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G = C62.51C23  order 288 = 25·32

46th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.51C23, D6:C4:9S3, D6:2(C4xS3), Dic32:1C2, C32:4(C4xD4), C6.48(S3xD4), C3:D12:2C4, C3:Dic3:11D4, Dic3:1(C4xS3), Dic3:C4:13S3, (C2xC12).198D6, C3:2(Dic3:5D4), C6.7(D4:2S3), C2.1(Dic3:D6), (C2xDic3).64D6, (C22xS3).33D6, C3:2(Dic3:4D4), C2.4(D12:S3), C6.D12:12C2, (C6xC12).229C22, C6.30(Q8:3S3), (C6xDic3).59C22, C2.17(C4xS32), (C2xC4).94S32, C6.16(S3xC2xC4), (S3xC6):3(C2xC4), (C2xS3xDic3):9C2, (C3xD6:C4):22C2, C22.31(C2xS32), (C3xC6).92(C2xD4), (C3xDic3):2(C2xC4), (S3xC2xC6).14C22, (C3xDic3:C4):12C2, (C2xC3:D12).7C2, (C3xC6).30(C4oD4), (C3xC6).15(C22xC4), (C2xC6).70(C22xS3), (C22xC3:S3).68C22, (C2xC3:Dic3).128C22, (C2xC4xC3:S3):12C2, (C2xC3:S3):7(C2xC4), SmallGroup(288,529)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C62.51C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xS3xDic3 — C62.51C23
Lower central
C32 — C3xC6 — C62.51C23
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.51C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, ede-1=a3d >

Subgroups: 850 in 215 conjugacy classes, 60 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, Dic3, C12, D6, D6, C2xC6, C2xC6, C42, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C4xD4, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, S3xC6, C2xC3:S3, C2xC3:S3, C62, C4xDic3, Dic3:C4, D6:C4, D6:C4, C3xC22:C4, C3xC4:C4, S3xC2xC4, C2xD12, C22xDic3, C2xC3:D4, S3xDic3, C3:D12, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, Dic3:4D4, Dic3:5D4, Dic32, C6.D12, C3xDic3:C4, C3xD6:C4, C2xS3xDic3, C2xC3:D12, C2xC4xC3:S3, C62.51C23
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, C23, D6, C22xC4, C2xD4, C4oD4, C4xS3, C22xS3, C4xD4, S32, S3xC2xC4, S3xD4, D4:2S3, Q8:3S3, C2xS32, Dic3:4D4, Dic3:5D4, D12:S3, C4xS32, Dic3:D6, C62.51C23

Smallest permutation representation of C62.51C23
►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 18 5 16 3 14)(2 13 6 17 4 15)(7 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C···4H4I4J4K4L6A···6F6G6H6I6J6K12A···12H12I···12N
order12222222333444···444446···66666612···1212···12
size1111661818224226···699992···244412124···412···12

48 irreducible representations

dim11111111122222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2C4S3S3D4D6D6D6C4oD4C4xS3C4xS3S32S3xD4D4:2S3Q8:3S3C2xS32D12:S3C4xS32Dic3:D6
kernelC62.51C23Dic32C6.D12C3xDic3:C4C3xD6:C4C2xS3xDic3C2xC3:D12C2xC4xC3:S3C3:D12Dic3:C4D6:C4C3:Dic3C2xDic3C2xC12C22xS3C3xC6Dic3D6C2xC4C6C6C6C22C2C2C2
# reps11111111811232124412111222

Matrix representation of C62.51C23 ►in GL8(F13)

10000000
01000000
001200000
000120000
000012000
000001200
00000001
0000001212
,
121000000
120000000
00100000
00010000
000012000
000001200
00000010
00000001
,
012000000
120000000
00010000
00100000
00000800
00005000
00000010
00000001
,
10000000
01000000
00010000
00100000
00000100
000012000
000000120
00000011
,
120000000
012000000
00100000
000120000
00005000
00000800
000000120
000000012

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

C62.51C23 in GAP, Magma, Sage, TeX 

C_6^2._{51}C_2^3
 
% in TeX
 
G:=Group("C6^2.51C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,529);
 
// by ID
 
G=gap.SmallGroup(288,529);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=e^2=b^3,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*c,e*d*e^-1=a^3*d>;
 
// generators/relations
 

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