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## G = C62⋊5C4order 144 = 24·32

### 3rd semidirect product of C62 and C4 acting via C4/C2=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62⋊5C4
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C2×C3⋊Dic3 — C62⋊5C4
 Lower central C32 — C3×C6 — C62⋊5C4
 Upper central C1 — C22 — C23

Generators and relations for C625C4
G = < a,b,c | a6=b6=c4=1, ab=ba, cac-1=a-1b3, cbc-1=b-1 >

Subgroups: 250 in 102 conjugacy classes, 51 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, C23, C32, Dic3, C2×C6, C2×C6, C22⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C22×C6, C3⋊Dic3, C62, C62, C62, C6.D4, C2×C3⋊Dic3, C2×C62, C625C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C327D4, C625C4

Smallest permutation representation of C625C4
On 72 points
Generators in S72
```(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 17 10 23 13 19)(2 18 11 24 14 20)(3 16 12 22 15 21)(4 31 34 30 8 25)(5 32 35 28 9 26)(6 33 36 29 7 27)(37 54 48 40 51 45)(38 49 43 41 52 46)(39 50 44 42 53 47)(55 67 62 58 70 65)(56 68 63 59 71 66)(57 69 64 60 72 61)
(1 55 32 37)(2 57 33 39)(3 59 31 41)(4 52 21 71)(5 54 19 67)(6 50 20 69)(7 42 24 60)(8 38 22 56)(9 40 23 58)(10 70 28 51)(11 72 29 53)(12 68 30 49)(13 62 26 48)(14 64 27 44)(15 66 25 46)(16 63 34 43)(17 65 35 45)(18 61 36 47)```

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

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

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

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 3D 4A 4B 4C 4D 6A ··· 6AB order 1 2 2 2 2 2 3 3 3 3 4 4 4 4 6 ··· 6 size 1 1 1 1 2 2 2 2 2 2 18 18 18 18 2 ··· 2

42 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 type + + + + + - + image C1 C2 C2 C4 S3 D4 Dic3 D6 C3⋊D4 kernel C62⋊5C4 C2×C3⋊Dic3 C2×C62 C62 C22×C6 C3×C6 C2×C6 C2×C6 C6 # reps 1 2 1 4 4 2 8 4 16

Matrix representation of C625C4 in GL4(𝔽13) generated by

 3 0 0 0 0 4 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 10 0 0 0 0 3 0 0 0 8 9
,
 0 1 0 0 12 0 0 0 0 0 2 8 0 0 1 11
`G:=sub<GL(4,GF(13))| [3,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,10,0,0,0,0,3,8,0,0,0,9],[0,12,0,0,1,0,0,0,0,0,2,1,0,0,8,11] >;`

C625C4 in GAP, Magma, Sage, TeX

`C_6^2\rtimes_5C_4`
`% in TeX`

`G:=Group("C6^2:5C4");`
`// GroupNames label`

`G:=SmallGroup(144,100);`
`// by ID`

`G=gap.SmallGroup(144,100);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,121,964,3461]);`
`// Polycyclic`

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

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