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G = C338D4order 216 = 23·33

5th semidirect product of C33 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C338D4, C325D12, C6.14S32, Dic3⋊(C3⋊S3), (C3×C6).33D6, (C3×Dic3)⋊1S3, C32(C12⋊S3), C31(C3⋊D12), C3210(C3⋊D4), (C32×Dic3)⋊1C2, (C32×C6).11C22, (C6×C3⋊S3)⋊3C2, (C2×C3⋊S3)⋊5S3, C6.6(C2×C3⋊S3), C2.6(S3×C3⋊S3), (C2×C33⋊C2)⋊2C2, SmallGroup(216,129)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C338D4
C1C3C32C33C32×C6C32×Dic3 — C338D4
C33C32×C6 — C338D4
C1C2

Generators and relations for C338D4
 G = < a,b,c,d,e | a3=b3=c3=d4=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, bd=db, ebe=b-1, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 852 in 136 conjugacy classes, 34 normal (14 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4, C22 [×2], S3 [×17], C6, C6 [×4], C6 [×5], D4, C32, C32 [×4], C32 [×4], Dic3, C12 [×4], D6 [×13], C2×C6, C3×S3 [×4], C3⋊S3 [×14], C3×C6, C3×C6 [×4], C3×C6 [×4], D12 [×4], C3⋊D4, C33, C3×Dic3 [×4], C3×C12, S3×C6 [×4], C2×C3⋊S3, C2×C3⋊S3 [×9], C3×C3⋊S3, C33⋊C2, C32×C6, C3⋊D12 [×4], C12⋊S3, C32×Dic3, C6×C3⋊S3, C2×C33⋊C2, C338D4
Quotients: C1, C2 [×3], C22, S3 [×5], D4, D6 [×5], C3⋊S3, D12 [×4], C3⋊D4, S32 [×4], C2×C3⋊S3, C3⋊D12 [×4], C12⋊S3, S3×C3⋊S3, C338D4

Smallest permutation representation of C338D4
On 36 points
Generators in S36
(1 29 8)(2 30 5)(3 31 6)(4 32 7)(9 33 27)(10 34 28)(11 35 25)(12 36 26)(13 22 18)(14 23 19)(15 24 20)(16 21 17)
(1 35 24)(2 36 21)(3 33 22)(4 34 23)(5 12 16)(6 9 13)(7 10 14)(8 11 15)(17 30 26)(18 31 27)(19 32 28)(20 29 25)
(1 8 29)(2 30 5)(3 6 31)(4 32 7)(9 27 33)(10 34 28)(11 25 35)(12 36 26)(13 18 22)(14 23 19)(15 20 24)(16 21 17)
(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)
(2 4)(5 32)(6 31)(7 30)(8 29)(9 18)(10 17)(11 20)(12 19)(13 27)(14 26)(15 25)(16 28)(21 34)(22 33)(23 36)(24 35)

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

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

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

C338D4 is a maximal subgroup of
S3×C3⋊D12  C3⋊S34D12  D6.3S32  Dic3.S32  C12.39S32  C12.40S32  C12.73S32  S3×C12⋊S3  C62.93D6  C3⋊S3×C3⋊D4  C6223D6
C338D4 is a maximal quotient of
C338D8  C3316SD16  C3317SD16  C338Q16  C62.78D6  C62.79D6  C62.80D6

33 conjugacy classes

class 1 2A2B2C3A···3E3F3G3H3I 4 6A···6E6F6G6H6I6J6K12A···12H
order12223···3333346···666666612···12
size1118542···2444462···2444418186···6

33 irreducible representations

dim111122222244
type+++++++++++
imageC1C2C2C2S3S3D4D6D12C3⋊D4S32C3⋊D12
kernelC338D4C32×Dic3C6×C3⋊S3C2×C33⋊C2C3×Dic3C2×C3⋊S3C33C3×C6C32C32C6C3
# reps111141158244

Matrix representation of C338D4 in GL8(ℤ)

10000000
01000000
00100000
00010000
00000-100
00001-100
00000010
00000001
,
10000000
01000000
00-1-10000
00100000
00000-100
00001-100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000000-1
0000001-1
,
0-1000000
10000000
00-100000
000-10000
0000-1000
00000-100
00000001
00000010
,
10000000
0-1000000
00100000
00-1-10000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,-1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C338D4 in GAP, Magma, Sage, TeX

C_3^3\rtimes_8D_4
% in TeX

G:=Group("C3^3:8D4");
// GroupNames label

G:=SmallGroup(216,129);
// by ID

G=gap.SmallGroup(216,129);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,24,73,201,730,5189]);
// 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,e*a*e=a^-1,b*c=c*b,b*d=d*b,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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