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G = C62.113C23  order 288 = 25·32

108th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.113C23, (S3×C6)⋊6D4, C3⋊Dic3⋊2D4, C23.16S32, C6.68(S3×D4), D6⋊4(C3⋊D4), C3⋊7(Dic3⋊D4), C6.D4⋊9S3, (C22×C6).73D6, C6.67(C4○D12), C32⋊13(C4⋊D4), (C2×Dic3).45D6, (C22×S3).49D6, C3⋊3(C23.14D6), C2.29(Dic3⋊D6), C6.54(D4⋊2S3), C6.D12⋊19C2, C62.C22⋊11C2, (C2×C62).32C22, C2.28(D6.3D6), (C6×Dic3).26C22, (C6×C3⋊D4)⋊9C2, (C2×C3⋊D4)⋊3S3, (C2×S3×Dic3)⋊20C2, C6.65(C2×C3⋊D4), C2.41(S3×C3⋊D4), C22.139(C2×S32), (C3×C6).159(C2×D4), (C2×C32⋊7D4)⋊2C2, (S3×C2×C6).45C22, (C2×C3⋊D12)⋊10C2, (C3×C6).84(C4○D4), (C3×C6.D4)⋊15C2, (C2×C6).132(C22×S3), (C22×C3⋊S3).32C22, (C2×C3⋊Dic3).69C22, SmallGroup(288,619)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.113C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.113C23
Lower central
C32 — C62 — C62.113C23
Upper central
C1 — C22 — C23

Generators and relations for C62.113C23
 G = < a,b,c,d,e | a6=b6=c2=e2=1, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece=a3b3c, ede=b3d >

Subgroups: 930 in 215 conjugacy classes, 50 normal (44 characteristic)
C1, 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, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C4⋊D4, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C62, C62, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, S3×Dic3, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, Dic3⋊D4, C23.14D6, C6.D12, C62.C22, C3×C6.D4, C2×S3×Dic3, C2×C3⋊D12, C6×C3⋊D4, C2×C32⋊7D4, C62.113C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, S32, C4○D12, S3×D4, D4⋊2S3, C2×C3⋊D4, C2×S32, Dic3⋊D4, C23.14D6, D6.3D6, S3×C3⋊D4, Dic3⋊D6, C62.113C23

Smallest permutation representation of C62.113C23
►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 48)(2 43)(3 44)(4 45)(5 46)(6 47)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(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 33 4 36)(2 32 5 35)(3 31 6 34)(7 30 10 27)(8 29 11 26)(9 28 12 25)(13 41 16 38)(14 40 17 37)(15 39 18 42)(19 44 22 47)(20 43 23 46)(21 48 24 45)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S12A···12F
order122222223334444446···66···66612···12
size11114663622466121218182···24···4121212···12

42 irreducible representations

dim1111111122222222224444444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D4⋊2S3C2×S32D6.3D6S3×C3⋊D4Dic3⋊D6
kernelC62.113C23C6.D12C62.C22C3×C6.D4C2×S3×Dic3C2×C3⋊D12C6×C3⋊D4C2×C32⋊7D4C6.D4C2×C3⋊D4C3⋊Dic3S3×C6C2×Dic3C22×S3C22×C6C3×C6D6C6C23C6C6C22C2C2C2
# reps1111111111223122441311222

Matrix representation of C62.113C23 ►in GL8(𝔽13)

120000000
012000000
00100000
00010000
00001000
00000100
00000001
0000001212
,
120000000
012000000
001200000
000120000
000012100
000012000
00000010
00000001
,
120000000
012000000
00520000
00180000
000001200
000012000
00000010
00000001
,
63000000
57000000
008110000
001250000
00001000
00000100
00000010
0000001212
,
63000000
107000000
001200000
00510000
000012000
000001200
00000010
00000001

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,1,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[6,5,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,8,12,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,12],[6,10,0,0,0,0,0,0,3,7,0,0,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1] >;
 

C62.113C23 in GAP, Magma, Sage, TeX 

C_6^2._{113}C_2^3
 
% in TeX
 
G:=Group("C6^2.113C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,619);
 
// by ID
 
G=gap.SmallGroup(288,619);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,590,219,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^6=c^2=e^2=1,d^2=a^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,c*d=d*c,e*c*e=a^3*b^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

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