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## G = C62.25C23order 288 = 25·32

### 20th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.25C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — D6⋊Dic3 — C62.25C23
 Lower central C32 — C3×C6 — C62.25C23
 Upper central C1 — C2×C4

Generators and relations for C62.25C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=a3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=b3c, ce=ec, de=ed >

Subgroups: 490 in 167 conjugacy classes, 68 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×6], C22, C22 [×4], S3 [×2], C6 [×2], C6 [×4], C6 [×5], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×2], Dic3 [×10], C12 [×4], C12 [×6], D6 [×2], D6 [×2], C2×C6 [×2], C2×C6 [×5], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C22×C4, C3×S3 [×2], C3×C6, C3×C6 [×2], C4×S3 [×4], C2×Dic3, C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×8], C22×S3, C22×C6, C42⋊C2, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C62, C4×Dic3, C4×Dic3 [×3], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×2], C6.D4 [×2], C4×C12, S3×C2×C4, C22×C12, S3×C12 [×4], C6×Dic3, C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C422S3, C23.26D6, D6⋊Dic3 [×2], Dic3⋊Dic3 [×2], Dic3×C12, C4×C3⋊Dic3, S3×C2×C12, C62.25C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, Dic3 [×4], D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×2], C2×Dic3 [×6], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4, C4○D12 [×4], C22×Dic3, S3×Dic3 [×2], C2×S32, C422S3, C23.26D6, D6.D6 [×2], C2×S3×Dic3, C62.25C23

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + - + + + + - + image C1 C2 C2 C2 C2 C2 C4 S3 S3 Dic3 D6 D6 D6 C4○D4 C4×S3 C4○D12 S32 S3×Dic3 C2×S32 D6.D6 kernel C62.25C23 D6⋊Dic3 Dic3⋊Dic3 Dic3×C12 C4×C3⋊Dic3 S3×C2×C12 S3×C12 C4×Dic3 S3×C2×C4 C4×S3 C2×Dic3 C2×C12 C22×S3 C3×C6 C12 C6 C2×C4 C4 C22 C2 # reps 1 2 2 1 1 1 8 1 1 4 3 2 1 4 4 16 1 2 1 4

Matrix representation of C62.25C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 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 12 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 2 7 0 0 0 0 7 11 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 12 12
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C62.25C23 in GAP, Magma, Sage, TeX

`C_6^2._{25}C_2^3`
`% in TeX`

`G:=Group("C6^2.25C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,503);`
`// by ID`

`G=gap.SmallGroup(288,503);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,64,100,1356,9414]);`
`// Polycyclic`

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

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