944784: $C_3^6.(C_3^2\wr C_2.D_4)$

Order 708588: $C_3^6.C_3^5.C_2^2$ x 2, $C_3^6.C_3^5.C_2^2$

Order 472392: $C_3^6.C_3.S_3^3$

Order 354294: $C_3^6.C_3^5.C_2$ x 2, $C_3^6.C_3^5.C_2$

Order 314928: $C_3^7.S_3^2:C_4$ x 2

Order 236196: $C_3^6.C_3^3.D_6$ x 2, $C_3^6.C_3^3.D_6$

Order 177147: $C_3^7.C_3^4$

Order 118098: $C_3^5.C_3.\He_3.C_6$ x 2, $C_3^6.C_3^4.C_2$

Order 59049: $C_3^6.C_3^4$

Order 39366: $C_3^6.C_3^3.C_2$ x 2

Order 19683: $C_3^7.C_3^2$ x 2

Order 13122: $C_3^6.C_3.S_3$ x 2

Order 6561: $C_3^6.C_3^2$ x 2

Order 2187: $C_3^7$

Order 729: $C_3^6$

Order 243: $C_3^5$

Order 81: $C_3^4$

Order 16: $C_2^2:C_4$

Order 9: $C_3^2$ x 2

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 1 (order at least 2834352) are shown, as well as all normal subgroups of any index.

Order 2834352: $C_3^7.(C_3^2\wr C_2.D_4)$

Order 1417176: $C_3^6.C_3^5.C_2^3$, $C_3^6.C_3^4:(C_4\times S_3)$

Order 944784: $C_3^6.(C_3^2\wr C_2.D_4)$

Order 708588: $C_3^6.C_3^5.C_2^2$ x 2, $C_3^6.C_3^5.C_2^2$

Order 472392: $C_3^6.C_3.S_3^3$

Order 354294: $C_3^6.C_3^5.C_2$, $C_3^6.C_3^5.C_2$

Order 314928: $C_3^7.S_3^2:C_4$

Order 236196: $C_3^6.C_3^3.D_6$ x 2, $C_3^6.C_3^3.D_6$

Order 177147: $C_3^7.C_3^4$

Order 118098: $C_3^5.C_3.\He_3.C_6$, $C_3^6.C_3^4.C_2$

Order 59049: $C_3^6.C_3^4$

Order 39366: $C_3^6.C_3^3.C_2$

Order 19683: $C_3^7.C_3^2$

Order 13122: $C_3^6.C_3.S_3$

Order 6561: $C_3^6.C_3^2$

Order 2187: $C_3^7$

Order 729: $C_3^6$

Order 243: $C_3^5$

Order 81: $C_3^4$

Order 16: $C_2^2:C_4$

Order 9: $C_3^2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2834352: $C_3^7.(C_3^2\wr C_2.D_4)$ ($C_1$)

Order 1417176: $C_3^6.C_3^4:(C_4\times S_3)$ ($C_2$) x 2, $C_3^6.C_3^5.C_2^3$ ($C_2$)

Order 708588: $C_3^6.C_3^5.C_2^2$ ($C_4$), $C_3^6.C_3^5.C_2^2$ ($C_2^2$), $C_3^6.C_3^5.C_2^2$ ($C_4$)

Order 472392: $C_3^6.C_3.S_3^3$ ($S_3$)

Order 354294: $C_3^6.C_3^5.C_2$ ($D_4$) x 2, $C_3^6.C_3^5.C_2$ ($C_2\times C_4$)

Order 236196: $C_3^6.C_3^3.D_6$ ($C_3:C_4$), $C_3^6.C_3^3.D_6$ ($D_6$), $C_3^6.C_3^3.D_6$ ($C_3:C_4$)

Order 177147: $C_3^7.C_3^4$ ($C_2^2:C_4$)

Order 118098: $C_3^5.C_3.\He_3.C_6$ ($C_3:D_4$) x 2, $C_3^6.C_3^4.C_2$ ($C_6:C_4$)

Order 59049: $C_3^6.C_3^4$ ($C_6.D_4$)

