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## G = C62.75C23order 288 = 25·32

### 70th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.75C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×D6⋊S3 — C62.75C23
 Lower central C32 — C62 — C62.75C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.75C23
G = < a,b,c,d,e | a6=b6=c2=d2=1, e2=a3, ab=ba, ac=ca, dad=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd=a3c, ce=ec, ede-1=b3d >

Subgroups: 618 in 169 conjugacy classes, 48 normal (44 characteristic)
C1, C2 [×3], C2 [×3], C3 [×2], C3, C4 [×5], C22, C22 [×7], S3 [×3], C6 [×6], C6 [×6], C2×C4, C2×C4 [×6], D4 [×2], C23 [×2], C32, Dic3 [×9], C12 [×5], D6 [×2], D6 [×5], C2×C6 [×2], C2×C6 [×8], C22⋊C4 [×3], C4⋊C4 [×2], C22×C4, C2×D4, C3×S3 [×3], C3×C6 [×3], C4×S3 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×5], C22×S3 [×2], C22×C6 [×2], C22.D4, C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12, S3×C6 [×2], S3×C6 [×5], C62, Dic3⋊C4 [×4], C4⋊Dic3, D6⋊C4, D6⋊C4 [×2], C6.D4 [×2], C3×C22⋊C4, S3×C2×C4, C2×C3⋊D4 [×2], C22×C12, D6⋊S3 [×2], S3×C12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], C23.9D6, C23.28D6, D6⋊Dic3 [×2], Dic3⋊Dic3, C3×D6⋊C4, C6.Dic6, C2×D6⋊S3, S3×C2×C12, C62.75C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], C3⋊D4 [×2], C22×S3 [×2], C22.D4, S32, C4○D12 [×3], S3×D4, D42S3, C2×C3⋊D4, C2×S32, C23.9D6, C23.28D6, D125S3, D6.D6, S3×C3⋊D4, C62.75C23

Smallest permutation representation of C62.75C23
On 96 points
Generators in S96
```(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)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 18 5 16 3 14)(2 13 6 17 4 15)(7 93 11 91 9 95)(8 94 12 92 10 96)(19 28 23 26 21 30)(20 29 24 27 22 25)(31 42 33 38 35 40)(32 37 34 39 36 41)(43 49 45 51 47 53)(44 50 46 52 48 54)(55 61 57 63 59 65)(56 62 58 64 60 66)(67 78 69 74 71 76)(68 73 70 75 72 77)(79 88 83 86 81 90)(80 89 84 87 82 85)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 55)(7 51)(8 52)(9 53)(10 54)(11 49)(12 50)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 69)(22 70)(23 71)(24 72)(25 73)(26 74)(27 75)(28 76)(29 77)(30 78)(31 79)(32 80)(33 81)(34 82)(35 83)(36 84)(37 85)(38 86)(39 87)(40 88)(41 89)(42 90)(43 91)(44 92)(45 93)(46 94)(47 95)(48 96)
(1 34)(2 33)(3 32)(4 31)(5 36)(6 35)(7 75)(8 74)(9 73)(10 78)(11 77)(12 76)(13 38)(14 37)(15 42)(16 41)(17 40)(18 39)(19 47)(20 46)(21 45)(22 44)(23 43)(24 48)(25 50)(26 49)(27 54)(28 53)(29 52)(30 51)(55 80)(56 79)(57 84)(58 83)(59 82)(60 81)(61 89)(62 88)(63 87)(64 86)(65 85)(66 90)(67 92)(68 91)(69 96)(70 95)(71 94)(72 93)
(1 23 4 20)(2 24 5 21)(3 19 6 22)(7 79 10 82)(8 80 11 83)(9 81 12 84)(13 27 16 30)(14 28 17 25)(15 29 18 26)(31 54 34 51)(32 49 35 52)(33 50 36 53)(37 45 40 48)(38 46 41 43)(39 47 42 44)(55 70 58 67)(56 71 59 68)(57 72 60 69)(61 75 64 78)(62 76 65 73)(63 77 66 74)(85 93 88 96)(86 94 89 91)(87 95 90 92)```

