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G = C62.100C23  order 288 = 25·32

95th non-split extension by C62 of C23 acting via C23/C2=C22

metabelian, supersoluble, monomial

Aliases: C62.100C23, C23.12S32, (C3×Dic3)⋊3D4, C6.163(S3×D4), C3⋊6(Dic3⋊D4), C3⋊2(D6⋊3D4), D6⋊Dic3⋊25C2, C6.D4⋊4S3, (C22×C6).65D6, C6.61(C4○D12), Dic3⋊2(C3⋊D4), C32⋊11(C4⋊D4), (C2×Dic3).39D6, (C22×S3).22D6, Dic3⋊Dic3⋊26C2, C2.28(Dic3⋊D6), C6.48(D4⋊2S3), (C2×C62).19C22, C2.22(D6.3D6), (C6×Dic3).40C22, (C2×C3⋊S3)⋊4D4, (C2×C3⋊D4)⋊1S3, (C6×C3⋊D4)⋊1C2, C6.59(C2×C3⋊D4), C2.36(S3×C3⋊D4), (C2×C3⋊D12)⋊9C2, C22.131(C2×S32), (C2×C6.D6)⋊2C2, (C3×C6).146(C2×D4), (C2×C32⋊7D4)⋊1C2, (S3×C2×C6).39C22, (C3×C6).76(C4○D4), (C3×C6.D4)⋊12C2, (C2×C6).119(C22×S3), (C22×C3⋊S3).30C22, (C2×C3⋊Dic3).62C22, SmallGroup(288,606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — C62.100C23
Chief series
C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.100C23
Lower central
C32 — C62 — C62.100C23
Upper central
C1 — C22 — C23

Generators and relations for C62.100C23
 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, dcd-1=b3c, ece=a3c, 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, Dic3, C12, 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, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C6.D6, C3⋊D12, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, Dic3⋊D4, D6⋊3D4, D6⋊Dic3, Dic3⋊Dic3, C3×C6.D4, C2×C6.D6, C2×C3⋊D12, C6×C3⋊D4, C2×C32⋊7D4, C62.100C23
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, D6⋊3D4, D6.3D6, S3×C3⋊D4, Dic3⋊D6, C62.100C23

Smallest permutation representation of C62.100C23
►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 36)(20 31)(21 32)(22 33)(23 34)(24 35)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G···6Q6R6S12A···12F
order122222223334444446···66···66612···12
size11114121818224666612362···24···4121212···12

42 irreducible representations

dim1111111122222222224444444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2S3S3D4D4D6D6D6C4○D4C3⋊D4C4○D12S32S3×D4D4⋊2S3C2×S32D6.3D6S3×C3⋊D4Dic3⋊D6
kernelC62.100C23D6⋊Dic3Dic3⋊Dic3C3×C6.D4C2×C6.D6C2×C3⋊D12C6×C3⋊D4C2×C32⋊7D4C6.D4C2×C3⋊D4C3×Dic3C2×C3⋊S3C2×Dic3C22×S3C22×C6C3×C6Dic3C6C23C6C6C22C2C2C2
# reps1111111111223122441311222

Matrix representation of C62.100C23 ►in GL8(𝔽13)

10000000
01000000
00100000
00010000
00000100
0000121200
000000120
000000012
,
120000000
012000000
001210000
001200000
00001000
00000100
000000120
000000012
,
610000000
37000000
000120000
001200000
000012000
000001200
000000012
000000120
,
10000000
412000000
001200000
000120000
000012000
00001100
00000001
000000120
,
610000000
37000000
00100000
00010000
00001000
00000100
00000010
000000012

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0],[1,4,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0],[6,3,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12] >;
 

C62.100C23 in GAP, Magma, Sage, TeX 

C_6^2._{100}C_2^3
 
% in TeX
 
G:=Group("C6^2.100C2^3");
 
// GroupNames label
 
G:=SmallGroup(288,606);
 
// by ID
 
G=gap.SmallGroup(288,606);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,254,219,1356,9414]);
 
// 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,d*c*d^-1=b^3*c,e*c*e=a^3*c,e*d*e=b^3*d>;
 
// generators/relations
 

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