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## G = C32×C3⋊D4order 216 = 23·33

### Direct product of C32 and C3⋊D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C32×C3⋊D4
 Chief series C1 — C3 — C6 — C3×C6 — C32×C6 — S3×C3×C6 — C32×C3⋊D4
 Lower central C3 — C6 — C32×C3⋊D4
 Upper central C1 — C3×C6 — C62

Generators and relations for C32×C3⋊D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 260 in 136 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, D4, C32, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C3×S3, C3×C6, C3×C6, C3×C6, C3⋊D4, C3×D4, C33, C3×Dic3, C3×C12, S3×C6, C62, C62, C62, S3×C32, C32×C6, C32×C6, C3×C3⋊D4, D4×C32, C32×Dic3, S3×C3×C6, C3×C62, C32×C3⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C32, D6, C2×C6, C3×S3, C3×C6, C3⋊D4, C3×D4, S3×C6, C62, S3×C32, C3×C3⋊D4, D4×C32, S3×C3×C6, C32×C3⋊D4

Smallest permutation representation of C32×C3⋊D4
On 36 points
Generators in S36
(1 12 20)(2 9 17)(3 10 18)(4 11 19)(5 28 16)(6 25 13)(7 26 14)(8 27 15)(21 29 35)(22 30 36)(23 31 33)(24 32 34)
(1 26 35)(2 27 36)(3 28 33)(4 25 34)(5 31 18)(6 32 19)(7 29 20)(8 30 17)(9 15 22)(10 16 23)(11 13 24)(12 14 21)
(1 29 14)(2 15 30)(3 31 16)(4 13 32)(5 10 33)(6 34 11)(7 12 35)(8 36 9)(17 27 22)(18 23 28)(19 25 24)(20 21 26)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)
(1 4)(2 3)(5 8)(6 7)(9 10)(11 12)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 26)(27 28)(29 32)(30 31)(33 36)(34 35)

G:=sub<Sym(36)| (1,12,20)(2,9,17)(3,10,18)(4,11,19)(5,28,16)(6,25,13)(7,26,14)(8,27,15)(21,29,35)(22,30,36)(23,31,33)(24,32,34), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,18)(6,32,19)(7,29,20)(8,30,17)(9,15,22)(10,16,23)(11,13,24)(12,14,21), (1,29,14)(2,15,30)(3,31,16)(4,13,32)(5,10,33)(6,34,11)(7,12,35)(8,36,9)(17,27,22)(18,23,28)(19,25,24)(20,21,26), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35)>;

G:=Group( (1,12,20)(2,9,17)(3,10,18)(4,11,19)(5,28,16)(6,25,13)(7,26,14)(8,27,15)(21,29,35)(22,30,36)(23,31,33)(24,32,34), (1,26,35)(2,27,36)(3,28,33)(4,25,34)(5,31,18)(6,32,19)(7,29,20)(8,30,17)(9,15,22)(10,16,23)(11,13,24)(12,14,21), (1,29,14)(2,15,30)(3,31,16)(4,13,32)(5,10,33)(6,34,11)(7,12,35)(8,36,9)(17,27,22)(18,23,28)(19,25,24)(20,21,26), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36), (1,4)(2,3)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,32)(30,31)(33,36)(34,35) );

G=PermutationGroup([[(1,12,20),(2,9,17),(3,10,18),(4,11,19),(5,28,16),(6,25,13),(7,26,14),(8,27,15),(21,29,35),(22,30,36),(23,31,33),(24,32,34)], [(1,26,35),(2,27,36),(3,28,33),(4,25,34),(5,31,18),(6,32,19),(7,29,20),(8,30,17),(9,15,22),(10,16,23),(11,13,24),(12,14,21)], [(1,29,14),(2,15,30),(3,31,16),(4,13,32),(5,10,33),(6,34,11),(7,12,35),(8,36,9),(17,27,22),(18,23,28),(19,25,24),(20,21,26)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36)], [(1,4),(2,3),(5,8),(6,7),(9,10),(11,12),(13,14),(15,16),(17,18),(19,20),(21,24),(22,23),(25,26),(27,28),(29,32),(30,31),(33,36),(34,35)]])

C32×C3⋊D4 is a maximal subgroup of   C62.90D6  C62.91D6  C6223D6  S3×D4×C32

81 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4 6A ··· 6H 6I ··· 6AQ 6AR ··· 6AY 12A ··· 12H order 1 2 2 2 3 ··· 3 3 ··· 3 4 6 ··· 6 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 2 6 1 ··· 1 2 ··· 2 6 1 ··· 1 2 ··· 2 6 ··· 6 6 ··· 6

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D4 D6 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×C3⋊D4 kernel C32×C3⋊D4 C32×Dic3 S3×C3×C6 C3×C62 C3×C3⋊D4 C3×Dic3 S3×C6 C62 C62 C33 C3×C6 C2×C6 C32 C32 C6 C3 # reps 1 1 1 1 8 8 8 8 1 1 1 8 2 8 8 16

Matrix representation of C32×C3⋊D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 9 0 0 0 0 9
,
 3 0 0 0 0 3 0 0 0 0 9 0 0 0 0 9
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 0 3
,
 1 11 0 0 1 12 0 0 0 0 0 1 0 0 12 0
,
 1 11 0 0 0 12 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,3],[1,1,0,0,11,12,0,0,0,0,0,12,0,0,1,0],[1,0,0,0,11,12,0,0,0,0,0,1,0,0,1,0] >;

C32×C3⋊D4 in GAP, Magma, Sage, TeX

C_3^2\times C_3\rtimes D_4
% in TeX

G:=Group("C3^2xC3:D4");
// GroupNames label

G:=SmallGroup(216,139);
// by ID

G=gap.SmallGroup(216,139);
# by ID

G:=PCGroup([6,-2,-2,-3,-3,-2,-3,457,5189]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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