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## G = C3×C32⋊7D4order 216 = 23·33

### Direct product of C3 and C32⋊7D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C32⋊7D4
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C3×C32⋊7D4
 Lower central C32 — C3×C6 — C3×C32⋊7D4
 Upper central C1 — C6 — C2×C6

Generators and relations for C3×C327D4
G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 360 in 136 conjugacy classes, 42 normal (18 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22, C22, S3 [×4], C6, C6 [×4], C6 [×18], D4, C32, C32 [×4], C32 [×4], Dic3 [×4], C12, D6 [×4], C2×C6, C2×C6 [×4], C2×C6 [×5], C3×S3 [×4], C3⋊S3, C3×C6, C3×C6 [×4], C3×C6 [×17], C3⋊D4 [×4], C3×D4, C33, C3×Dic3 [×4], C3⋊Dic3, S3×C6 [×4], C2×C3⋊S3, C62, C62 [×4], C62 [×4], C3×C3⋊S3, C32×C6, C32×C6, C3×C3⋊D4 [×4], C327D4, C3×C3⋊Dic3, C6×C3⋊S3, C3×C62, C3×C327D4
Quotients: C1, C2 [×3], C3, C22, S3 [×4], C6 [×3], D4, D6 [×4], C2×C6, C3×S3 [×4], C3⋊S3, C3⋊D4 [×4], C3×D4, S3×C6 [×4], C2×C3⋊S3, C3×C3⋊S3, C3×C3⋊D4 [×4], C327D4, C6×C3⋊S3, C3×C327D4

Smallest permutation representation of C3×C327D4
On 36 points
Generators in S36
(1 27 9)(2 28 10)(3 25 11)(4 26 12)(5 23 20)(6 24 17)(7 21 18)(8 22 19)(13 34 30)(14 35 31)(15 36 32)(16 33 29)
(1 16 7)(2 8 13)(3 14 5)(4 6 15)(9 29 18)(10 19 30)(11 31 20)(12 17 32)(21 27 33)(22 34 28)(23 25 35)(24 36 26)
(1 21 29)(2 30 22)(3 23 31)(4 32 24)(5 35 11)(6 12 36)(7 33 9)(8 10 34)(13 19 28)(14 25 20)(15 17 26)(16 27 18)
(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 12)(10 11)(13 14)(15 16)(17 18)(19 20)(21 24)(22 23)(25 28)(26 27)(29 32)(30 31)(33 36)(34 35)

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

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

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

C3×C327D4 is a maximal subgroup of   C3×S3×C3⋊D4  C62.91D6  C62.93D6  C6223D6  C62.96D6  C6224D6  C3×D4×C3⋊S3

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C ··· 3N 4 6A 6B 6C ··· 6AN 6AO 6AP 12A 12B order 1 2 2 2 3 3 3 ··· 3 4 6 6 6 ··· 6 6 6 12 12 size 1 1 2 18 1 1 2 ··· 2 18 1 1 2 ··· 2 18 18 18 18

63 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 C3×C32⋊7D4 C3×C3⋊Dic3 C6×C3⋊S3 C3×C62 C32⋊7D4 C3⋊Dic3 C2×C3⋊S3 C62 C62 C33 C3×C6 C2×C6 C32 C32 C6 C3 # reps 1 1 1 1 2 2 2 2 4 1 4 8 8 2 8 16

Matrix representation of C3×C327D4 in GL4(𝔽13) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 0 3 0 0 0 0 3 0 0 0 12 9
,
 1 0 0 0 0 1 0 0 0 0 9 0 0 0 1 3
,
 0 1 0 0 12 0 0 0 0 0 8 4 0 0 7 5
,
 0 1 0 0 1 0 0 0 0 0 8 4 0 0 7 5
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,3,0,0,0,0,3,12,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,9,1,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,8,7,0,0,4,5],[0,1,0,0,1,0,0,0,0,0,8,7,0,0,4,5] >;

C3×C327D4 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_7D_4
% in TeX

G:=Group("C3xC3^2:7D4");
// GroupNames label

G:=SmallGroup(216,144);
// by ID

G=gap.SmallGroup(216,144);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-3,-3,169,1444,5189]);
// 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,d*b*d^-1=e*b*e=b^-1,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
// generators/relations

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