Copied to
clipboard

## G = C62.117C23order 288 = 25·32

### 112nd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.117C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.117C23
 Lower central C32 — C62 — C62.117C23
 Upper central C1 — C22 — C23

Generators and relations for C62.117C23
G = < a,b,c,d,e | a6=b6=e2=1, c2=d2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece=a3c, ede=a3b3d >

Subgroups: 770 in 183 conjugacy classes, 46 normal (14 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C22×C6, C22.D4, C3×Dic3, C3⋊Dic3, C2×C3⋊S3, C2×C3⋊S3, C62, C62, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4, C6.D6, C6×Dic3, C2×C3⋊Dic3, C327D4, C22×C3⋊S3, C2×C62, C23.9D6, C6.D12, Dic3⋊Dic3, C3×C6.D4, C2×C6.D6, C2×C327D4, C62.117C23
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, C22×S3, C22.D4, S32, C4○D12, S3×D4, D42S3, C2×S32, C23.9D6, D6.3D6, Dic3⋊D6, C62.117C23

Smallest permutation representation of C62.117C23
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 31 16 38)(2 32 17 39)(3 33 18 40)(4 34 13 41)(5 35 14 42)(6 36 15 37)(7 27 43 20)(8 28 44 21)(9 29 45 22)(10 30 46 23)(11 25 47 24)(12 26 48 19)
(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 20)(2 21)(3 22)(4 23)(5 24)(6 19)(7 34)(8 35)(9 36)(10 31)(11 32)(12 33)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(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), (1,18,5,16,3,14)(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), (1,31,16,38)(2,32,17,39)(3,33,18,40)(4,34,13,41)(5,35,14,42)(6,36,15,37)(7,27,43,20)(8,28,44,21)(9,29,45,22)(10,30,46,23)(11,25,47,24)(12,26,48,19), (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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(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), (1,18,5,16,3,14)(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), (1,31,16,38)(2,32,17,39)(3,33,18,40)(4,34,13,41)(5,35,14,42)(6,36,15,37)(7,27,43,20)(8,28,44,21)(9,29,45,22)(10,30,46,23)(11,25,47,24)(12,26,48,19), (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), (1,20)(2,21)(3,22)(4,23)(5,24)(6,19)(7,34)(8,35)(9,36)(10,31)(11,32)(12,33)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(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)], [(1,18,5,16,3,14),(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)], [(1,31,16,38),(2,32,17,39),(3,33,18,40),(4,34,13,41),(5,35,14,42),(6,36,15,37),(7,27,43,20),(8,28,44,21),(9,29,45,22),(10,30,46,23),(11,25,47,24),(12,26,48,19)], [(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)], [(1,20),(2,21),(3,22),(4,23),(5,24),(6,19),(7,34),(8,35),(9,36),(10,31),(11,32),(12,33),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(37,45),(38,46),(39,47),(40,48),(41,43),(42,44)]])`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G ··· 6Q 12A ··· 12H order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 ··· 6 12 ··· 12 size 1 1 1 1 4 18 18 2 2 4 6 6 6 6 12 12 36 2 ··· 2 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 S3 D4 D6 D6 C4○D4 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D6.3D6 Dic3⋊D6 kernel C62.117C23 C6.D12 Dic3⋊Dic3 C3×C6.D4 C2×C6.D6 C2×C32⋊7D4 C6.D4 C2×C3⋊S3 C2×Dic3 C22×C6 C3×C6 C6 C23 C6 C6 C22 C2 C2 # reps 1 1 2 2 1 1 2 2 4 2 4 8 1 2 2 1 4 2

Matrix representation of C62.117C23 in GL8(𝔽13)

 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 12 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 1 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 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 1 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 8 0 0 0 0 0 0 0 0 8
,
 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 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 5 8
,
 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 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 12

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

C62.117C23 in GAP, Magma, Sage, TeX

`C_6^2._{117}C_2^3`
`% in TeX`

`G:=Group("C6^2.117C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,623);`
`// by ID`

`G=gap.SmallGroup(288,623);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,254,303,1356,9414]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^6=e^2=1,c^2=d^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=a^3*c,e*d*e=a^3*b^3*d>;`
`// generators/relations`

׿
×
𝔽