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G = C62.77C23  order 288 = 25·32

72nd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.77C23, Dic32:3C2, D6:C4:17S3, (C2xDic6):6S3, C6.144(S3xD4), D6:Dic3:33C2, (C6xDic6):11C2, (C2xC12).229D6, C6.15(C4oD12), (C2xDic3).31D6, (C3xDic3).10D4, (C22xS3).13D6, C32:6(C4.4D4), C6.D12:18C2, C6.11D12:11C2, C6.11(D4:2S3), C3:2(C12.23D4), (C6xC12).187C22, C6.16(Q8:3S3), Dic3.2(C3:D4), C2.18(D12:S3), C3:6(C23.11D6), C2.17(D6.6D6), (C6xDic3).83C22, (C2xC4).32S32, (C3xD6:C4):13C2, C2.18(S3xC3:D4), C6.39(C2xC3:D4), C22.115(C2xS32), (C3xC6).104(C2xD4), (S3xC2xC6).28C22, (C2xC3:D12).9C2, (C3xC6).47(C4oD4), (C2xC6).96(C22xS3), (C22xC3:S3).22C22, (C2xC3:Dic3).54C22, SmallGroup(288,555)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.77C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — C2xC3:D12 — C62.77C23
Lower central
C32 — C62 — C62.77C23
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.77C23
 G = < a,b,c,d,e | a6=b6=d2=1, c2=e2=b3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd=ece-1=b3c, ede-1=a3d >

Subgroups: 730 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, Q8, C23, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C42, C22:C4, C2xD4, C2xQ8, C3xS3, C3:S3, C3xC6, Dic6, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C3xQ8, C22xS3, C22xS3, C22xC6, C4.4D4, C3xDic3, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C62, C4xDic3, D6:C4, D6:C4, C6.D4, C3xC22:C4, C2xDic6, C2xD12, C2xC3:D4, C6xQ8, C3:D12, C3xDic6, C6xDic3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, C23.11D6, C12.23D4, Dic32, D6:Dic3, C6.D12, C3xD6:C4, C6.11D12, C2xC3:D12, C6xDic6, C62.77C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C3:D4, C22xS3, C4.4D4, S32, C4oD12, S3xD4, D4:2S3, Q8:3S3, C2xC3:D4, C2xS32, C23.11D6, C12.23D4, D12:S3, D6.6D6, S3xC3:D4, C62.77C23

Smallest permutation representation of C62.77C23
►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 15 3 17 5 13)(2 16 4 18 6 14)(7 44 11 48 9 46)(8 45 12 43 10 47)(19 29 21 25 23 27)(20 30 22 26 24 28)(31 37 35 41 33 39)(32 38 36 42 34 40)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I6J6K12A···12H12I···12N
order122222333444444446···66666612···1212···12
size11111236224466661218182···244412124···412···12

42 irreducible representations

dim1111111122222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4oD4C3:D4C4oD12S32S3xD4D4:2S3Q8:3S3C2xS32D12:S3D6.6D6S3xC3:D4
kernelC62.77C23Dic32D6:Dic3C6.D12C3xD6:C4C6.11D12C2xC3:D12C6xDic6D6:C4C2xDic6C3xDic3C2xDic3C2xC12C22xS3C3xC6Dic3C6C2xC4C6C6C6C22C2C2C2
# reps1111111111232144411121222

Matrix representation of C62.77C23 ►in GL8(F13)

120000000
012000000
00100000
00010000
00001000
00000100
000000121
000000120
,
10000000
01000000
001200000
000120000
000012100
000012000
00000010
00000001
,
120000000
012000000
00050000
00500000
000001200
000012000
00000010
00000001
,
120000000
121000000
00080000
00500000
000012000
000001200
00000001
00000010
,
111000000
012000000
00010000
001200000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [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],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.77C23 in GAP, Magma, Sage, TeX 

C_6^2._{77}C_2^3
 
% in TeX
 
G:=Group("C6^2.77C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,555);
 
// by ID
 
G=gap.SmallGroup(288,555);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,422,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=d^2=1,c^2=e^2=b^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d=e*c*e^-1=b^3*c,e*d*e^-1=a^3*d>;
 
// generators/relations
 

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