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## G = C62.54C23order 288 = 25·32

### 49th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.54C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.54C23
 Lower central C32 — C62 — C62.54C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.54C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3c, 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], D12 [×2], C2×Dic3 [×2], C2×Dic3 [×8], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], 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, Dic3⋊C4 [×3], D6⋊C4 [×3], C6.D4 [×3], C3×C4⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C6×D4, S3×Dic3 [×2], C3×D12 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6 [×2], D6.D4, C23.23D6, D6⋊Dic3 [×3], C3×Dic3⋊C4, C6.Dic6, C2×S3×Dic3, C6×D12, C62.54C23
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, S3×D4, D42S3 [×2], Q83S3, C2×C3⋊D4, C2×S32, D6.D4, C23.23D6, D125S3, D12⋊S3, S3×C3⋊D4, C62.54C23

Smallest permutation representation of C62.54C23
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 78 10 75)(8 77 11 74)(9 76 12 73)(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 80 58 83)(56 79 59 82)(57 84 60 81)(61 89 64 86)(62 88 65 85)(63 87 66 90)(67 92 70 95)(68 91 71 94)(69 96 72 93)
(1 23 4 20)(2 24 5 21)(3 19 6 22)(7 82 10 79)(8 83 11 80)(9 84 12 81)(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 67 58 70)(56 68 59 71)(57 69 60 72)(61 78 64 75)(62 73 65 76)(63 74 66 77)(85 96 88 93)(86 91 89 94)(87 92 90 95)```

`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)(7,78,10,75)(8,77,11,74)(9,76,12,73)(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)(55,80,58,83)(56,79,59,82)(57,84,60,81)(61,89,64,86)(62,88,65,85)(63,87,66,90)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,23,4,20)(2,24,5,21)(3,19,6,22)(7,82,10,79)(8,83,11,80)(9,84,12,81)(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)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,78,64,75)(62,73,65,76)(63,74,66,77)(85,96,88,93)(86,91,89,94)(87,92,90,95)>;`

`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)(7,78,10,75)(8,77,11,74)(9,76,12,73)(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)(55,80,58,83)(56,79,59,82)(57,84,60,81)(61,89,64,86)(62,88,65,85)(63,87,66,90)(67,92,70,95)(68,91,71,94)(69,96,72,93), (1,23,4,20)(2,24,5,21)(3,19,6,22)(7,82,10,79)(8,83,11,80)(9,84,12,81)(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)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,78,64,75)(62,73,65,76)(63,74,66,77)(85,96,88,93)(86,91,89,94)(87,92,90,95) );`

`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),(7,78,10,75),(8,77,11,74),(9,76,12,73),(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),(55,80,58,83),(56,79,59,82),(57,84,60,81),(61,89,64,86),(62,88,65,85),(63,87,66,90),(67,92,70,95),(68,91,71,94),(69,96,72,93)], [(1,23,4,20),(2,24,5,21),(3,19,6,22),(7,82,10,79),(8,83,11,80),(9,84,12,81),(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),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,78,64,75),(62,73,65,76),(63,74,66,77),(85,96,88,93),(86,91,89,94),(87,92,90,95)])`

42 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 12A ··· 12H 12I 12J 12K 12L 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 12 ··· 12 12 12 12 12 size 1 1 1 1 6 6 12 2 2 4 4 6 6 12 18 18 36 2 ··· 2 4 4 4 12 12 12 12 4 ··· 4 12 12 12 12

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 S3 S3 D4 D6 D6 D6 C4○D4 C3⋊D4 C4○D12 S32 S3×D4 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊5S3 D12⋊S3 S3×C3⋊D4 kernel C62.54C23 D6⋊Dic3 C3×Dic3⋊C4 C6.Dic6 C2×S3×Dic3 C6×D12 Dic3⋊C4 C2×D12 S3×C6 C2×Dic3 C2×C12 C22×S3 C3×C6 D6 C6 C2×C4 C6 C6 C6 C22 C2 C2 C2 # reps 1 3 1 1 1 1 1 1 2 2 2 2 4 4 4 1 1 2 1 1 2 2 2

Matrix representation of C62.54C23 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 12 0
,
 9 7 0 0 0 0 9 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 0 1
,
 7 4 0 0 0 0 7 6 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 0 8
,
 5 3 0 0 0 0 0 8 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 3 7 0 0 0 0 6 10

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12,0,0,0,0,1,0],[9,9,0,0,0,0,7,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[7,7,0,0,0,0,4,6,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[5,0,0,0,0,0,3,8,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,6,0,0,0,0,7,10] >;`

C62.54C23 in GAP, Magma, Sage, TeX

`C_6^2._{54}C_2^3`
`% in TeX`

`G:=Group("C6^2.54C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,532);`
`// by ID`

`G=gap.SmallGroup(288,532);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,254,219,58,1356,9414]);`
`// Polycyclic`

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

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