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G = C62.82C23  order 288 = 25·32

77th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.82C23, D6:C4:11S3, C6.52(S3xD4), C3:Dic3:12D4, C3:2(Dic3:D4), (C2xC12).202D6, C32:9(C4:D4), C6.34(C4oD12), (C2xDic3).33D6, C62.C22:4C2, (C22xS3).17D6, C2.15(D6:D6), C2.12(Dic3:D6), (C6xC12).238C22, C2.18(D6.D6), (C6xDic3).17C22, (C2xC4).97S32, (C2xC3:S3):9D4, (C3xD6:C4):8C2, (C2xD6:S3):4C2, (C2xC3:D12):5C2, C22.120(C2xS32), (C3xC6).108(C2xD4), (S3xC2xC6).32C22, (C3xC6).51(C4oD4), (C2xC6).101(C22xS3), (C22xC3:S3).73C22, (C2xC3:Dic3).136C22, (C2xC4xC3:S3):14C2, SmallGroup(288,560)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.82C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — S3xC2xC6 — C2xD6:S3 — C62.82C23
Lower central
C32 — C62 — C62.82C23
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.82C23
 G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 978 in 215 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, D4, C23, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xD4, C3xS3, C3:S3, C3xC6, C4xS3, D12, C2xDic3, C2xDic3, C3:D4, C2xC12, C2xC12, C22xS3, C22xS3, C22xC6, C4:D4, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C62, Dic3:C4, D6:C4, C3xC22:C4, S3xC2xC4, C2xD12, C2xC3:D4, D6:S3, C3:D12, C6xDic3, C4xC3:S3, C2xC3:Dic3, C6xC12, S3xC2xC6, C22xC3:S3, Dic3:D4, C62.C22, C3xD6:C4, C2xD6:S3, C2xC3:D12, C2xC4xC3:S3, C62.82C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2xD4, C4oD4, C22xS3, C4:D4, S32, C4oD12, S3xD4, C2xS32, Dic3:D4, D6.D6, D6:D6, Dic3:D6, C62.82C23

Smallest permutation representation of C62.82C23
►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 47 9 43 11 45)(8 48 10 44 12 46)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim11111122222222444444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C4oD4C4oD12S32S3xD4C2xS32D6.D6D6:D6Dic3:D6
kernelC62.82C23C62.C22C3xD6:C4C2xD6:S3C2xC3:D12C2xC4xC3:S3D6:C4C3:Dic3C2xC3:S3C2xDic3C2xC12C22xS3C3xC6C6C2xC4C6C22C2C2C2
# reps11212122222228141222

Matrix representation of C62.82C23 ►in GL8(Z)

-10000000
0-1000000
00100000
00010000
0000-1100
0000-1000
00000010
00000001
,
-10000000
0-1000000
00010000
00-1-10000
00001000
00000100
000000-10
0000000-1
,
0-1000000
-10000000
00-100000
00110000
0000-1000
00000-100
000000-12
00000001
,
-10000000
01000000
00100000
00010000
00000100
00001000
0000001-2
0000000-1
,
01000000
-10000000
00-100000
000-10000
0000-1000
00000-100
00000010
0000001-1

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

C62.82C23 in GAP, Magma, Sage, TeX 

C_6^2._{82}C_2^3
 
% in TeX
 
G:=Group("C6^2.82C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,560);
 
// by ID
 
G=gap.SmallGroup(288,560);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,422,219,58,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=1,e^2=a^3,a*b=b*a,a*c=c*a,d*a*d=a^-1,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,d*c*d=a^3*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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