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## G = C62.5D6order 432 = 24·33

### 5th non-split extension by C62 of D6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C2×He3 — C62.5D6
 Chief series C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C2×C32⋊C12 — C62.5D6
 Lower central He3 — C2×He3 — C62.5D6
 Upper central C1 — C22

Generators and relations for C62.5D6
G = < a,b,c,d | a6=b6=d2=1, c6=a3b3, ab=ba, cac-1=dad=a-1b4, cbc-1=b-1, bd=db, dcd=b3c5 >

Subgroups: 839 in 151 conjugacy classes, 33 normal (11 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C62, C62, D6⋊C4, C6.D4, He3⋊C2, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C32⋊C12, C2×He3⋊C2, C2×He3⋊C2, C22×He3, D6⋊Dic3, C2×C32⋊C12, C22×He3⋊C2, C62.5D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, S32, D6⋊C4, C6.D6, C3⋊D12, C32⋊D6, C6.D12, He3⋊(C2×C4), He33D4, C62.5D6

Smallest permutation representation of C62.5D6
On 72 points
Generators in S72
```(1 35 44 59 48 31)(2 52 61 60 6 17)(3 57 70 49 11 14)(4 34 39 50 47 26)(5 20 9 51 64 55)(7 29 38 53 42 25)(8 58 67 54 12 23)(10 28 45 56 41 32)(13 37 21 69 36 65)(15 63 30 71 19 43)(16 68 27 72 24 40)(18 46 22 62 33 66)
(1 14 40 53 64 33)(2 34 65 54 41 15)(3 16 42 55 66 35)(4 36 67 56 43 17)(5 18 44 57 68 25)(6 26 69 58 45 19)(7 20 46 59 70 27)(8 28 71 60 47 21)(9 22 48 49 72 29)(10 30 61 50 37 23)(11 24 38 51 62 31)(12 32 63 52 39 13)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)
(1 55)(2 8)(3 53)(4 6)(5 51)(7 49)(9 59)(10 12)(11 57)(13 23)(14 66)(15 21)(16 64)(17 19)(18 62)(20 72)(22 70)(24 68)(25 38)(26 36)(27 48)(28 34)(29 46)(30 32)(31 44)(33 42)(35 40)(37 39)(41 47)(43 45)(50 52)(54 60)(56 58)(61 63)(65 71)(67 69)```

`G:=sub<Sym(72)| (1,35,44,59,48,31)(2,52,61,60,6,17)(3,57,70,49,11,14)(4,34,39,50,47,26)(5,20,9,51,64,55)(7,29,38,53,42,25)(8,58,67,54,12,23)(10,28,45,56,41,32)(13,37,21,69,36,65)(15,63,30,71,19,43)(16,68,27,72,24,40)(18,46,22,62,33,66), (1,14,40,53,64,33)(2,34,65,54,41,15)(3,16,42,55,66,35)(4,36,67,56,43,17)(5,18,44,57,68,25)(6,26,69,58,45,19)(7,20,46,59,70,27)(8,28,71,60,47,21)(9,22,48,49,72,29)(10,30,61,50,37,23)(11,24,38,51,62,31)(12,32,63,52,39,13), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,55)(2,8)(3,53)(4,6)(5,51)(7,49)(9,59)(10,12)(11,57)(13,23)(14,66)(15,21)(16,64)(17,19)(18,62)(20,72)(22,70)(24,68)(25,38)(26,36)(27,48)(28,34)(29,46)(30,32)(31,44)(33,42)(35,40)(37,39)(41,47)(43,45)(50,52)(54,60)(56,58)(61,63)(65,71)(67,69)>;`

`G:=Group( (1,35,44,59,48,31)(2,52,61,60,6,17)(3,57,70,49,11,14)(4,34,39,50,47,26)(5,20,9,51,64,55)(7,29,38,53,42,25)(8,58,67,54,12,23)(10,28,45,56,41,32)(13,37,21,69,36,65)(15,63,30,71,19,43)(16,68,27,72,24,40)(18,46,22,62,33,66), (1,14,40,53,64,33)(2,34,65,54,41,15)(3,16,42,55,66,35)(4,36,67,56,43,17)(5,18,44,57,68,25)(6,26,69,58,45,19)(7,20,46,59,70,27)(8,28,71,60,47,21)(9,22,48,49,72,29)(10,30,61,50,37,23)(11,24,38,51,62,31)(12,32,63,52,39,13), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72), (1,55)(2,8)(3,53)(4,6)(5,51)(7,49)(9,59)(10,12)(11,57)(13,23)(14,66)(15,21)(16,64)(17,19)(18,62)(20,72)(22,70)(24,68)(25,38)(26,36)(27,48)(28,34)(29,46)(30,32)(31,44)(33,42)(35,40)(37,39)(41,47)(43,45)(50,52)(54,60)(56,58)(61,63)(65,71)(67,69) );`