Order 39366: $C_3^6.C_3^3.C_2$ ($\SOPlus(4,2)$) x 2

Order 19683: $C_3^7.C_3^2$ ($S_3^2:C_4$) x 2

Order 13122: $C_3^6.C_3.S_3$ ($S_3^2:S_3$) x 2

Order 6561: $C_3^6.C_3^2$ ($(C_3\times S_3^2):C_4$) x 2

Order 2187: $C_3^7$ ($C_3^2\wr C_2.D_4$)

Order 729: $C_3^6$ ($C_3^3:S_3^2:C_4$)

Order 243: $C_3^5$ ($C_3^4.S_3^2:C_4$)

Order 81: $C_3^4$ ($C_3^5.S_3^2:C_4$)

Order 9: $C_3^2$ ($C_3^4.C_3^5.C_2^3.C_2$) x 2

Order 1: $C_1$ ($C_3^7.(C_3^2\wr C_2.D_4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2834352: $C_3^7.(C_3^2\wr C_2.D_4)$ ($C_1$)

Order 1417176: $C_3^6.C_3^5.C_2^3$ ($C_2$), $C_3^6.C_3^4:(C_4\times S_3)$ ($C_2$)

Order 708588: $C_3^6.C_3^5.C_2^2$ ($C_4$), $C_3^6.C_3^5.C_2^2$ ($C_2^2$), $C_3^6.C_3^5.C_2^2$ ($C_4$)

Order 472392: $C_3^6.C_3.S_3^3$ ($S_3$)

Order 354294: $C_3^6.C_3^5.C_2$ ($D_4$), $C_3^6.C_3^5.C_2$ ($C_2\times C_4$)

Order 236196: $C_3^6.C_3^3.D_6$ ($C_3:C_4$), $C_3^6.C_3^3.D_6$ ($D_6$), $C_3^6.C_3^3.D_6$ ($C_3:C_4$)

Order 177147: $C_3^7.C_3^4$ ($C_2^2:C_4$)

Order 118098: $C_3^5.C_3.\He_3.C_6$ ($C_3:D_4$), $C_3^6.C_3^4.C_2$ ($C_6:C_4$)

Order 59049: $C_3^6.C_3^4$ ($C_6.D_4$)

Order 39366: $C_3^6.C_3^3.C_2$ ($\SOPlus(4,2)$)

Order 19683: $C_3^7.C_3^2$ ($S_3^2:C_4$)

Order 13122: $C_3^6.C_3.S_3$ ($S_3^2:S_3$)

Order 6561: $C_3^6.C_3^2$ ($(C_3\times S_3^2):C_4$)

Order 2187: $C_3^7$ ($C_3^2\wr C_2.D_4$)

Order 729: $C_3^6$ ($C_3^3:S_3^2:C_4$)

Order 243: $C_3^5$ ($C_3^4.S_3^2:C_4$)

Order 81: $C_3^4$ ($C_3^5.S_3^2:C_4$)

Order 9: $C_3^2$ ($C_3^4.C_3^5.C_2^3.C_2$)

Order 1: $C_1$ ($C_3^7.(C_3^2\wr C_2.D_4)$)

Series

Derived series

$C_3^7.(C_3^2\wr C_2.D_4)$

$\rhd$

$C_3^6.C_3^5.C_2$

$\rhd$

$C_3^6.C_3^4$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^7.(C_3^2\wr C_2.D_4)$

$\rhd$

$C_3^6.C_3^5.C_2^3$

$\rhd$

$C_3^6.C_3^5.C_2^2$

$\rhd$

$C_3^6.C_3^5.C_2$

$\rhd$

$C_3^7.C_3^4$

$\rhd$

$C_3^6.C_3^4$

$\rhd$

$C_3^6.C_3^2$

$\rhd$

$C_3^6$

$\rhd$

$C_3^4$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_3^7.(C_3^2\wr C_2.D_4)$

$\rhd$

$C_3^6.C_3^5.C_2$

$\rhd$

$C_3^7.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 4 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $298 \times 298$ character table is not available for this group.

Rational character table

The $196 \times 196$ rational character table is not available for this group.