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## G = C62.48C23order 288 = 25·32

### 43rd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C62.48C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — S3×C2×C6 — C2×S3×Dic3 — C62.48C23
 Lower central C32 — C3×C6 — C62.48C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.48C23
G = < a,b,c,d,e | a6=b6=c2=1, d2=b3, e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=b3d >

Subgroups: 522 in 167 conjugacy classes, 58 normal (44 characteristic)
C1, C2 [×3], C2 [×2], C3 [×2], C3, C4 [×8], C22, C22 [×4], S3 [×2], C6 [×6], C6 [×5], C2×C4, C2×C4 [×9], C23, C32, Dic3 [×2], Dic3 [×12], C12 [×8], 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 [×3], C4×S3 [×4], C2×Dic3 [×3], C2×Dic3 [×10], C2×C12 [×2], C2×C12 [×4], C22×S3, C22×C6, C42⋊C2, C3×Dic3 [×2], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, S3×C6 [×2], S3×C6 [×2], C62, C4×Dic3 [×5], Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C22×Dic3, S3×Dic3 [×4], C6×Dic3 [×3], C2×C3⋊Dic3 [×2], C6×C12, S3×C2×C6, C23.16D6, C4⋊C47S3, Dic32, D6⋊Dic3, Dic3⋊Dic3, C3×Dic3⋊C4, C3×D6⋊C4, C4×C3⋊Dic3, C2×S3×Dic3, C62.48C23
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×2], C2×C4 [×6], C23, D6 [×6], C22×C4, C4○D4 [×2], C4×S3 [×4], C22×S3 [×2], C42⋊C2, S32, S3×C2×C4 [×2], D42S3 [×3], Q83S3, C2×S32, C23.16D6, C4⋊C47S3, D12⋊S3, C4×S32, D6.4D6, C62.48C23

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

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 3C 4A 4B 4C ··· 4H 4I 4J 4K 4L 4M 4N 6A ··· 6F 6G 6H 6I 6J 6K 12A ··· 12H 12I ··· 12N 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 12 ··· 12 12 ··· 12 size 1 1 1 1 6 6 2 2 4 2 2 6 ··· 6 9 9 9 9 18 18 2 ··· 2 4 4 4 12 12 4 ··· 4 12 ··· 12

48 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 C4 S3 S3 D6 D6 D6 C4○D4 C4×S3 C4×S3 S32 D4⋊2S3 Q8⋊3S3 C2×S32 D12⋊S3 C4×S32 D6.4D6 kernel C62.48C23 Dic32 D6⋊Dic3 Dic3⋊Dic3 C3×Dic3⋊C4 C3×D6⋊C4 C4×C3⋊Dic3 C2×S3×Dic3 S3×Dic3 Dic3⋊C4 D6⋊C4 C2×Dic3 C2×C12 C22×S3 C3×C6 Dic3 D6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 8 1 1 3 2 1 4 4 4 1 3 1 1 2 2 2

Matrix representation of C62.48C23 in GL6(𝔽13)

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

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

C62.48C23 in GAP, Magma, Sage, TeX

`C_6^2._{48}C_2^3`
`% in TeX`

`G:=Group("C6^2.48C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,526);`
`// by ID`

`G=gap.SmallGroup(288,526);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,422,219,58,1356,9414]);`
`// Polycyclic`

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

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