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G = C62.125C23  order 288 = 25·32

120th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.125C23, C23⋊2S32, (S3×C6)⋊7D4, (C22×C6)⋊6D6, D6⋊6(C3⋊D4), (C2×Dic3)⋊6D6, C6.173(S3×D4), C32⋊7C22≀C2, C3⋊3(C23⋊2D6), D6⋊Dic3⋊37C2, (C2×C62)⋊3C22, (C22×S3).52D6, C6.D12⋊23C2, C2.35(Dic3⋊D6), (C6×Dic3)⋊16C22, (C2×C3⋊S3)⋊5D4, (C22×S32)⋊3C2, (C2×C3⋊D4)⋊9S3, (C6×C3⋊D4)⋊14C2, C6.68(C2×C3⋊D4), C2.46(S3×C3⋊D4), C22.148(C2×S32), (C3×C6).171(C2×D4), (C2×C32⋊7D4)⋊6C2, (S3×C2×C6).51C22, (C2×C3⋊Dic3)⋊6C22, (C2×C6).144(C22×S3), (C22×C3⋊S3).36C22, SmallGroup(288,631)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.125C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C22×S32 — C62.125C23
Lower central
C32 — C62 — C62.125C23
Upper central
C1 — C22 — C23

Generators and relations for C62.125C23
 G = < a,b,c,d,e | a6=b6=c2=d2=e2=1, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=b3c, ede=a3d >

Subgroups: 1346 in 287 conjugacy classes, 54 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22≀C2, C3×Dic3, C3⋊Dic3, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, D6⋊C4, C6.D4, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×C23, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, C2×S32, S3×C2×C6, C22×C3⋊S3, C2×C62, C23⋊2D6, D6⋊Dic3, C6.D12, C6×C3⋊D4, C2×C32⋊7D4, C22×S32, C62.125C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C22≀C2, S32, S3×D4, C2×C3⋊D4, C2×S32, C23⋊2D6, S3×C3⋊D4, Dic3⋊D6, C62.125C23

Smallest permutation representation of C62.125C23
►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)(2 42)(3 37)(4 38)(5 39)(6 40)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 48)(20 43)(21 44)(22 45)(23 46)(24 47)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J3A3B3C4A4B4C6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222222223334446···66···6666612121212
size11114666618182241212362···24···41212121212121212

42 irreducible representations

dim111111222222244444
type++++++++++++++++
imageC1C2C2C2C2C2S3D4D4D6D6D6C3⋊D4S32S3×D4C2×S32S3×C3⋊D4Dic3⋊D6
kernelC62.125C23D6⋊Dic3C6.D12C6×C3⋊D4C2×C32⋊7D4C22×S32C2×C3⋊D4S3×C6C2×C3⋊S3C2×Dic3C22×S3C22×C6D6C23C6C22C2C2
# reps121211242222814142

Matrix representation of C62.125C23 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
000012100
000012000
000000120
000000012
,
120000000
012000000
00010000
0012120000
00001000
00000100
000000120
000000012
,
10000000
112000000
00100000
0012120000
00001000
00000100
00000001
00000010
,
120000000
012000000
00100000
00010000
00000100
00001000
000000012
000000120
,
111000000
012000000
001200000
000120000
000012000
000001200
00000063
000000107

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],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[1,0,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,6,10,0,0,0,0,0,0,3,7] >;
 

C62.125C23 in GAP, Magma, Sage, TeX 

C_6^2._{125}C_2^3
 
% in TeX
 
G:=Group("C6^2.125C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,631);
 
// by ID
 
G=gap.SmallGroup(288,631);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,135,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=d^2=e^2=1,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,c*d=d*c,e*c*e=b^3*c,e*d*e=a^3*d>;
 
// generators/relations
 

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