`G:=sub<Sym(96)| (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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,93,11,91,9,95)(8,94,12,92,10,96)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,42,33,38,35,40)(32,37,34,39,36,41)(43,49,45,51,47,53)(44,50,46,52,48,54)(55,61,57,63,59,65)(56,62,58,64,60,66)(67,78,69,74,71,76)(68,73,70,75,72,77)(79,88,83,86,81,90)(80,89,84,87,82,85), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,75)(8,74)(9,73)(10,78)(11,77)(12,76)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,50)(26,49)(27,54)(28,53)(29,52)(30,51)(55,80)(56,79)(57,84)(58,83)(59,82)(60,81)(61,89)(62,88)(63,87)(64,86)(65,85)(66,90)(67,92)(68,91)(69,96)(70,95)(71,94)(72,93), (1,23,4,20)(2,24,5,21)(3,19,6,22)(7,79,10,82)(8,80,11,83)(9,81,12,84)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,54,34,51)(32,49,35,52)(33,50,36,53)(37,45,40,48)(38,46,41,43)(39,47,42,44)(55,70,58,67)(56,71,59,68)(57,72,60,69)(61,75,64,78)(62,76,65,73)(63,77,66,74)(85,93,88,96)(86,94,89,91)(87,95,90,92)>;`

`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)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,18,5,16,3,14)(2,13,6,17,4,15)(7,93,11,91,9,95)(8,94,12,92,10,96)(19,28,23,26,21,30)(20,29,24,27,22,25)(31,42,33,38,35,40)(32,37,34,39,36,41)(43,49,45,51,47,53)(44,50,46,52,48,54)(55,61,57,63,59,65)(56,62,58,64,60,66)(67,78,69,74,71,76)(68,73,70,75,72,77)(79,88,83,86,81,90)(80,89,84,87,82,85), (1,56)(2,57)(3,58)(4,59)(5,60)(6,55)(7,51)(8,52)(9,53)(10,54)(11,49)(12,50)(13,61)(14,62)(15,63)(16,64)(17,65)(18,66)(19,67)(20,68)(21,69)(22,70)(23,71)(24,72)(25,73)(26,74)(27,75)(28,76)(29,77)(30,78)(31,79)(32,80)(33,81)(34,82)(35,83)(36,84)(37,85)(38,86)(39,87)(40,88)(41,89)(42,90)(43,91)(44,92)(45,93)(46,94)(47,95)(48,96), (1,34)(2,33)(3,32)(4,31)(5,36)(6,35)(7,75)(8,74)(9,73)(10,78)(11,77)(12,76)(13,38)(14,37)(15,42)(16,41)(17,40)(18,39)(19,47)(20,46)(21,45)(22,44)(23,43)(24,48)(25,50)(26,49)(27,54)(28,53)(29,52)(30,51)(55,80)(56,79)(57,84)(58,83)(59,82)(60,81)(61,89)(62,88)(63,87)(64,86)(65,85)(66,90)(67,92)(68,91)(69,96)(70,95)(71,94)(72,93), (1,23,4,20)(2,24,5,21)(3,19,6,22)(7,79,10,82)(8,80,11,83)(9,81,12,84)(13,27,16,30)(14,28,17,25)(15,29,18,26)(31,54,34,51)(32,49,35,52)(33,50,36,53)(37,45,40,48)(38,46,41,43)(39,47,42,44)(55,70,58,67)(56,71,59,68)(57,72,60,69)(61,75,64,78)(62,76,65,73)(63,77,66,74)(85,93,88,96)(86,94,89,91)(87,95,90,92) );`

