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G = C62.116C23order 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

C1C3xC6 — C62.116C23
C1C3C32C3xC6C62C6xDic3C2xC6.D6 — C62.116C23
C32C3xC6 — C62.116C23
C1C22C23

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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