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G = C12:3S32  order 432 = 24·33

3rd semidirect product of C12 and S32 acting via S32/C32=C22

metabelian, supersoluble, monomial

Aliases: C12:3S32, C3:S3:3D12, C3:4(S3xD12), (C3xC12):14D6, C33:16(C2xD4), C3:Dic3:20D6, C4:(C32:4D6), C12:S3:13S3, C32:12(S3xD4), C32:7(C2xD12), C33:9D4:4C2, C3:3(D6:D6), (C32xC12):6C22, (C32xC6).69C23, C6.98(C2xS32), (C3xC3:S3):9D4, (C4xC3:S3):10S3, (C12xC3:S3):9C2, (C2xC3:S3):11D6, (C6xC3:S3):12C22, (C3xC12:S3):13C2, (C2xC32:4D6):3C2, C2.6(C2xC32:4D6), (C3xC6).119(C22xS3), (C3xC3:Dic3):15C22, SmallGroup(432,691)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — C12:3S32
Chief series
C1 — C3 — C32 — C33 — C32xC6 — C6xC3:S3 — C2xC32:4D6 — C12:3S32
Lower central
C33 — C32xC6 — C12:3S32
Upper central
C1 — C2 — C4

Generators and relations for C12:3S32
 G = < a,b,c,d,e | a3=b3=c2=d12=e2=1, ab=ba, cac=a-1, ad=da, ae=ea, cbc=ebe=b-1, bd=db, cd=dc, ce=ec, ede=d-1 >

Subgroups: 1752 in 270 conjugacy classes, 51 normal (19 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2xC4, D4, C23, C32, C32, C32, Dic3, C12, C12, C12, D6, C2xC6, C2xD4, C3xS3, C3:S3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, D12, C3:D4, C2xC12, C3xD4, C22xS3, C33, C3xDic3, C3:Dic3, C3xC12, C3xC12, C3xC12, S32, S3xC6, C2xC3:S3, C2xC3:S3, C2xD12, S3xD4, C3xC3:S3, C3xC3:S3, C32xC6, D6:S3, C3:D12, S3xC12, C3xD12, C4xC3:S3, C12:S3, C2xS32, C3xC3:Dic3, C32xC12, C32:4D6, C6xC3:S3, C6xC3:S3, S3xD12, D6:D6, C33:9D4, C12xC3:S3, C3xC12:S3, C2xC32:4D6, C12:3S32
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C22xS3, S32, C2xD12, S3xD4, C2xS32, C32:4D6, S3xD12, D6:D6, C2xC32:4D6, C12:3S32

Smallest permutation representation of C12:3S32
►On 48 points
Generators in S48
(1 5 9)(2 6 10)(3 7 11)(4 8 12)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C3D···3H4A4B6A6B6C6D···6H6I6J6K6L6M6N12A12B12C···12N12O12P
order122222223333···3446666···6666666121212···121212
size1199181818182224···42182224···4181836363636224···41818

48 irreducible representations

dim11111222222244444444
type++++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6D6D12S32S3xD4C2xS32C32:4D6S3xD12D6:D6C2xC32:4D6C12:3S32
kernelC12:3S32C33:9D4C12xC3:S3C3xC12:S3C2xC32:4D6C4xC3:S3C12:S3C3xC3:S3C3:Dic3C3xC12C2xC3:S3C3:S3C12C32C6C4C3C3C2C1
# reps12122122135432324224

Matrix representation of C12:3S32 ►in GL8(F13)

10000000
01000000
00100000
00010000
00001000
00000100
000000012
000000112
,
10000000
01000000
00100000
00010000
00000100
0000121200
00000010
00000001
,
120000000
012000000
001200000
000120000
00001000
0000121200
00000001
00000010
,
1010000000
123000000
000120000
00110000
00001000
00000100
00000010
00000001
,
06000000
110000000
00010000
00100000
000012000
00001100
000000120
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[10,12,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,11,0,0,0,0,0,0,6,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12] >;
 

C12:3S32 in GAP, Magma, Sage, TeX 

C_{12}\rtimes_3S_3^2
 
% in TeX
 
G:=Group("C12:3S3^2");
 
// GroupNames label
 
G:=SmallGroup(432,691);
 
// by ID
 
G=gap.SmallGroup(432,691);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,135,58,1124,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^3=b^3=c^2=d^12=e^2=1,a*b=b*a,c*a*c=a^-1,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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