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G = C33⋊9D8  order 432 = 24·33

6th semidirect product of C33 and D8 acting via D8/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊9D8, C32⋊6D24, C12.33S32, C12⋊S3⋊9S3, (C3×C6).44D12, C3⋊3(C3⋊D24), C32⋊4C8⋊11S3, C32⋊7(D4⋊S3), (C3×C12).124D6, C3⋊1(C32⋊2D8), (C32×C6).47D4, C6.7(D6⋊S3), C2.3(C33⋊9D4), C6.37(C3⋊D12), C4.1(C32⋊4D6), (C32×C12).20C22, (C3×C12⋊S3)⋊4C2, (C3×C32⋊4C8)⋊3C2, (C3×C6).65(C3⋊D4), SmallGroup(432,457)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊9D8
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C3×C12⋊S3 — C33⋊9D8
Lower central
C33 — C32×C6 — C32×C12 — C33⋊9D8
Upper central
C1 — C2 — C4

Generators and relations for C33⋊9D8
 G = < a,b,c,d,e | a3=b3=c3=d8=e2=1, ab=ba, ac=ca, ad=da, eae=a-1, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 808 in 134 conjugacy classes, 31 normal (15 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, S3, C6, C6, C6, C8, D4, C32, C32, C32, C12, C12, C12, D6, C2×C6, D8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C33, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, D24, D4⋊S3, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×D12, C12⋊S3, C32×C12, C6×C3⋊S3, C32⋊2D8, C3⋊D24, C3×C32⋊4C8, C3×C12⋊S3, C33⋊9D8
Quotients: C1, C2, C22, S3, D4, D6, D8, D12, C3⋊D4, S32, D24, D4⋊S3, D6⋊S3, C3⋊D12, C32⋊4D6, C32⋊2D8, C3⋊D24, C33⋊9D4, C33⋊9D8

Smallest permutation representation of C33⋊9D8
►On 48 points
Generators in S48
(1 32 46)(2 25 47)(3 26 48)(4 27 41)(5 28 42)(6 29 43)(7 30 44)(8 31 45)(9 23 35)(10 24 36)(11 17 37)(12 18 38)(13 19 39)(14 20 40)(15 21 33)(16 22 34)

(1 32 46)(2 47 25)(3 26 48)(4 41 27)(5 28 42)(6 43 29)(7 30 44)(8 45 31)(9 23 35)(10 36 24)(11 17 37)(12 38 18)(13 19 39)(14 40 20)(15 21 33)(16 34 22)

(1 46 32)(2 25 47)(3 48 26)(4 27 41)(5 42 28)(6 29 43)(7 44 30)(8 31 45)(9 23 35)(10 36 24)(11 17 37)(12 38 18)(13 19 39)(14 40 20)(15 21 33)(16 34 22)

(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 37 38 39 40)(41 42 43 44 45 46 47 48)

(1 18)(2 17)(3 24)(4 23)(5 22)(6 21)(7 20)(8 19)(9 27)(10 26)(11 25)(12 32)(13 31)(14 30)(15 29)(16 28)(33 43)(34 42)(35 41)(36 48)(37 47)(38 46)(39 45)(40 44)
 
G:=sub<Sym(48)| (1,32,46)(2,25,47)(3,26,48)(4,27,41)(5,28,42)(6,29,43)(7,30,44)(8,31,45)(9,23,35)(10,24,36)(11,17,37)(12,18,38)(13,19,39)(14,20,40)(15,21,33)(16,22,34), (1,32,46)(2,47,25)(3,26,48)(4,41,27)(5,28,42)(6,43,29)(7,30,44)(8,45,31)(9,23,35)(10,36,24)(11,17,37)(12,38,18)(13,19,39)(14,40,20)(15,21,33)(16,34,22), (1,46,32)(2,25,47)(3,48,26)(4,27,41)(5,42,28)(6,29,43)(7,44,30)(8,31,45)(9,23,35)(10,36,24)(11,17,37)(12,38,18)(13,19,39)(14,40,20)(15,21,33)(16,34,22), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44)>;
 
