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G = C62.53C23order 288 = 25·32

48th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.53C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C2×C6.D6 — C62.53C23
 Lower central C32 — C3×C6 — C62.53C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.53C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 786 in 211 conjugacy classes, 66 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×8], C22, C22 [×6], S3 [×12], C6 [×6], C6 [×3], C2×C4, C2×C4 [×13], C23, C32, Dic3 [×4], Dic3 [×6], C12 [×10], D6 [×18], C2×C6 [×2], C2×C6, C4⋊C4 [×4], C22×C4 [×3], C3⋊S3 [×4], C3×C6 [×3], C4×S3 [×20], C2×Dic3 [×4], C2×Dic3 [×3], C2×C12 [×2], C2×C12 [×5], C22×S3 [×3], C2×C4⋊C4, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3, C3×C12, C2×C3⋊S3 [×6], C62, Dic3⋊C4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], C3×C4⋊C4 [×2], S3×C2×C4 [×7], C6.D6 [×4], C6.D6 [×2], C6×Dic3 [×4], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, S3×C4⋊C4 [×2], Dic3⋊Dic3 [×2], C3×Dic3⋊C4 [×2], C2×C6.D6 [×2], C2×C4×C3⋊S3, C62.53C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×6], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×4], C22×S3 [×2], C2×C4⋊C4, S32, S3×C2×C4 [×2], S3×D4 [×2], S3×Q8 [×2], C2×S32, S3×C4⋊C4 [×2], Dic3.D6, C4×S32, Dic3⋊D6, C62.53C23

Smallest permutation representation of C62.53C23
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 36 16 41)(2 31 17 42)(3 32 18 37)(4 33 13 38)(5 34 14 39)(6 35 15 40)(7 21 45 30)(8 22 46 25)(9 23 47 26)(10 24 48 27)(11 19 43 28)(12 20 44 29)
(1 41 16 36)(2 40 17 35)(3 39 18 34)(4 38 13 33)(5 37 14 32)(6 42 15 31)(7 30 45 21)(8 29 46 20)(9 28 47 19)(10 27 48 24)(11 26 43 23)(12 25 44 22)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 33 10 36)(8 34 11 31)(9 35 12 32)(13 27 16 30)(14 28 17 25)(15 29 18 26)(37 47 40 44)(38 48 41 45)(39 43 42 46)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 3C 4A 4B 4C ··· 4J 4K 4L 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 2 2 2 2 3 3 3 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 9 9 9 9 2 2 4 2 2 6 ··· 6 18 18 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + - + + + + - + + image C1 C2 C2 C2 C2 C4 S3 D4 Q8 D6 D6 C4×S3 S32 S3×D4 S3×Q8 C2×S32 Dic3.D6 C4×S32 Dic3⋊D6 kernel C62.53C23 Dic3⋊Dic3 C3×Dic3⋊C4 C2×C6.D6 C2×C4×C3⋊S3 C6.D6 Dic3⋊C4 C2×C3⋊S3 C2×C3⋊S3 C2×Dic3 C2×C12 Dic3 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 2 2 1 8 2 2 2 4 2 8 1 2 2 1 2 2 2

Matrix representation of C62.53C23 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 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 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 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
,
 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 7 1 0 0 0 0 0 0 2 6 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 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 6 12 0 0 0 0 0 0 11 7 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 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 12 3 0 0 0 0 0 0 8 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1

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

C62.53C23 in GAP, Magma, Sage, TeX

`C_6^2._{53}C_2^3`
`% in TeX`

`G:=Group("C6^2.53C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,531);`
`// by ID`

`G=gap.SmallGroup(288,531);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,120,422,219,58,1356,9414]);`
`// Polycyclic`

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

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