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G = C33⋊18SD16  order 432 = 24·33

10th semidirect product of C33 and SD16 acting via SD16/C4=C22

metabelian, supersoluble, monomial

Aliases: C33⋊18SD16, C12.34S32, (C3×C6).45D12, C32⋊4Q8⋊9S3, C12⋊S3.5S3, C32⋊4C8⋊12S3, (C3×C12).125D6, (C32×C6).48D4, C32⋊9(C24⋊C2), C6.8(D6⋊S3), C32⋊9(D4.S3), C2.4(C33⋊9D4), C3⋊3(C32⋊5SD16), C3⋊1(Dic6⋊S3), C3⋊3(D12.S3), C6.38(C3⋊D12), C32⋊9(Q8⋊2S3), C4.2(C32⋊4D6), (C32×C12).21C22, (C3×C32⋊4C8)⋊6C2, (C3×C32⋊4Q8)⋊4C2, (C3×C12⋊S3).4C2, (C3×C6).66(C3⋊D4), SmallGroup(432,458)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32×C12 — C33⋊18SD16
Chief series
C1 — C3 — C32 — C33 — C32×C6 — C32×C12 — C3×C12⋊S3 — C33⋊18SD16
Lower central
C33 — C32×C6 — C32×C12 — C33⋊18SD16
Upper central
C1 — C2 — C4

Generators and relations for C33⋊18SD16
 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=d3 >

Subgroups: 632 in 122 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, D4.S3, Q8⋊2S3, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×Dic6, C3×D12, C32⋊4Q8, C12⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, Dic6⋊S3, D12.S3, C32⋊5SD16, C3×C32⋊4C8, C3×C32⋊4Q8, C3×C12⋊S3, C33⋊18SD16
Quotients: C1, C2, C22, S3, D4, D6, SD16, D12, C3⋊D4, S32, C24⋊C2, D4.S3, Q8⋊2S3, D6⋊S3, C3⋊D12, C32⋊4D6, Dic6⋊S3, D12.S3, C32⋊5SD16, C33⋊9D4, C33⋊18SD16

Smallest permutation representation of C33⋊18SD16
►On 48 points
Generators in S48
(1 15 37)(2 16 38)(3 9 39)(4 10 40)(5 11 33)(6 12 34)(7 13 35)(8 14 36)(17 31 45)(18 32 46)(19 25 47)(20 26 48)(21 27 41)(22 28 42)(23 29 43)(24 30 44)

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

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

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

45 conjugacy classes

class 1 2A2B3A3B3C3D···3H4A4B6A6B6C6D···6H6I6J8A8B12A12B12C···12N12O12P24A24B24C24D
order1223333···3446666···66688121212···12121224242424
size11362224···42362224···436361818224···4363618181818

45 irreducible representations

dim111122222222244444444444
type+++++++++++-+-+-+
imageC1C2C2C2S3S3S3D4D6SD16D12C3⋊D4C24⋊C2S32D4.S3Q8⋊2S3D6⋊S3C3⋊D12C32⋊4D6Dic6⋊S3D12.S3C32⋊5SD16C33⋊9D4C33⋊18SD16
kernelC33⋊18SD16C3×C32⋊4C8C3×C32⋊4Q8C3×C12⋊S3C32⋊4C8C32⋊4Q8C12⋊S3C32×C6C3×C12C33C3×C6C3×C6C32C12C32C32C6C6C4C3C3C3C2C1
# reps111111113224431112222224

Matrix representation of C33⋊18SD16 ►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
,
6331000000
022000000
007200000
000720000
000014500
0000195900
0000004811
0000003625
,
4230000000
4131000000
00010000
00100000
0000104100
0000516300
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],[63,0,0,0,0,0,0,0,31,22,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,14,19,0,0,0,0,0,0,5,59,0,0,0,0,0,0,0,0,48,36,0,0,0,0,0,0,11,25],[42,41,0,0,0,0,0,0,30,31,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,10,51,0,0,0,0,0,0,41,63,0,0,0,0,0,0,0,0,30,13,0,0,0,0,0,0,60,43] >;
 

C33⋊18SD16 in GAP, Magma, Sage, TeX 

C_3^3\rtimes_{18}{\rm SD}_{16}
 
% in TeX
 
G:=Group("C3^3:18SD16");
 
// GroupNames label
 
G:=SmallGroup(432,458);
 
// by ID
 
G=gap.SmallGroup(432,458);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,1124,571,2028,14118]);
 
// 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^3>;
 
// generators/relations
 

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