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G = C62.116C23  order 288 = 25·32

111st non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.116C23, C62:8(C2xC4), C23.33S32, C6.70(S3xD4), (C2xDic3):12D6, (C22xC6).76D6, C2.5(Dic3:D6), C6.D4:11S3, C6.D12:16C2, (C6xDic3):14C22, C22:3(C6.D6), (C2xC62).35C22, (C2xC6):8(C4xS3), C6.39(S3xC2xC4), C3:2(S3xC22:C4), (C22xC3:S3):5C4, (C2xC3:S3).62D4, C22.57(C2xS32), C32:8(C2xC22:C4), C3:S3:3(C22:C4), (C3xC6).162(C2xD4), (C23xC3:S3).2C2, (C2xC6.D6):14C2, (C3xC6).71(C22xC4), C2.16(C2xC6.D6), (C3xC6.D4):20C2, (C2xC6).135(C22xS3), (C22xC3:S3).77C22, (C2xC3:S3):15(C2xC4), SmallGroup(288,622)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C62.116C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — C2xC6.D6 — C62.116C23
Lower central
C32 — C3xC6 — C62.116C23
Upper central
C1 — C22 — C23

Generators and relations for C62.116C23
 G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=b3d >

Subgroups: 1538 in 331 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C2xC4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C22xC4, C24, C3:S3, C3:S3, C3xC6, C3xC6, C3xC6, C4xS3, C2xDic3, C2xC12, C22xS3, C22xC6, C22xC6, C2xC22:C4, C3xDic3, C2xC3:S3, C2xC3:S3, C62, C62, C62, D6:C4, C6.D4, C3xC22:C4, S3xC2xC4, S3xC23, C6.D6, C6xDic3, C22xC3:S3, C22xC3:S3, C22xC3:S3, C2xC62, S3xC22:C4, C6.D12, C3xC6.D4, C2xC6.D6, C23xC3:S3, C62.116C23
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22:C4, C22xC4, C2xD4, C4xS3, C22xS3, C2xC22:C4, S32, S3xC2xC4, S3xD4, C6.D6, C2xS32, S3xC22:C4, C2xC6.D6, Dic3:D6, C62.116C23

Permutation representations of C62.116C23
►On 24 points - transitive group 24T673
Generators in S24
(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)

(1 17 5 15 3 13)(2 18 6 16 4 14)(7 19 9 21 11 23)(8 20 10 22 12 24)

(1 10 4 7)(2 11 5 8)(3 12 6 9)(13 22 16 19)(14 23 17 20)(15 24 18 21)

(1 21 4 24)(2 20 5 23)(3 19 6 22)(7 18 10 15)(8 17 11 14)(9 16 12 13)

(7 21)(8 22)(9 23)(10 24)(11 19)(12 20)
 
G:=sub<Sym(24)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20)>;
 
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), (1,17,5,15,3,13)(2,18,6,16,4,14)(7,19,9,21,11,23)(8,20,10,22,12,24), (1,10,4,7)(2,11,5,8)(3,12,6,9)(13,22,16,19)(14,23,17,20)(15,24,18,21), (1,21,4,24)(2,20,5,23)(3,19,6,22)(7,18,10,15)(8,17,11,14)(9,16,12,13), (7,21)(8,22)(9,23)(10,24)(11,19)(12,20) );
 
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)], [(1,17,5,15,3,13),(2,18,6,16,4,14),(7,19,9,21,11,23),(8,20,10,22,12,24)], [(1,10,4,7),(2,11,5,8),(3,12,6,9),(13,22,16,19),(14,23,17,20),(15,24,18,21)], [(1,21,4,24),(2,20,5,23),(3,19,6,22),(7,18,10,15),(8,17,11,14),(9,16,12,13)], [(7,21),(8,22),(9,23),(10,24),(11,19),(12,20)]])
 
G:=TransitiveGroup(24,673);
 

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K3A3B3C4A···4H6A···6F6G···6Q12A···12H
order1222222222223334···46···66···612···12
size111122999918182246···62···24···412···12

48 irreducible representations

dim1111112222244444
type++++++++++++++
imageC1C2C2C2C2C4S3D4D6D6C4xS3S32S3xD4C6.D6C2xS32Dic3:D6
kernelC62.116C23C6.D12C3xC6.D4C2xC6.D6C23xC3:S3C22xC3:S3C6.D4C2xC3:S3C2xDic3C22xC6C2xC6C23C6C22C22C2
# reps1222182442814214

Matrix representation of C62.116C23 ►in GL6(F13)

1120000
100000
0012000
0001200
000010
000001
,
100000
010000
0012000
0001200
0000121
0000120
,
800000
080000
0011000
0051200
000001
000010
,
050000
500000
0012300
008100
000010
000001
,
1200000
0120000
001000
0051200
000010
000001

G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,5,0,0,0,0,10,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,12,8,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
 

C62.116C23 in GAP, Magma, Sage, TeX 

C_6^2._{116}C_2^3
 
% in TeX
 
G:=Group("C6^2.116C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,622);
 
// by ID
 
G=gap.SmallGroup(288,622);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,64,422,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=d^2=a^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=b^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

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