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G = C62:5D4  order 288 = 25·32

2nd semidirect product of C62 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial

Aliases: C62:5D4, C62.119C23, (S3xC6):16D4, (C2xC6):11D12, C23.27S32, D6:8(C3:D4), (S3xC23):5S3, (C2xDic3):4D6, C6.171(S3xD4), C6.86(C2xD12), C32:5C22wrC2, C3:5(D6:D4), D6:Dic3:17C2, C3:1(C24:4S3), C6.D4:13S3, (C6xDic3):3C22, (C22xS3).70D6, (C22xC6).119D6, C22:4(C3:D12), (C2xC62).38C22, (S3xC22xC6):2C2, (C2xC6):5(C3:D4), C2.43(S3xC3:D4), C6.23(C2xC3:D4), C22.142(C2xS32), (C3xC6).165(C2xD4), (C2xC32:7D4):3C2, (S3xC2xC6).85C22, (C2xC3:D12):11C2, (C3xC6.D4):9C2, C2.24(C2xC3:D12), (C22xC3:S3):2C22, (C2xC3:Dic3):5C22, (C2xC6).138(C22xS3), SmallGroup(288,625)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62:5D4
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xC3:D12 — C62:5D4
Lower central
C32 — C62 — C62:5D4
Upper central
C1 — C22 — C23

Generators and relations for C62:5D4
 G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, cac-1=dad=a-1b3, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 1186 in 287 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2xC4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2xC6, C2xC6, C2xC6, C22:C4, C2xD4, C24, C3xS3, C3:S3, C3xC6, C3xC6, C3xC6, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C22xS3, C22xS3, C22xC6, C22xC6, C22wrC2, C3xDic3, C3:Dic3, S3xC6, S3xC6, C2xC3:S3, C62, C62, C62, D6:C4, C6.D4, C6.D4, C3xC22:C4, C2xD12, C2xC3:D4, S3xC23, C23xC6, C3:D12, C6xDic3, C2xC3:Dic3, C32:7D4, S3xC2xC6, S3xC2xC6, C22xC3:S3, C2xC62, D6:D4, C24:4S3, D6:Dic3, C3xC6.D4, C2xC3:D12, C2xC32:7D4, S3xC22xC6, C62:5D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, D12, C3:D4, C22xS3, C22wrC2, S32, C2xD12, S3xD4, C2xC3:D4, C3:D12, C2xS32, D6:D4, C24:4S3, C2xC3:D12, S3xC3:D4, C62:5D4

Smallest permutation representation of C62:5D4
►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 31 3 33 5 35)(2 32 4 34 6 36)(7 14 9 16 11 18)(8 15 10 17 12 13)(19 40 23 38 21 42)(20 41 24 39 22 37)(25 46 29 44 27 48)(26 47 30 45 28 43)

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6J6K···6S6T···6AA12A12B12C12D
order122222222223334446···66···66···612121212
size1111226666362241212362···24···46···612121212

48 irreducible representations

dim111111222222222244444
type++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C3:D4D12C3:D4S32S3xD4C3:D12C2xS32S3xC3:D4
kernelC62:5D4D6:Dic3C3xC6.D4C2xC3:D12C2xC32:7D4S3xC22xC6C6.D4S3xC23S3xC6C62C2xDic3C22xS3C22xC6D6C2xC6C2xC6C23C6C22C22C2
# reps121211114222284412214

Matrix representation of C62:5D4 ►in GL8(Z)

0-1000000
-10000000
00100000
00010000
00001000
00000100
000000-11
000000-10
,
-10000000
0-1000000
00100000
00010000
0000-1100
0000-1000
00000010
00000001
,
0-1000000
10000000
00-1-20000
00110000
0000-1000
00000-100
00000001
00000010
,
-10000000
01000000
00-100000
00110000
00000100
00001000
00000001
00000010

G:=sub<GL(8,Integers())| [0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,0,0,0,0,0,0,1,0],[-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,-1,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62:5D4 in GAP, Magma, Sage, TeX 

C_6^2\rtimes_5D_4
 
% in TeX
 
G:=Group("C6^2:5D4");
 
// GroupNames label
 
G:=SmallGroup(288,625);
 
// by ID
 
G=gap.SmallGroup(288,625);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^3,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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