G:=Group( (1,32,46)(2,25,47)(3,26,48)(4,27,41)(5,28,42)(6,29,43)(7,30,44)(8,31,45)(9,23,35)(10,24,36)(11,17,37)(12,18,38)(13,19,39)(14,20,40)(15,21,33)(16,22,34), (1,32,46)(2,47,25)(3,26,48)(4,41,27)(5,28,42)(6,43,29)(7,30,44)(8,45,31)(9,23,35)(10,36,24)(11,17,37)(12,38,18)(13,19,39)(14,40,20)(15,21,33)(16,34,22), (1,46,32)(2,25,47)(3,48,26)(4,27,41)(5,42,28)(6,29,43)(7,44,30)(8,31,45)(9,23,35)(10,36,24)(11,17,37)(12,38,18)(13,19,39)(14,40,20)(15,21,33)(16,34,22), (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,18)(2,17)(3,24)(4,23)(5,22)(6,21)(7,20)(8,19)(9,27)(10,26)(11,25)(12,32)(13,31)(14,30)(15,29)(16,28)(33,43)(34,42)(35,41)(36,48)(37,47)(38,46)(39,45)(40,44) );
 
G=PermutationGroup([[(1,32,46),(2,25,47),(3,26,48),(4,27,41),(5,28,42),(6,29,43),(7,30,44),(8,31,45),(9,23,35),(10,24,36),(11,17,37),(12,18,38),(13,19,39),(14,20,40),(15,21,33),(16,22,34)], [(1,32,46),(2,47,25),(3,26,48),(4,41,27),(5,28,42),(6,43,29),(7,30,44),(8,45,31),(9,23,35),(10,36,24),(11,17,37),(12,38,18),(13,19,39),(14,40,20),(15,21,33),(16,34,22)], [(1,46,32),(2,25,47),(3,48,26),(4,27,41),(5,42,28),(6,29,43),(7,44,30),(8,31,45),(9,23,35),(10,36,24),(11,17,37),(12,38,18),(13,19,39),(14,40,20),(15,21,33),(16,34,22)], [(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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,18),(2,17),(3,24),(4,23),(5,22),(6,21),(7,20),(8,19),(9,27),(10,26),(11,25),(12,32),(13,31),(14,30),(15,29),(16,28),(33,43),(34,42),(35,41),(36,48),(37,47),(38,46),(39,45),(40,44)]])
 

45 conjugacy classes

class 1 2A2B2C3A3B3C3D···3H 4 6A6B6C6D···6H6I6J6K6L8A8B12A12B12C···12N24A24B24C24D
order12223333···346666···6666688121212···1224242424
size1136362224···422224···4363636361818224···418181818

45 irreducible representations

dim11122222222444444444
type++++++++++++-++
imageC1C2C2S3S3D4D6D8D12C3⋊D4D24S32D4⋊S3D6⋊S3C3⋊D12C32⋊4D6C32⋊2D8C3⋊D24C33⋊9D4C33⋊9D8
kernelC33⋊9D8C3×C32⋊4C8C3×C12⋊S3C32⋊4C8C12⋊S3C32×C6C3×C12C33C3×C6C3×C6C32C12C32C6C6C4C3C3C2C1
# reps11212132244321222424

Matrix representation of C33⋊9D8 ►in GL8(𝔽73)

10000000
01000000
000720000
001720000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
00001000
00000100
0000007272
00000010
,
10000000
01000000
00100000
00010000
000007200
000017200
00000010
00000001
,
2848000000
313000000
007200000
000720000
0000511200
0000632200
0000001022
0000001263
,
101000000
4763000000
000720000
007200000
0000136200
000026000
0000003060
0000001343

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[28,3,0,0,0,0,0,0,48,13,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,51,63,0,0,0,0,0,0,12,22,0,0,0,0,0,0,0,0,10,12,0,0,0,0,0,0,22,63],[10,47,0,0,0,0,0,0,1,63,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,13,2,0,0,0,0,0,0,62,60,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43] >;
 

C33⋊9D8 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_9D_8
 
% in TeX
 
G:=Group("C3^3:9D8");
 
// GroupNames label
 
G:=SmallGroup(432,457);
 
// by ID
 
G=gap.SmallGroup(432,457);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e=a^-1,b*c=c*b,d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=e*c*e=c^-1,e*d*e=d^-1>;
 
// generators/relations
 

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