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G = C62.31D4order 288 = 25·32

15th non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.31D4, C23.8S32, (C2xC6).9D12, (C2xDic3):Dic3, (C6xDic3):1C4, C6.41(D6:C4), C62:5C4:1C2, C6.D4:1S3, C32:4(C23:C4), C62.34(C2xC4), (C22xC6).54D6, (C22xS3):2Dic3, (C2xC62).7C22, C22.5(S3xDic3), C2.11(D6:Dic3), C3:3(C23.6D6), C3:1(C23.7D6), C6.11(C6.D4), C22.3(D6:S3), C22.8(C3:D12), (S3xC2xC6):1C4, (C2xC6).70(C4xS3), (C2xC3:D4).1S3, (C6xC3:D4).4C2, (C2xC6).6(C2xDic3), (C3xC6.D4):1C2, (C2xC6).12(C3:D4), (C3xC6).39(C22:C4), SmallGroup(288,228)

Series: Derived Chief Lower central Upper central

C1C62 — C62.31D4
C1C3C32C3xC6C62C2xC62C6xC3:D4 — C62.31D4
C32C3xC6C62 — C62.31D4
C1C2C23

Generators and relations for C62.31D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=a-1b3, dad=ab3, cbc-1=dbd=b-1, dcd=a3c-1 >

Subgroups: 474 in 121 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C2xD4, C3xS3, C3xC6, C3xC6, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C23:C4, C3xDic3, C3:Dic3, S3xC6, C62, C62, C6.D4, C6.D4, C3xC22:C4, C2xC3:D4, C6xD4, C6xDic3, C6xDic3, C3xC3:D4, C2xC3:Dic3, S3xC2xC6, C2xC62, C23.6D6, C23.7D6, C3xC6.D4, C62:5C4, C6xC3:D4, C62.31D4
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, Dic3, D6, C22:C4, C4xS3, D12, C2xDic3, C3:D4, C23:C4, S32, D6:C4, C6.D4, S3xDic3, D6:S3, C3:D12, C23.6D6, C23.7D6, D6:Dic3, C62.31D4

Permutation representations of C62.31D4
On 24 points - transitive group 24T581
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 4 2 5 3 6)(7 11 9 10 8 12)(13 14 15 16 17 18)(19 24 23 22 21 20)
(1 19 5 22)(2 21 6 24)(3 23 4 20)(7 13)(8 15)(9 17)(10 16)(11 18)(12 14)
(1 13)(2 17)(3 15)(4 18)(5 16)(6 14)(7 22)(8 20)(9 24)(10 19)(11 23)(12 21)

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

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

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

G:=TransitiveGroup(24,581);

On 24 points - transitive group 24T614
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 16 5 14 3 18)(2 17 6 15 4 13)(7 20 9 22 11 24)(8 21 10 23 12 19)
(2 13)(3 5)(4 17)(6 15)(7 10 22 19)(8 24 23 9)(11 12 20 21)(16 18)
(1 7)(2 23)(3 9)(4 19)(5 11)(6 21)(8 15)(10 17)(12 13)(14 22)(16 24)(18 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), (1,16,5,14,3,18)(2,17,6,15,4,13)(7,20,9,22,11,24)(8,21,10,23,12,19), (2,13)(3,5)(4,17)(6,15)(7,10,22,19)(8,24,23,9)(11,12,20,21)(16,18), (1,7)(2,23)(3,9)(4,19)(5,11)(6,21)(8,15)(10,17)(12,13)(14,22)(16,24)(18,20)>;

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

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

G:=TransitiveGroup(24,614);

39 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E6A···6F6G···6Q6R6S12A···12F
order122222333444446···66···66612···12
size112221222412121236362···24···4121212···12

39 irreducible representations

dim11111122222222244444444
type+++++++--++++--+
imageC1C2C2C2C4C4S3S3D4Dic3Dic3D6C4xS3D12C3:D4C23:C4S32S3xDic3D6:S3C3:D12C23.6D6C23.7D6C62.31D4
kernelC62.31D4C3xC6.D4C62:5C4C6xC3:D4C6xDic3S3xC2xC6C6.D4C2xC3:D4C62C2xDic3C22xS3C22xC6C2xC6C2xC6C2xC6C32C23C22C22C22C3C3C1
# reps11112211211222611111224

Matrix representation of C62.31D4 in GL4(F7) generated by

5232
1434
2233
0002
,
1526
0302
3301
0005
,
2052
4204
3331
1630
,
4145
2204
3424
4456
G:=sub<GL(4,GF(7))| [5,1,2,0,2,4,2,0,3,3,3,0,2,4,3,2],[1,0,3,0,5,3,3,0,2,0,0,0,6,2,1,5],[2,4,3,1,0,2,3,6,5,0,3,3,2,4,1,0],[4,2,3,4,1,2,4,4,4,0,2,5,5,4,4,6] >;

C62.31D4 in GAP, Magma, Sage, TeX

C_6^2._{31}D_4
% in TeX

G:=Group("C6^2.31D4");
// GroupNames label

G:=SmallGroup(288,228);
// by ID

G=gap.SmallGroup(288,228);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,36,422,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d=a*b^3,c*b*c^-1=d*b*d=b^-1,d*c*d=a^3*c^-1>;
// generators/relations

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