Abstract group 236196.lh: $C_3^6.C_3\wr C_2^2$

• Label: 236196.lh
• Order: \( 2^{2} \cdot 3^{10} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3 \)
• $\card{Z(G)}$: \( 3 \)
• $\card{\Aut(G)}$: \( 2^{4} \cdot 3^{13} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{4} \)
• Perm deg.: $36$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_3^6.C_3\wr C_2^2$
Order:	\(236196\)\(\medspace = 2^{2} \cdot 3^{10} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_3^5.C_3^5.C_6^3.C_2$, of order \(25509168\)\(\medspace = 2^{4} \cdot 3^{13} \)
Composition factors:	$C_2$ x 2, $C_3$ x 10
Derived length:	$3$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	1215	18224	70956	40824	104976	236196
Conjugacy classes	1	3	145	29	198	57	433
Divisions	1	3	81	17	38	11	151
Autjugacy classes	1	2	23	8	16	7	57

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	12	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid c^{3}=d^{9}=e^{3}=f^{3}=g^{3}=h^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,34,3,35,2,36)(4,9,17,21,29,33)(5,7,18,19,30,31)(6,8,16,20,28,32)(10,14,12,15,11,13) \!\cdots\! \rangle$
Transitive group:	36T25106				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:	$(C_3^6.C_3^3)$ . $D_6$ (2)	$(C_3^6.C_3^2)$ . $S_3^2$	$(C_3^6.C_3^2)$ . $S_3^2$	$C_3^5$ . $(C_3^3:S_3^2)$	all 19

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

Homology

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

Subgroups

There are 48 normal subgroups (22 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_3$	$G/Z \simeq$ $C_3^5.C_3^4.C_2^2$
Commutator:	$G' \simeq$ $C_3^6.C_3^3$	$G/G' \simeq$ $C_2\times C_6$
Frattini:	$\Phi \simeq$ $C_3^6$	$G/\Phi \simeq$ $C_3\wr C_2^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6.C_3^4$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^6.C_3\wr C_2^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^4$	$G/\operatorname{soc} \simeq$ $C_3^4:S_3^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6.C_3^4$

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 27 (order at least 8748) are shown, as well as all normal subgroups of any index.

Order 236196: $C_3^6.C_3\wr C_2^2$

Order 118098: $C_3^6.C_3^4.C_2$ x 2, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^5.C_3^3.C_6.C_2$ x 2, $C_3^5.C_3^3.C_6.C_2$, $C_3^5.\He_3.D_6$

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

Order 39366: $C_3^6.C_3^3.C_2$ x 4, $C_3^4.C_3^3.C_3.C_6$ x 4, $C_3^5.\He_3.C_6$ x 2, $C_3^6.C_3^3.C_2$ x 2, $C_3^5.\He_3.C_6$ x 2, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$

Order 26244: $C_3^5.C_3^3.C_2^2$ x 2, $C_3^6.S_3^2$ x 2, $C_3^5.(C_{18}\times S_3)$, $C_3^4:D_9^2$

Order 19683: $C_3^6.C_3^3$ x 5, $C_3^6.C_3^3$ x 4, $C_3^6.C_3^3$ x 4, $C_3^6.C_3^3$ x 2, $C_3^5.C_3^4$

Order 13122: $C_3^5.C_3^3.C_2$ x 4, $C_3^4.C_3^3.C_6$ x 4, $C_3^5.C_3^3.C_2$ x 4, $C_3^6.C_3.C_6$ x 4, $C_3^5.C_3^3.C_2$ x 4, $C_3^6.C_3.C_6$ x 2, $C_3^3.C_3^5.C_2$ x 2, $C_3^6.C_3.S_3$ x 2, $C_3^3.C_3^4.C_6$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^4.C_3^3.C_6$ x 2, $C_3^5.C_3^3.C_2$ x 2, $C_3^4.\He_3.C_6$ x 2, $C_3^6.C_3.S_3$, $C_3^3.C_3^4.C_6$, $C_3^6.C_3.C_6$, $C_3^4.C_3^4.C_2$, $C_3^5.C_3.C_{18}$, $C_3^4.C_3^4.C_2$

