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G = C62.23C23  order 288 = 25·32

18th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.23C23, D6⋊C4⋊8S3, C6.46(S3×D4), Dic3⋊C4⋊9S3, D6⋊Dic3⋊30C2, (C2×C12).188D6, (C2×Dic3).9D6, (C22×S3).2D6, Dic3⋊Dic3⋊1C2, C6.21(C4○D12), C2.7(Dic3⋊D6), C3⋊1(C23.9D6), C6.3(D4⋊2S3), C3⋊2(D6.D4), C2.8(D12⋊S3), C6.D12⋊10C2, (C6×C12).214C22, C6.23(Q8⋊3S3), C2.9(D6.D6), (C6×Dic3).5C22, C32⋊3(C22.D4), (C2×C4).89S32, (C3×D6⋊C4)⋊3C2, (C2×C3⋊S3).53D4, C22.81(C2×S32), (C3×C6).80(C2×D4), (S3×C2×C6).2C22, (C3×Dic3⋊C4)⋊5C2, (C2×C3⋊D12).3C2, (C3×C6).11(C4○D4), (C2×C6).42(C22×S3), (C22×C3⋊S3).64C22, (C2×C3⋊Dic3).116C22, (C2×C4×C3⋊S3)⋊11C2, SmallGroup(288,501)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.23C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.23C23
Lower central
C32 — C62 — C62.23C23
Upper central
C1 — C22 — C2×C4

Generators and relations for C62.23C23
 G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 183 conjugacy classes, 46 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C3⋊D12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C23.9D6, D6.D4, D6⋊Dic3, C6.D12, Dic3⋊Dic3, C3×Dic3⋊C4, C3×D6⋊C4, C2×C3⋊D12, C2×C4×C3⋊S3, C62.23C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, S32, C4○D12, S3×D4, D4⋊2S3, Q8⋊3S3, C2×S32, C23.9D6, D6.D4, D12⋊S3, D6.D6, Dic3⋊D6, C62.23C23

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G6A···6F6G6H6I6J6K12A···12H12I···12N
order122222233344444446···66666612···1212···12
size11111218182242212121218182···244412124···412···12

42 irreducible representations

dim111111112222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2C2C2S3S3D4D6D6D6C4○D4C4○D12S32S3×D4D4⋊2S3Q8⋊3S3C2×S32D12⋊S3D6.D6Dic3⋊D6
kernelC62.23C23D6⋊Dic3C6.D12Dic3⋊Dic3C3×Dic3⋊C4C3×D6⋊C4C2×C3⋊D12C2×C4×C3⋊S3Dic3⋊C4D6⋊C4C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C6C2×C4C6C6C6C22C2C2C2
# reps111111111123214812111222

Matrix representation of C62.23C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
000012100
000012000
000000120
000000012
,
10000000
01000000
001210000
001200000
00001000
00000100
000000120
000000012
,
85000000
35000000
000120000
001200000
000012000
000001200
00000073
000000106
,
120000000
111000000
00100000
00010000
00000100
00001000
00000037
000000610
,
80000000
08000000
001200000
000120000
000012000
000001200
00000001
00000010

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[8,3,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,7,10,0,0,0,0,0,0,3,6],[12,11,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,0,0,7,10],[8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;
 

C62.23C23 in GAP, Magma, Sage, TeX 

C_6^2._{23}C_2^3
 
% in TeX
 
G:=Group("C6^2.23C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,501);
 
// by ID
 
G=gap.SmallGroup(288,501);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,120,422,219,142,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=1,d^2=b^3,e^2=a^3*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,d*c*d^-1=a^3*c,e*c*e^-1=b^3*c,e*d*e^-1=b^3*d>;
 
// generators/relations
 

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