Abstract group 1296.3015: $C_6^3.C_6$

• Label: 1296.3015
• Order: \( 2^{4} \cdot 3^{4} \)
• Exponent: \( 2 \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \cdot 3^{2} \)
• $\card{Z(G)}$: \( 2 \cdot 3 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 3^{2} \)
• Perm deg.: $18$
• Trans deg.: $36$
• Rank: $2$

Group information

Description:	$C_6^3.C_6$
Order:	\(1296\)\(\medspace = 2^{4} \cdot 3^{4} \)
Exponent:	\(18\)\(\medspace = 2 \cdot 3^{2} \)
Automorphism group:	$C_6^2.C_3^3.C_2^3$, of order \(7776\)\(\medspace = 2^{5} \cdot 3^{5} \)
Composition factors:	$C_2$ x 4, $C_3$ x 4
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	9	18
Elements	1	31	26	374	216	648	1296
Conjugacy classes	1	7	9	67	12	24	120
Divisions	1	7	5	35	6	12	66
Autjugacy classes	1	5	7	27	2	3	45

Dimension	1	2	3	4	6	12
Irr. complex chars.	36	18	44	0	22	0	120
Irr. rational chars.	4	18	4	8	22	10	66

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$36$
Rank:	$2$
Inequivalent generating pairs:	$24$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	12	12
Arbitrary	5	8	8

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{18}=c^{2}=d^{6}=[a,c]=[a,d]=[c,d]=1, b^{a}=b^{7}, c^{b}=d^{3}, d^{b}=cd^{5} \rangle$
Permutation group:	Degree $18$ $\langle(6,7), (17,18), (2,4,3)(8,9,11,10,13,15,14,12,16), (9,12,13)(11,15,16), (8,10,14)(9,13,12)(11,15,16), (1,2)(3,4), (1,3)(2,4), (5,6,7)\rangle$
Transitive group:	36T1966				more information
Direct product:	$C_2$ $\, \times\, $ $S_3$ $\, \times\, $ $(C_3^2.A_4)$
Semidirect product:	$C_3$ $\,\rtimes\,$ $(C_6^2.A_4)$	$(C_6^3.C_3)$ $\,\rtimes\,$ $C_2$	$(C_6^2.C_6)$ $\,\rtimes\,$ $C_6$ (3)	$(C_6^2.C_6)$ $\,\rtimes\,$ $C_6$ (6)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_6^3$ . $C_6$	$C_6^2$ . $C_6^2$	$(S_3\times C_6^2)$ . $C_6$ (2)	$C_6^2$ . $(C_6\times S_3)$	all 24

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

Homology

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

Subgroups

There are 2132 subgroups in 464 conjugacy classes, 70 normal (35 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_6$	$G/Z \simeq$ $C_6^2:C_6$
Commutator:	$G' \simeq$ $C_6^2$	$G/G' \simeq$ $C_6^2$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_6\times S_3\times A_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_6^3$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_6^3.C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6^2$	$G/\operatorname{soc} \simeq$ $C_3\times C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9:C_3^2$

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 1296: $C_6^3.C_6$

Order 648: $S_3\times C_3^2.A_4$ x 2, $C_6^3.C_3$

Order 432: $C_2^2:C_9\times D_6$ x 3, $D_6\times C_6^2$, $C_6^2.A_4$

Order 324: $C_3^2.C_6^2$, $C_3^3.A_4$

Order 216: $C_6^2.C_6$ x 7, $C_6^2.C_6$ x 6, $S_3\times C_6^2$ x 6, $C_6^2.C_6$ x 3, $C_6^3$

Order 162: $C_3^3.C_6$ x 2, $C_{18}:C_3^2$

Order 144: $C_2^3:C_{18}$ x 3, $C_6^2:C_2^2$ x 2, $C_2^2\times C_6^2$

Order 108: $C_3^2\times D_6$ x 10, $C_3^2.A_4$ x 5, $C_3\times C_6^2$ x 3, $S_3\times C_{18}$ x 3, $C_3^2.A_4$ x 3, $C_{18}:C_6$

Order 81: $C_9:C_3^2$

Order 72: $C_6\times D_6$ x 20, $C_2^2:C_{18}$ x 12, $C_2\times C_6^2$ x 11

Order 54: $C_9:C_6$ x 7, $S_3\times C_9$ x 6, $S_3\times C_3^2$ x 4, $C_3\times C_{18}$ x 3, $C_3^2\times C_6$ x 3

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$

Order 36: $C_6\times S_3$ x 38, $C_6^2$ x 33, $C_2^2:C_9$ x 6, $C_2\times C_{18}$ x 3