Order 8748: $C_3^5.(C_6\times S_3)$ x 2, $C_3^2.C_3^4.C_6.C_2$ x 2, $C_3^5.S_3^2$ x 2, $C_3^5.S_3^2$ x 2, $C_3^3:D_9^2$ x 2, $C_3^6.C_6.C_2$, $C_3^5:D_{18}$

Order 6561: $C_3^6.C_3^2$ x 4, $C_3^6.C_3^2$ x 2

Order 2187: $C_3\times C_3^5.C_3$ x 4

Order 729: $C_3^6$

Order 243: $C_3^5$ x 2

Order 81: $C_3^4$

Order 27: $C_3^3$ x 4

Order 9: $C_3^2$ x 6

Order 4: $C_2^2$

Order 3: $C_3$ x 4

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 27 (order at least 8748) are shown, as well as all normal subgroups of any index.

Order 236196: $C_3^6.C_3\wr C_2^2$

Order 118098: $C_3^6.C_3^4.C_2$, $C_3^6.C_3^4.C_2$

Order 78732: $C_3^5.C_3^3.C_6.C_2$, $C_3^5.C_3^3.C_6.C_2$, $C_3^5.\He_3.D_6$

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

Order 39366: $C_3^6.C_3^3.C_2$ x 2, $C_3^4.C_3^3.C_3.C_6$ x 2, $C_3^5.\He_3.C_6$, $C_3^6.C_3^3.C_2$, $C_3^5.\He_3.C_6$, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$, $C_3^6.C_3^3.C_2$, $C_3^5.C_3^3.C_6$

Order 26244: $C_3^5.(C_{18}\times S_3)$, $C_3^5.C_3^3.C_2^2$, $C_3^6.S_3^2$, $C_3^4:D_9^2$

Order 19683: $C_3^6.C_3^3$ x 4, $C_3^6.C_3^3$ x 3, $C_3^6.C_3^3$ x 2, $C_3^6.C_3^3$ x 2, $C_3^5.C_3^4$

Order 13122: $C_3^5.C_3^3.C_2$ x 2, $C_3^4.C_3^3.C_6$ x 2, $C_3^5.C_3^3.C_2$ x 2, $C_3^3.C_3^4.C_6$ x 2, $C_3^6.C_3.C_6$ x 2, $C_3^5.C_3^3.C_2$ x 2, $C_3^6.C_3.S_3$, $C_3^6.C_3.C_6$, $C_3^3.C_3^4.C_6$, $C_3^3.C_3^5.C_2$, $C_3^6.C_3.C_6$, $C_3^4.C_3^4.C_2$, $C_3^5.C_3.C_{18}$, $C_3^4.C_3^4.C_2$, $C_3^6.C_3.S_3$, $C_3^6.C_3.C_6$, $C_3^4.C_3^3.C_6$, $C_3^5.C_3^3.C_2$, $C_3^4.\He_3.C_6$

Order 8748: $C_3^5.(C_6\times S_3)$, $C_3^2.C_3^4.C_6.C_2$, $C_3^6.C_6.C_2$, $C_3^5.S_3^2$, $C_3^5.S_3^2$, $C_3^3:D_9^2$, $C_3^5:D_{18}$

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

Order 2187: $C_3\times C_3^5.C_3$ x 3

Order 729: $C_3^6$

Order 243: $C_3^5$

Order 81: $C_3^4$

Order 27: $C_3^3$ x 3

Order 9: $C_3^2$ x 4

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 236196: $C_3^6.C_3\wr C_2^2$ ($C_1$)

Order 118098: $C_3^6.C_3^4.C_2$ ($C_2$) x 2, $C_3^6.C_3^4.C_2$ ($C_2$)

Order 78732: $C_3^5.\He_3.D_6$ ($C_3$)

