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G = C62.58C23  order 288 = 25·32

53rd non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.58C23, C6.10(S3xQ8), (C2xDic6):5S3, C6.141(S3xD4), Dic3:C4:18S3, C3:1(D6:3Q8), C3:6(D6:Q8), (C6xDic6):10C2, (C2xC12).227D6, (C3xDic3).7D4, C6.11(C4oD12), C32:9(C22:Q8), (C2xDic3).23D6, Dic3:Dic3:25C2, (C6xC12).183C22, C6.12(Q8:3S3), C62.C22:19C2, C6.D12.8C2, C6.11D12.5C2, Dic3.9(C3:D4), C2.15(D6.6D6), (C6xDic3).37C22, C2.11(Dic3.D6), (C2xC4).24S32, (C2xC3:S3):1Q8, (C3xC6).96(C2xD4), C2.15(S3xC3:D4), C6.35(C2xC3:D4), (C3xC6).28(C2xQ8), C22.105(C2xS32), (C3xDic3:C4):16C2, (C3xC6).35(C4oD4), (C2xC6).77(C22xS3), (C2xC6.D6).3C2, (C22xC3:S3).15C22, (C2xC3:Dic3).42C22, SmallGroup(288,536)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.58C23
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C6xDic3 — C2xC6.D6 — C62.58C23
Lower central
C32 — C62 — C62.58C23
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.58C23
 G = < a,b,c,d,e | a6=b6=1, c2=d2=e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=a3d >

Subgroups: 666 in 165 conjugacy classes, 50 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, Q8, C23, C32, Dic3, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C4:C4, C22xC4, C2xQ8, C3:S3, C3xC6, Dic6, C4xS3, C2xDic3, C2xDic3, C2xC12, C2xC12, C3xQ8, C22xS3, C22:Q8, C3xDic3, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C2xC3:S3, C62, Dic3:C4, Dic3:C4, C4:Dic3, D6:C4, C3xC4:C4, C2xDic6, S3xC2xC4, C6xQ8, C6.D6, C3xDic6, C6xDic3, C2xC3:Dic3, C6xC12, C22xC3:S3, D6:Q8, D6:3Q8, C6.D12, Dic3:Dic3, C62.C22, C3xDic3:C4, C6.11D12, C2xC6.D6, C6xDic6, C62.58C23
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, C2xD4, C2xQ8, C4oD4, C3:D4, C22xS3, C22:Q8, S32, C4oD12, S3xD4, S3xQ8, Q8:3S3, C2xC3:D4, C2xS32, D6:Q8, D6:3Q8, Dic3.D6, D6.6D6, S3xC3:D4, C62.58C23

Smallest permutation representation of C62.58C23
►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 41 16 34)(2 42 17 35)(3 37 18 36)(4 38 13 31)(5 39 14 32)(6 40 15 33)(7 20 43 27)(8 21 44 28)(9 22 45 29)(10 23 46 30)(11 24 47 25)(12 19 48 26)

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

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

42 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A···12H12I···12P
order122222333444444446···666612···1212···12
size11111818224466661212362···24444···412···12

42 irreducible representations

dim1111111122222222244444444
type+++++++++++-++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4Q8D6D6C4oD4C3:D4C4oD12S32S3xD4S3xQ8Q8:3S3C2xS32Dic3.D6D6.6D6S3xC3:D4
kernelC62.58C23C6.D12Dic3:Dic3C62.C22C3xDic3:C4C6.11D12C2xC6.D6C6xDic6Dic3:C4C2xDic6C3xDic3C2xC3:S3C2xDic3C2xC12C3xC6Dic3C6C2xC4C6C6C6C22C2C2C2
# reps1111111111224224411211222

Matrix representation of C62.58C23 ►in GL8(F13)

10000000
01000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
001210000
001200000
00001000
00000100
00000010
00000001
,
50000000
08000000
00010000
00100000
00001000
00000100
000000120
000000012
,
50000000
05000000
00100000
00010000
00000100
00001000
000000120
00000001
,
01000000
120000000
001200000
000120000
000012000
000001200
00000001
00000010

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

C62.58C23 in GAP, Magma, Sage, TeX 

C_6^2._{58}C_2^3
 
% in TeX
 
G:=Group("C6^2.58C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,536);
 
// by ID
 
G=gap.SmallGroup(288,536);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,135,100,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=1,c^2=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^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=b^3*c,e*d*e^-1=a^3*d>;
 
// generators/relations
 

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