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G = C62:23D6  order 432 = 24·33

4th semidirect product of C62 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, rational

Aliases: C62:23D6, (S3xC6):6D6, C33:26(C2xD4), C3:Dic3:16D6, (C3xDic3):5D6, C32:18(S3xD4), C32:7D4:6S3, C33:C2:4D4, C3:3(Dic3:D6), C33:8D4:10C2, C33:7D4:10C2, (C3xC62):5C22, (C32xC6).64C23, (C32xDic3):8C22, (C2xC6):6S32, C3:3(D4xC3:S3), C6.74(C2xS32), D6:4(C2xC3:S3), (C2xC3:S3):17D6, (C3xC3:D4):4S3, C22:5(S3xC3:S3), C33:8(C2xC4):9C2, C3:D4:2(C3:S3), (S3xC3xC6):14C22, Dic3:2(C2xC3:S3), (C6xC3:S3):11C22, (C32xC3:D4):8C2, (C3xC32:7D4):4C2, C6.27(C22xC3:S3), (C3xC3:Dic3):6C22, (C3xC6).114(C22xS3), (C22xC33:C2):3C2, (C2xC33:C2):11C22, (C2xS3xC3:S3):10C2, (C2xC6):5(C2xC3:S3), C2.27(C2xS3xC3:S3), SmallGroup(432,686)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C32xC6 — C62:23D6
Chief series
C1 — C3 — C32 — C33 — C32xC6 — S3xC3xC6 — C2xS3xC3:S3 — C62:23D6
Lower central
C33 — C32xC6 — C62:23D6
Upper central
C1 — C2 — C22

Generators and relations for C62:23D6
 G = < a,b,c,d | a6=b6=c6=d2=1, ab=ba, cac-1=a-1b3, dad=a-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 3296 in 452 conjugacy classes, 70 normal (32 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C32, C32, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C2xD4, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, D12, C3:D4, C3:D4, C3xD4, C22xS3, C33, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S32, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C62, C62, S3xD4, S3xC32, C3xC3:S3, C33:C2, C33:C2, C32xC6, C32xC6, C6.D6, C3:D12, C3xC3:D4, C3xC3:D4, C4xC3:S3, C12:S3, C32:7D4, C32:7D4, D4xC32, C2xS32, C22xC3:S3, C32xDic3, C3xC3:Dic3, S3xC3:S3, S3xC3xC6, C6xC3:S3, C2xC33:C2, C2xC33:C2, C3xC62, Dic3:D6, D4xC3:S3, C33:8(C2xC4), C33:7D4, C33:8D4, C32xC3:D4, C3xC32:7D4, C2xS3xC3:S3, C22xC33:C2, C62:23D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C3:S3, C22xS3, S32, C2xC3:S3, S3xD4, C2xS32, C22xC3:S3, S3xC3:S3, Dic3:D6, D4xC3:S3, C2xS3xC3:S3, C62:23D6

Smallest permutation representation of C62:23D6
►On 36 points
Generators in S36
(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 15 8 11 5 16)(2 13 9 12 6 17)(3 14 7 10 4 18)(19 26 33 22 29 36)(20 27 34 23 30 31)(21 28 35 24 25 32)

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

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

51 conjugacy classes

class 1 2A2B2C2D2E2F2G3A···3E3F3G3H3I4A4B6A···6E6F···6V6W6X6Y6Z6AA12A12B12C12D12E
order122222223···33333446···66···6666661212121212
size1126182727542···244446182···24···412121212361212121236

51 irreducible representations

dim11111111222222224444
type++++++++++++++++++++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6D6D6S32S3xD4C2xS32Dic3:D6
kernelC62:23D6C33:8(C2xC4)C33:7D4C33:8D4C32xC3:D4C3xC32:7D4C2xS3xC3:S3C22xC33:C2C3xC3:D4C32:7D4C33:C2C3xDic3C3:Dic3S3xC6C2xC3:S3C62C2xC6C32C6C3
# reps11111111412414154548

Matrix representation of C62:23D6 ►in GL8(F13)

10000000
312000000
000120000
00110000
000011200
00001000
000000120
000000012
,
120000000
012000000
00010000
0012120000
000001200
000011200
00000010
00000001
,
18000000
012000000
00100000
0012120000
00000100
00001000
000000121
000000120
,
10000000
01000000
00100000
0012120000
00000100
00001000
000000120
000000121

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

C62:23D6 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_{23}D_6
 
% in TeX
 
G:=Group("C6^2:23D6");
 
// GroupNames label
 
G:=SmallGroup(432,686);
 
// by ID
 
G=gap.SmallGroup(432,686);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,254,135,571,2028,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a^-1,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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