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G = C62:5Dic3  order 432 = 24·33

4th semidirect product of C62 and Dic3 acting via Dic3/C2=S3

non-abelian, soluble, monomial

Aliases: C62:5Dic3, (C3xA4):C12, (C6xA4).C6, C6.7S4:C3, C6.9(C3xS4), (C3xC6).6S4, C32:A4:1C4, (C2xC62).7S3, C32:1(A4:C4), C22:(C32:C12), C23.(C32:C6), C2.1(C62:S3), C3.3(C3xA4:C4), (C2xC32:A4).1C2, (C22xC6).6(C3xS3), (C2xC6).4(C3xDic3), SmallGroup(432,251)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C3xA4 — C62:5Dic3
Chief series
C1 — C22 — C2xC6 — C3xA4 — C6xA4 — C2xC32:A4 — C62:5Dic3
Lower central
C3xA4 — C62:5Dic3
Upper central
C1 — C2

Generators and relations for C62:5Dic3
 G = < a,b,c,d | a6=b6=c6=1, d2=c3, cac-1=ab=ba, dad-1=a4b3, cbc-1=a3b4, dbd-1=a3b2, dcd-1=c-1 >

Subgroups: 449 in 82 conjugacy classes, 18 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, C6, C6, C2xC4, C23, C32, C32, Dic3, C12, A4, C2xC6, C2xC6, C22:C4, C3xC6, C3xC6, C2xDic3, C2xC12, C2xA4, C22xC6, C22xC6, He3, C3xDic3, C3:Dic3, C3xA4, C3xA4, C62, C62, C6.D4, C3xC22:C4, A4:C4, C2xHe3, C6xDic3, C6xA4, C6xA4, C2xC62, C32:C12, C32:A4, C3xC6.D4, C6.7S4, C2xC32:A4, C62:5Dic3
Quotients: C1, C2, C3, C4, S3, C6, Dic3, C12, C3xS3, S4, C3xDic3, A4:C4, C32:C6, C3xS4, C32:C12, C3xA4:C4, C62:S3, C62:5Dic3

Smallest permutation representation of C62:5Dic3
►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 4 5 2 3 6)(7 10 12 8 9 11)(13 17 15 16 14 18)(19 24 21 23 20 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)

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

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

38 conjugacy classes

class 1 2A2B2C3A3B3C3D3E3F4A4B4C4D6A6B···6G6H···6M6N6O6P12A···12H
order1222333333444466···66···666612···12
size11332332424241818181823···36···624242418···18

38 irreducible representations

dim11111122223333666666
type+++-++-+-
imageC1C2C3C4C6C12S3Dic3C3xS3C3xDic3S4A4:C4C3xS4C3xA4:C4C32:C6C32:C12C62:S3C62:S3C62:5Dic3C62:5Dic3
kernelC62:5Dic3C2xC32:A4C6.7S4C32:A4C6xA4C3xA4C2xC62C62C22xC6C2xC6C3xC6C32C6C3C23C22C2C2C1C1
# reps11222411222244111212

Matrix representation of C62:5Dic3 ►in GL9(F13)

400000000
040000000
009000000
000100000
000130000
000409000
000000100
000000130
000000409
,
1200000000
010000000
0012000000
000900000
000090000
000009000
000000300
000000030
000000003
,
0120000000
0012000000
1200000000
000120000
0000121000
0000120000
000000102
0000000012
0000000112
,
800000000
008000000
080000000
000000100
000000010
000000001
000100000
000010000
000001000

G:=sub<GL(9,GF(13))| [4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,1,1,4,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,9],[12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,3],[0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,2,12,12],[8,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0] >;
 

C62:5Dic3 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_5{\rm Dic}_3
 
% in TeX
 
G:=Group("C6^2:5Dic3");
 
// GroupNames label
 
G:=SmallGroup(432,251);
 
// by ID
 
G=gap.SmallGroup(432,251);
 
# by ID
 
G:=PCGroup([7,-2,-3,-2,-3,-3,-2,2,42,675,682,2524,9077,782,5298,1350]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^6=1,d^2=c^3,c*a*c^-1=a*b=b*a,d*a*d^-1=a^4*b^3,c*b*c^-1=a^3*b^4,d*b*d^-1=a^3*b^2,d*c*d^-1=c^-1>;
 
// generators/relations
 

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