`G=PermutationGroup([[(1,35,44,59,48,31),(2,52,61,60,6,17),(3,57,70,49,11,14),(4,34,39,50,47,26),(5,20,9,51,64,55),(7,29,38,53,42,25),(8,58,67,54,12,23),(10,28,45,56,41,32),(13,37,21,69,36,65),(15,63,30,71,19,43),(16,68,27,72,24,40),(18,46,22,62,33,66)], [(1,14,40,53,64,33),(2,34,65,54,41,15),(3,16,42,55,66,35),(4,36,67,56,43,17),(5,18,44,57,68,25),(6,26,69,58,45,19),(7,20,46,59,70,27),(8,28,71,60,47,21),(9,22,48,49,72,29),(10,30,61,50,37,23),(11,24,38,51,62,31),(12,32,63,52,39,13)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72)], [(1,55),(2,8),(3,53),(4,6),(5,51),(7,49),(9,59),(10,12),(11,57),(13,23),(14,66),(15,21),(16,64),(17,19),(18,62),(20,72),(22,70),(24,68),(25,38),(26,36),(27,48),(28,34),(29,46),(30,32),(31,44),(33,42),(35,40),(37,39),(41,47),(43,45),(50,52),(54,60),(56,58),(61,63),(65,71),(67,69)]])`

38 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A 6B 6C 6D ··· 6I 6J 6K 6L 6M 6N 6O 6P 12A ··· 12H order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 6 6 6 ··· 6 6 6 6 6 6 6 6 12 ··· 12 size 1 1 1 1 18 18 2 6 6 12 18 18 18 18 2 2 2 6 ··· 6 12 12 12 18 18 18 18 18 ··· 18

38 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 4 4 4 6 6 6 type + + + + + + + + + + + - image C1 C2 C2 C4 S3 D4 D6 C4×S3 D12 C3⋊D4 S32 C6.D6 C3⋊D12 C32⋊D6 He3⋊(C2×C4) He3⋊3D4 kernel C62.5D6 C2×C32⋊C12 C22×He3⋊C2 C2×He3⋊C2 C2×C3⋊Dic3 C2×He3 C62 C3×C6 C3×C6 C3×C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 2 1 4 2 2 2 4 4 4 1 1 2 2 2 4

Matrix representation of C62.5D6 in GL10(𝔽13)

 1 12 0 0 0 0 0 0 0 0 1 0 0 0 0 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 0 0 0 0 0 9 0 0 0 0 0 0 0 0 12 12 2 0 0 0 0 0 0 0 1 4 1 0 0 0 0 0 0 0 0 0 0 4 4 5 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 10 9 9
,
 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 10
,
 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 12 12 2 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 11 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0
,
 12 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 12 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12 12 2 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 11 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 12

`G:=sub<GL(10,GF(13))| [1,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,9,12,4,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,4,3,10,0,0,0,0,0,0,0,4,0,9,0,0,0,0,0,0,0,5,0,9],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,10],[0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,0,12,0,0,0,0,12,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,2,0,1,0,0,0],[12,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,0,2,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,11,0,12] >;`

C62.5D6 in GAP, Magma, Sage, TeX

`C_6^2._5D_6`
`% in TeX`

`G:=Group("C6^2.5D6");`
`// GroupNames label`

`G:=SmallGroup(432,98);`
`// by ID`

`G=gap.SmallGroup(432,98);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,571,4037,537,14118,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^6=b^6=d^2=1,c^6=a^3*b^3,a*b=b*a,c*a*c^-1=d*a*d=a^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d=b^3*c^5>;`
`// generators/relations`

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