Order 59049: $C_3^6.C_3^4$ ($C_2^2$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_6$) x 2, $C_3^6.C_3^3.C_2$ ($S_3$) x 2, $C_3^6.C_3^3.C_2$ ($C_6$), $C_3^5.C_3^3.C_6$ ($S_3$)

Order 19683: $C_3^6.C_3^3$ ($D_6$) x 3, $C_3^6.C_3^3$ ($C_2\times C_6$)

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

Order 6561: $C_3^6.C_3^2$ ($C_6\times S_3$) x 2, $C_3^6.C_3^2$ ($S_3^2$) x 2, $C_3^6.C_3^2$ ($C_6\times S_3$), $C_3^6.C_3^2$ ($S_3^2$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3\times S_3^2$) x 3, $C_3\times C_3^5.C_3$ ($C_3:S_3^2$)

Order 729: $C_3^6$ ($C_3\wr C_2^2$)

Order 243: $C_3^5$ ($C_3^3:S_3^2$) x 2

Order 81: $C_3^4$ ($C_3^4:S_3^2$)

Order 27: $C_3^3$ ($C_3^4.C_3^3.C_2^2$) x 2, $C_3^3$ ($C_3^3.C_3^4.C_2^2$), $C_3^3$ ($C_3^3.C_3^4.C_2^2$)

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

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

Order 1: $C_1$ ($C_3^6.C_3\wr C_2^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 236196: $C_3^6.C_3\wr C_2^2$ ($C_1$)

Order 118098: $C_3^6.C_3^4.C_2$ ($C_2$), $C_3^6.C_3^4.C_2$ ($C_2$)

Order 78732: $C_3^5.\He_3.D_6$ ($C_3$)

Order 59049: $C_3^6.C_3^4$ ($C_2^2$)

Order 39366: $C_3^6.C_3^3.C_2$ ($C_6$), $C_3^6.C_3^3.C_2$ ($S_3$), $C_3^6.C_3^3.C_2$ ($C_6$), $C_3^5.C_3^3.C_6$ ($S_3$)

Order 19683: $C_3^6.C_3^3$ ($D_6$) x 2, $C_3^6.C_3^3$ ($C_2\times C_6$)

Order 13122: $C_3^6.C_3.S_3$ ($C_3\times S_3$), $C_3^6.C_3.S_3$ ($C_3\times S_3$)

Order 6561: $C_3^6.C_3^2$ ($C_6\times S_3$), $C_3^6.C_3^2$ ($S_3^2$), $C_3^6.C_3^2$ ($C_6\times S_3$), $C_3^6.C_3^2$ ($S_3^2$)

Order 2187: $C_3\times C_3^5.C_3$ ($C_3\times S_3^2$) x 2, $C_3\times C_3^5.C_3$ ($C_3:S_3^2$)

Order 729: $C_3^6$ ($C_3\wr C_2^2$)

Order 243: $C_3^5$ ($C_3^3:S_3^2$)

Order 81: $C_3^4$ ($C_3^4:S_3^2$)

Order 27: $C_3^3$ ($C_3^3.C_3^4.C_2^2$), $C_3^3$ ($C_3^3.C_3^4.C_2^2$), $C_3^3$ ($C_3^4.C_3^3.C_2^2$)

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

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

Order 1: $C_1$ ($C_3^6.C_3\wr C_2^2$)

Series

Derived series

$C_3^6.C_3\wr C_2^2$

$\rhd$

$C_3^6.C_3^3$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6.C_3\wr C_2^2$

$\rhd$

$C_3^6.C_3^4.C_2$

$\rhd$

$C_3^6.C_3^4$

$\rhd$

$C_3^6.C_3^3$

$\rhd$

$C_3^6.C_3^2$

$\rhd$

$C_3\times C_3^5.C_3$

$\rhd$

$C_3^6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^6.C_3\wr C_2^2$

$\rhd$

$C_3^6.C_3^3$

Upper central series

$C_1$

$\lhd$

$C_3$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

The $433 \times 433$ character table is not available for this group.

Rational character table

The $151 \times 151$ rational character table is not available for this group.