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G = C62.57D4  order 288 = 25·32

41st non-split extension by C62 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.57D4, C62.104C23, C23.31S32, (C2xC6).15D12, C6.82(C2xD12), D6:Dic3:15C2, C6.D4:6S3, (C22xC6).68D6, (C2xDic3).41D6, (C22xS3).25D6, Dic3:Dic3:35C2, C6.67(D4:2S3), (C2xC62).23C22, C2.15(D6.4D6), C3:5(C23.21D6), C3:2(C23.23D6), (C6xDic3).24C22, C22.11(C3:D12), C32:11(C22.D4), (C6xC3:D4).7C2, (C2xC3:D4).4S3, C6.19(C2xC3:D4), C22.134(C2xS32), (C3xC6).150(C2xD4), (S3xC2xC6).42C22, (C3xC6).79(C4oD4), C2.22(C2xC3:D12), (C3xC6.D4):7C2, (C2xC6).23(C3:D4), (C22xC3:Dic3):2C2, (C2xC6).123(C22xS3), (C2xC3:Dic3).144C22, SmallGroup(288,610)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.57D4
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — D6:Dic3 — C62.57D4
Lower central
C32 — C62 — C62.57D4
Upper central
C1 — C22 — C23

Generators and relations for C62.57D4
 G = < a,b,c,d | a6=b6=c4=1, d2=b3, ab=ba, cac-1=a-1b3, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 626 in 183 conjugacy classes, 52 normal (26 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3xC6, C3xC6, C3xC6, C2xDic3, C2xDic3, C2xDic3, C3:D4, C2xC12, C3xD4, C22xS3, C22xC6, C22xC6, C22.D4, C3xDic3, C3:Dic3, S3xC6, C62, C62, C62, Dic3:C4, C4:Dic3, D6:C4, C6.D4, C6.D4, C3xC22:C4, C22xDic3, C2xC3:D4, C6xD4, C6xDic3, C6xDic3, C3xC3:D4, C2xC3:Dic3, C2xC3:Dic3, S3xC2xC6, C2xC62, C23.21D6, C23.23D6, D6:Dic3, Dic3:Dic3, C3xC6.D4, C6xC3:D4, C22xC3:Dic3, C62.57D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, D12, C3:D4, C22xS3, C22.D4, S32, C2xD12, D4:2S3, C2xC3:D4, C3:D12, C2xS32, C23.21D6, C23.23D6, D6.4D6, C2xC3:D12, C62.57D4

Smallest permutation representation of C62.57D4
►On 48 points
Generators in S48
(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)(37 38 39 40 41 42)(43 44 45 46 47 48)

(1 13 3 15 5 17)(2 14 4 16 6 18)(7 48 9 44 11 46)(8 43 10 45 12 47)(19 25 23 29 21 27)(20 26 24 30 22 28)(31 41 35 39 33 37)(32 42 36 40 34 38)

(1 27 18 30)(2 22 13 19)(3 25 14 28)(4 20 15 23)(5 29 16 26)(6 24 17 21)(7 39 47 42)(8 36 48 33)(9 37 43 40)(10 34 44 31)(11 41 45 38)(12 32 46 35)

(1 10 15 47)(2 9 16 46)(3 8 17 45)(4 7 18 44)(5 12 13 43)(6 11 14 48)(19 35 29 37)(20 34 30 42)(21 33 25 41)(22 32 26 40)(23 31 27 39)(24 36 28 38)
 
G:=sub<Sym(48)| (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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,13,3,15,5,17)(2,14,4,16,6,18)(7,48,9,44,11,46)(8,43,10,45,12,47)(19,25,23,29,21,27)(20,26,24,30,22,28)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,27,18,30)(2,22,13,19)(3,25,14,28)(4,20,15,23)(5,29,16,26)(6,24,17,21)(7,39,47,42)(8,36,48,33)(9,37,43,40)(10,34,44,31)(11,41,45,38)(12,32,46,35), (1,10,15,47)(2,9,16,46)(3,8,17,45)(4,7,18,44)(5,12,13,43)(6,11,14,48)(19,35,29,37)(20,34,30,42)(21,33,25,41)(22,32,26,40)(23,31,27,39)(24,36,28,38)>;
 
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)(37,38,39,40,41,42)(43,44,45,46,47,48), (1,13,3,15,5,17)(2,14,4,16,6,18)(7,48,9,44,11,46)(8,43,10,45,12,47)(19,25,23,29,21,27)(20,26,24,30,22,28)(31,41,35,39,33,37)(32,42,36,40,34,38), (1,27,18,30)(2,22,13,19)(3,25,14,28)(4,20,15,23)(5,29,16,26)(6,24,17,21)(7,39,47,42)(8,36,48,33)(9,37,43,40)(10,34,44,31)(11,41,45,38)(12,32,46,35), (1,10,15,47)(2,9,16,46)(3,8,17,45)(4,7,18,44)(5,12,13,43)(6,11,14,48)(19,35,29,37)(20,34,30,42)(21,33,25,41)(22,32,26,40)(23,31,27,39)(24,36,28,38) );
 
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),(37,38,39,40,41,42),(43,44,45,46,47,48)], [(1,13,3,15,5,17),(2,14,4,16,6,18),(7,48,9,44,11,46),(8,43,10,45,12,47),(19,25,23,29,21,27),(20,26,24,30,22,28),(31,41,35,39,33,37),(32,42,36,40,34,38)], [(1,27,18,30),(2,22,13,19),(3,25,14,28),(4,20,15,23),(5,29,16,26),(6,24,17,21),(7,39,47,42),(8,36,48,33),(9,37,43,40),(10,34,44,31),(11,41,45,38),(12,32,46,35)], [(1,10,15,47),(2,9,16,46),(3,8,17,45),(4,7,18,44),(5,12,13,43),(6,11,14,48),(19,35,29,37),(20,34,30,42),(21,33,25,41),(22,32,26,40),(23,31,27,39),(24,36,28,38)]])
 

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G···6Q6R6S12A···12F
order122222233344444446···66···66612···12
size11112212224121212181818182···24···4121212···12

42 irreducible representations

dim11111122222222244444
type++++++++++++++-++-
imageC1C2C2C2C2C2S3S3D4D6D6D6C4oD4D12C3:D4S32D4:2S3C3:D12C2xS32D6.4D6
kernelC62.57D4D6:Dic3Dic3:Dic3C3xC6.D4C6xC3:D4C22xC3:Dic3C6.D4C2xC3:D4C62C2xDic3C22xS3C22xC6C3xC6C2xC6C2xC6C23C6C22C22C2
# reps12211111231244414214

Matrix representation of C62.57D4 ►in GL8(F13)

10000000
1012000000
001120000
00100000
00001000
00000100
00000010
00000001
,
120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
128000000
31000000
00010000
00100000
00001200
0000121200
00000010
00000001
,
50000000
05000000
00010000
00100000
000012000
00001100
00000001
00000010

G:=sub<GL(8,GF(13))| [1,10,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0],[12,3,0,0,0,0,0,0,8,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,2,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.57D4 in GAP, Magma, Sage, TeX 

C_6^2._{57}D_4
 
% in TeX
 
G:=Group("C6^2.57D4");
 
// GroupNames label
 
G:=SmallGroup(288,610);
 
// by ID
 
G=gap.SmallGroup(288,610);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,176,422,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=1,d^2=b^3,a*b=b*a,c*a*c^-1=a^-1*b^3,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^3*c^-1>;
 
// generators/relations
 

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