Order 27: $C_9:C_3$ x 5, $C_3\times C_9$ x 3, $C_3^3$

Order 24: $C_2^2\times C_6$ x 25, $C_2\times D_6$ x 6

Order 18: $C_3\times C_6$ x 27, $C_3\times S_3$ x 12, $C_{18}$ x 12

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 61, $D_6$ x 10

Order 9: $C_9$ x 6, $C_3^2$ x 5

Order 8: $C_2^3$ x 7

Order 6: $C_6$ x 35, $S_3$ x 4

Order 4: $C_2^2$ x 13

Order 3: $C_3$ x 5

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1296: $C_6^3.C_6$

Order 648: $C_6^3.C_3$, $S_3\times C_3^2.A_4$

Order 432: $D_6\times C_6^2$, $C_6^2.A_4$, $C_2^2:C_9\times D_6$

Order 324: $C_3^2.C_6^2$, $C_3^3.A_4$

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

Order 162: $C_{18}:C_3^2$, $C_3^3.C_6$

Order 144: $C_6^2:C_2^2$ x 2, $C_2^2\times C_6^2$, $C_2^3:C_{18}$

Order 108: $C_3^2\times D_6$ x 6, $C_3\times C_6^2$ x 3, $C_3^2.A_4$ x 3, $C_{18}:C_6$, $S_3\times C_{18}$, $C_3^2.A_4$

Order 81: $C_9:C_3^2$

Order 72: $C_2\times C_6^2$ x 9, $C_6\times D_6$ x 8, $C_2^2:C_{18}$ x 3

Order 54: $C_9:C_6$ x 4, $C_3^2\times C_6$ x 3, $S_3\times C_3^2$ x 2, $C_3\times C_{18}$, $S_3\times C_9$

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$

Order 36: $C_6^2$ x 21, $C_6\times S_3$ x 12, $C_2^2:C_9$ x 2, $C_2\times C_{18}$

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

Order 24: $C_2^2\times C_6$ x 13, $C_2\times D_6$ x 4

Order 18: $C_3\times C_6$ x 17, $C_3\times S_3$ x 4, $C_{18}$ x 3

Order 16: $C_2^4$

Order 12: $C_2\times C_6$ x 27, $D_6$ x 6

Order 9: $C_3^2$ x 5, $C_9$ x 2

Order 8: $C_2^3$ x 5

Order 6: $C_6$ x 19, $S_3$ x 2

Order 4: $C_2^2$ x 9

Order 3: $C_3$ x 5

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1296: $C_6^3.C_6$ ($C_1$)

Order 648: $S_3\times C_3^2.A_4$ ($C_2$) x 2, $C_6^3.C_3$ ($C_2$)

Order 432: $C_2^2:C_9\times D_6$ ($C_3$) x 3, $D_6\times C_6^2$ ($C_3$)

Order 324: $C_3^3.A_4$ ($C_2^2$)

Order 216: $C_6^2.C_6$ ($C_6$) x 6, $C_6^2.C_6$ ($C_6$) x 3, $S_3\times C_6^2$ ($C_6$) x 2, $C_6^3$ ($C_6$), $C_6^2.C_6$ ($S_3$)

Order 144: $C_6^2:C_2^2$ ($C_3^2$)

Order 108: $C_3^2.A_4$ ($C_2\times C_6$) x 3, $C_3\times C_6^2$ ($C_2\times C_6$), $C_3^2\times D_6$ ($A_4$), $C_3^2.A_4$ ($D_6$)

Order 72: $C_2^2:C_{18}$ ($C_3\times S_3$) x 3, $C_6\times D_6$ ($C_3\times C_6$) x 2, $C_2\times C_6^2$ ($C_3\times C_6$), $C_2\times C_6^2$ ($C_3\times S_3$)

Order 54: $S_3\times C_3^2$ ($C_2\times A_4$) x 2, $C_3^2\times C_6$ ($C_2\times A_4$)

Order 48: $C_2^2\times D_6$ ($C_9:C_3$)

Order 36: $C_2^2:C_9$ ($C_6\times S_3$) x 3, $C_6^2$ ($C_6^2$), $C_6^2$ ($C_6\times S_3$), $C_6\times S_3$ ($C_3\times A_4$)

Order 27: $C_3^3$ ($C_2^2\times A_4$)

Order 24: $C_2\times D_6$ ($C_9:C_6$) x 2, $C_2^2\times C_6$ ($S_3\times C_3^2$), $C_2^2\times C_6$ ($C_9:C_6$)

Order 18: $C_3\times S_3$ ($C_6\times A_4$) x 2, $C_3\times C_6$ ($C_6\times A_4$), $C_3\times C_6$ ($S_3\times A_4$)