`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),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,18,5,16,3,14),(2,13,6,17,4,15),(7,93,11,91,9,95),(8,94,12,92,10,96),(19,28,23,26,21,30),(20,29,24,27,22,25),(31,42,33,38,35,40),(32,37,34,39,36,41),(43,49,45,51,47,53),(44,50,46,52,48,54),(55,61,57,63,59,65),(56,62,58,64,60,66),(67,78,69,74,71,76),(68,73,70,75,72,77),(79,88,83,86,81,90),(80,89,84,87,82,85)], [(1,56),(2,57),(3,58),(4,59),(5,60),(6,55),(7,51),(8,52),(9,53),(10,54),(11,49),(12,50),(13,61),(14,62),(15,63),(16,64),(17,65),(18,66),(19,67),(20,68),(21,69),(22,70),(23,71),(24,72),(25,73),(26,74),(27,75),(28,76),(29,77),(30,78),(31,79),(32,80),(33,81),(34,82),(35,83),(36,84),(37,85),(38,86),(39,87),(40,88),(41,89),(42,90),(43,91),(44,92),(45,93),(46,94),(47,95),(48,96)], [(1,34),(2,33),(3,32),(4,31),(5,36),(6,35),(7,75),(8,74),(9,73),(10,78),(11,77),(12,76),(13,38),(14,37),(15,42),(16,41),(17,40),(18,39),(19,47),(20,46),(21,45),(22,44),(23,43),(24,48),(25,50),(26,49),(27,54),(28,53),(29,52),(30,51),(55,80),(56,79),(57,84),(58,83),(59,82),(60,81),(61,89),(62,88),(63,87),(64,86),(65,85),(66,90),(67,92),(68,91),(69,96),(70,95),(71,94),(72,93)], [(1,23,4,20),(2,24,5,21),(3,19,6,22),(7,79,10,82),(8,80,11,83),(9,81,12,84),(13,27,16,30),(14,28,17,25),(15,29,18,26),(31,54,34,51),(32,49,35,52),(33,50,36,53),(37,45,40,48),(38,46,41,43),(39,47,42,44),(55,70,58,67),(56,71,59,68),(57,72,60,69),(61,75,64,78),(62,76,65,73),(63,77,66,74),(85,93,88,96),(86,94,89,91),(87,95,90,92)])`

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 6A ··· 6F 6G 6H 6I 6J 6K 6L 6M 6N 6O 12A 12B 12C 12D 12E ··· 12J 12K 12L 12M 12N 12O 12P order 1 2 2 2 2 2 2 3 3 3 4 4 4 4 4 4 4 6 ··· 6 6 6 6 6 6 6 6 6 6 12 12 12 12 12 ··· 12 12 12 12 12 12 12 size 1 1 1 1 6 6 12 2 2 4 2 2 6 6 12 36 36 2 ··· 2 4 4 4 6 6 6 6 12 12 2 2 2 2 4 ··· 4 6 6 6 6 12 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 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 S3 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 C2×S32 D12⋊5S3 D6.D6 S3×C3⋊D4 kernel C62.75C23 D6⋊Dic3 Dic3⋊Dic3 C3×D6⋊C4 C6.Dic6 C2×D6⋊S3 S3×C2×C12 D6⋊C4 S3×C2×C4 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 1 1 1 1 1 1 1 2 2 2 2 4 4 12 1 1 1 1 2 2 2

Matrix representation of C62.75C23 in GL6(𝔽13)

 0 12 0 0 0 0 1 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 2 4 0 0 0 0 9 11 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 10 3 0 0 0 0 6 3 0 0 0 0 0 0 7 3 0 0 0 0 10 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 2 9 0 0 0 0 4 11 0 0 0 0 0 0 1 0 0 0 0 0 0 1

`G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[2,9,0,0,0,0,4,11,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[10,6,0,0,0,0,3,3,0,0,0,0,0,0,7,10,0,0,0,0,3,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,2,4,0,0,0,0,9,11,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;`

C62.75C23 in GAP, Magma, Sage, TeX

`C_6^2._{75}C_2^3`
`% in TeX`

`G:=Group("C6^2.75C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,553);`
`// by ID`

`G=gap.SmallGroup(288,553);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,58,1356,9414]);`
`// Polycyclic`

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

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