Abstract group 2916.ld: $C_3^5:D_6$

• Label: 2916.ld
• Order: \( 2^{2} \cdot 3^{6} \)
• Exponent: \( 2 \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{10} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{11} \cdot 3^{4} \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$C_3^5:D_6$
Order:	\(2916\)\(\medspace = 2^{2} \cdot 3^{6} \)
Exponent:	\(6\)\(\medspace = 2 \cdot 3 \)
Automorphism group:	$(C_3:S_3)^3.S_4\wr C_3.C_2$, of order \(483729408\)\(\medspace = 2^{13} \cdot 3^{10} \)
Composition factors:	$C_2$ x 2, $C_3$ x 6
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), metabelian, and an A-group.

Group statistics

Order	1	2	3	6
Elements	1	243	728	1944	2916
Conjugacy classes	1	3	188	12	204
Divisions	1	3	124	12	140
Autjugacy classes	1	1	3	1	6

Dimension	1	2	4	8
Irr. complex chars.	4	24	176	0	204
Irr. rational chars.	4	24	48	64	140

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$7$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{6}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}=[a,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $18$ $\langle(2,3,5)(7,9,11)(8,10,12)(14,15,17), (2,3)(4,6)(7,8,9,10,11,12)(13,14)(15,16)(17,18), (1,2,4,3,6,5)(9,11)(10,12)(13,14)(15,18)(16,17)\rangle$
Transitive group:	36T4240				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_3^5$ $\,\rtimes\,$ $D_6$	$C_3^4$ $\,\rtimes\,$ $S_3^2$	$C_3^6$ $\,\rtimes\,$ $C_2^2$	$(C_3^4:C_6)$ $\,\rtimes\,$ $S_3$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Aut. group:	$\Aut(C_{12}^2.D_6)$	$\Aut(D_6^2.D_6)$	$\Aut(C_2^9.(C_4\times D_{10}))$	$\Aut(C_4^4.\PSOPlus(4,3))$

Elements of the group are displayed as permutations of degree 18.

Homology

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

Subgroups

There are 111416 subgroups in 17560 conjugacy classes, 235 normal (3 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^5:D_6$
Commutator:	$G' \simeq$ $C_3^6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^5:D_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_3^5:D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6$

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

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Subgroup information

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 2916: $C_3^5:D_6$

Order 1458: $C_3^5:C_6$ x 3

Order 972: $C_3^3:S_3^2$ x 12

Order 729: $C_3^6$

Order 486: $C_3^4:C_6$ x 72, $C_3^4:C_6$ x 12

Order 324: $C_3^2:S_3^2$ x 48, $C_3^2:S_3^2$ x 3

Order 243: $C_3^5$ x 124

Order 162: $C_3^3:C_6$ x 288, $C_3^2\wr C_2$ x 222, $C_3^3:S_3$ x 3

Order 108: $C_3:S_3^2$ x 64, $C_3:S_3^2$ x 24

Order 81: $C_3^4$ x 2971

Order 54: $C_3^2:C_6$ x 888, $C_3^2:S_3$ x 72, $S_3\times C_3^2$ x 72

Order 36: $S_3^2$ x 48, $C_6:S_3$ x 3

Order 27: $C_3^3$ x 8920

Order 18: $C_3\times S_3$ x 288, $C_3:S_3$ x 222, $C_3\times C_6$ x 3

Order 12: $D_6$ x 12

Order 9: $C_3^2$ x 2971

Order 6: $S_3$ x 72, $C_6$ x 12

Order 4: $C_2^2$

Order 3: $C_3$ x 124

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 2916: $C_3^5:D_6$

Order 1458: $C_3^5:C_6$

Order 972: $C_3^3:S_3^2$

Order 729: $C_3^6$

Order 486: $C_3^4:C_6$ x 2, $C_3^4:C_6$

Order 324: $C_3^2:S_3^2$, $C_3^2:S_3^2$

Order 243: $C_3^5$ x 3

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

Order 108: $C_3:S_3^2$, $C_3:S_3^2$

Order 81: $C_3^4$ x 10

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

Order 36: $C_6:S_3$, $S_3^2$

Order 27: $C_3^3$ x 13

Order 18: $C_3:S_3$ x 4, $C_3\times S_3$ x 2, $C_3\times C_6$

Order 12: $D_6$

Order 9: $C_3^2$ x 10

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

Order 4: $C_2^2$

Order 3: $C_3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2916: $C_3^5:D_6$ ($C_1$)

Order 1458: $C_3^5:C_6$ ($C_2$) x 3

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

Order 486: $C_3^4:C_6$ ($S_3$) x 12

Order 243: $C_3^5$ ($D_6$) x 12

Order 162: $C_3^3:S_3$ ($C_3:S_3$) x 3

Order 81: $C_3^4$ ($S_3^2$) x 48, $C_3^4$ ($C_6:S_3$) x 3

Order 27: $C_3^3$ ($C_3:S_3^2$) x 64, $C_3^3$ ($C_3:S_3^2$) x 24

Order 9: $C_3^2$ ($C_3^2:S_3^2$) x 48, $C_3^2$ ($C_3^2:S_3^2$) x 3

Order 3: $C_3$ ($C_3^3:S_3^2$) x 12

Order 1: $C_1$ ($C_3^5:D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2916: $C_3^5:D_6$ ($C_1$)

Order 1458: $C_3^5:C_6$ ($C_2$)

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

Order 486: $C_3^4:C_6$ ($S_3$)

Order 243: $C_3^5$ ($D_6$)

Order 162: $C_3^3:S_3$ ($C_3:S_3$)

Order 81: $C_3^4$ ($C_6:S_3$), $C_3^4$ ($S_3^2$)

Order 27: $C_3^3$ ($C_3:S_3^2$), $C_3^3$ ($C_3:S_3^2$)

Order 9: $C_3^2$ ($C_3^2:S_3^2$), $C_3^2$ ($C_3^2:S_3^2$)

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

Order 1: $C_1$ ($C_3^5:D_6$)

Series

Derived series

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^5:D_6$

$\rhd$

$C_3^5:C_6$

$\rhd$

$C_3^6$

$\rhd$

$C_3^5$

$\rhd$

$C_3^4$

$\rhd$

$C_3^3$

$\rhd$

$C_3^2$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_3^5:D_6$

$\rhd$

$C_3^6$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 2 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 $204 \times 204$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $140 \times 140$ rational character table (warning: may be slow to load).