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## G = C62.100C23order 288 = 25·32

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

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 [×3], C2 [×4], C3 [×2], C3, C4 [×5], C22, C22 [×10], S3 [×7], C6 [×6], C6 [×8], C2×C4 [×6], D4 [×6], C23, C23 [×2], C32, Dic3 [×2], Dic3 [×6], C12 [×4], D6 [×15], C2×C6 [×2], C2×C6 [×14], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3, C3⋊S3 [×2], C3×C6 [×3], C3×C6, C4×S3 [×4], D12 [×2], C2×Dic3 [×3], C2×Dic3 [×3], C3⋊D4 [×12], C2×C12 [×3], C3×D4 [×2], C22×S3, C22×S3 [×3], C22×C6 [×2], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3, S3×C6 [×3], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×3], Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, S3×C2×C4 [×2], C2×D12, C2×C3⋊D4, C2×C3⋊D4 [×4], C6×D4, C6.D6 [×2], C3⋊D12 [×2], C6×Dic3 [×3], C3×C3⋊D4 [×2], C2×C3⋊Dic3, C327D4 [×2], S3×C2×C6, C22×C3⋊S3, C2×C62, Dic3⋊D4, D63D4, D6⋊Dic3, Dic3⋊Dic3, C3×C6.D4, C2×C6.D6, C2×C3⋊D12, C6×C3⋊D4, C2×C327D4, C62.100C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C4○D12, S3×D4 [×3], D42S3, C2×C3⋊D4, C2×S32, Dic3⋊D4, D63D4, 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 43 9 45 11 47)(8 44 10 46 12 48)(19 30 23 28 21 26)(20 25 24 29 22 27)(31 40 33 42 35 38)(32 41 34 37 36 39)
(1 10)(2 11)(3 12)(4 7)(5 8)(6 9)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 34)(20 35)(21 36)(22 31)(23 32)(24 33)(25 42)(26 37)(27 38)(28 39)(29 40)(30 41)
(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 33)(8 34)(9 35)(10 36)(11 31)(12 32)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(37 44)(38 45)(39 46)(40 47)(41 48)(42 43)```

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 S3 S3 D4 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D6.3D6 S3×C3⋊D4 Dic3⋊D6 kernel C62.100C23 D6⋊Dic3 Dic3⋊Dic3 C3×C6.D4 C2×C6.D6 C2×C3⋊D12 C6×C3⋊D4 C2×C32⋊7D4 C6.D4 C2×C3⋊D4 C3×Dic3 C2×C3⋊S3 C2×Dic3 C22×S3 C22×C6 C3×C6 Dic3 C6 C23 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 3 1 2 2 4 4 1 3 1 1 2 2 2

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

 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 1 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 0 0 12 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
,
 6 10 0 0 0 0 0 0 3 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 0 0 0 0 0 0 0 4 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 0 1 0 0 0 0 0 0 12 0
,
 6 10 0 0 0 0 0 0 3 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

`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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