Order 12: $C_2\times C_6$ ($C_3^2\times D_6$), $C_2\times C_6$ ($C_{18}:C_6$), $D_6$ ($C_3^2.A_4$)

Order 9: $C_3^2$ ($C_2^2:C_6^2$), $C_3^2$ ($A_4\times D_6$)

Order 8: $C_2^3$ ($C_3^3.C_6$)

Order 6: $S_3$ ($C_6^2.C_6$) x 2, $C_6$ ($C_6^2:C_6$), $C_6$ ($C_6^2.C_6$)

Order 4: $C_2^2$ ($C_3^2.C_6^2$)

Order 3: $C_3$ ($C_6\times S_3\times A_4$), $C_3$ ($C_6^2.A_4$)

Order 2: $C_2$ ($S_3\times C_3^2.A_4$)

Order 1: $C_1$ ($C_6^3.C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1296: $C_6^3.C_6$ ($C_1$)

Order 648: $C_6^3.C_3$ ($C_2$), $S_3\times C_3^2.A_4$ ($C_2$)

Order 432: $D_6\times C_6^2$ ($C_3$), $C_2^2:C_9\times D_6$ ($C_3$)

Order 324: $C_3^3.A_4$ ($C_2^2$)

Order 216: $C_6^2.C_6$ ($C_6$), $C_6^3$ ($C_6$), $S_3\times C_6^2$ ($C_6$), $C_6^2.C_6$ ($S_3$), $C_6^2.C_6$ ($C_6$)

Order 144: $C_6^2:C_2^2$ ($C_3^2$)

Order 108: $C_3\times C_6^2$ ($C_2\times C_6$), $C_3^2\times D_6$ ($A_4$), $C_3^2.A_4$ ($D_6$), $C_3^2.A_4$ ($C_2\times C_6$)

Order 72: $C_2\times C_6^2$ ($C_3\times C_6$), $C_2\times C_6^2$ ($C_3\times S_3$), $C_6\times D_6$ ($C_3\times C_6$), $C_2^2:C_{18}$ ($C_3\times S_3$)

Order 54: $C_3^2\times C_6$ ($C_2\times A_4$), $S_3\times C_3^2$ ($C_2\times A_4$)

Order 48: $C_2^2\times D_6$ ($C_9:C_3$)

Order 36: $C_2^2:C_9$ ($C_6\times S_3$), $C_6^2$ ($C_6^2$), $C_6^2$ ($C_6\times S_3$), $C_6\times S_3$ ($C_3\times A_4$)

Order 27: $C_3^3$ ($C_2^2\times A_4$)

Order 24: $C_2^2\times C_6$ ($S_3\times C_3^2$), $C_2^2\times C_6$ ($C_9:C_6$), $C_2\times D_6$ ($C_9:C_6$)

Order 18: $C_3\times C_6$ ($C_6\times A_4$), $C_3\times C_6$ ($S_3\times A_4$), $C_3\times S_3$ ($C_6\times A_4$)

Order 12: $C_2\times C_6$ ($C_3^2\times D_6$), $C_2\times C_6$ ($C_{18}:C_6$), $D_6$ ($C_3^2.A_4$)

Order 9: $C_3^2$ ($C_2^2:C_6^2$), $C_3^2$ ($A_4\times D_6$)

Order 8: $C_2^3$ ($C_3^3.C_6$)

Order 6: $C_6$ ($C_6^2:C_6$), $C_6$ ($C_6^2.C_6$), $S_3$ ($C_6^2.C_6$)

Order 4: $C_2^2$ ($C_3^2.C_6^2$)

Order 3: $C_3$ ($C_6\times S_3\times A_4$), $C_3$ ($C_6^2.A_4$)

Order 2: $C_2$ ($S_3\times C_3^2.A_4$)

Order 1: $C_1$ ($C_6^3.C_6$)

Series

Derived series

$C_6^3.C_6$

$\rhd$

$C_6^2$

$\rhd$

$C_1$

Chief series

$C_6^3.C_6$

$\rhd$

$S_3\times C_3^2.A_4$

$\rhd$

$C_6^2.C_6$

$\rhd$

$C_3^2.A_4$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_6^3.C_6$

$\rhd$

$C_6^2$

$\rhd$

$C_2\times C_6$

Upper central series

$C_1$

$\lhd$

$C_6$

$\lhd$

$C_3\times C_6$

Supergroups

This group is a maximal subgroup of 8 larger groups in the database.

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

Character theory

Complex character table

See the $120 \times 120$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $66 \times 66$ rational character table.