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## G = C62.9C23order 288 = 25·32

### 4th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.9C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic32 — C62.9C23
 Lower central C32 — C62 — C62.9C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.9C23
G = < a,b,c,d,e | a6=b6=1, c2=b3, d2=e2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ece-1=b3c, ede-1=a3d >

Subgroups: 490 in 149 conjugacy classes, 54 normal (44 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×10], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], Q8 [×4], C32, Dic3 [×4], Dic3 [×11], C12 [×9], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×4], C2×Q8 [×2], C3×C6 [×3], Dic6 [×6], C2×Dic3 [×4], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C4⋊Q8, C3×Dic3 [×4], C3×Dic3 [×2], C3⋊Dic3 [×2], C3⋊Dic3, C3×C12, C62, C4×Dic3 [×2], Dic3⋊C4, Dic3⋊C4 [×6], C4⋊Dic3, C3×C4⋊C4, C2×Dic6, C2×Dic6 [×2], C6×Q8, C322Q8 [×2], C3×Dic6 [×2], C6×Dic3 [×4], C2×C3⋊Dic3 [×2], C6×C12, C12⋊Q8, Dic3⋊Q8, Dic32, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C6.Dic6, C2×C322Q8, C6×Dic6, C62.9C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], Q8 [×4], C23, D6 [×6], C2×D4, C2×Q8 [×2], Dic6 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊Q8, S32, C2×Dic6, S3×D4, S3×Q8 [×3], C2×C3⋊D4, C2×S32, C12⋊Q8, Dic3⋊Q8, S3×Dic6, Dic3.D6, S3×C3⋊D4, C62.9C23

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

42 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6F 6G 6H 6I 12A ··· 12H 12I ··· 12P order 1 2 2 2 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 6 6 6 12 ··· 12 12 ··· 12 size 1 1 1 1 2 2 4 4 6 6 6 6 12 12 18 18 36 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 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 C2 S3 S3 D4 Q8 Q8 D6 D6 Dic6 C3⋊D4 S32 S3×D4 S3×Q8 C2×S32 S3×Dic6 Dic3.D6 S3×C3⋊D4 kernel C62.9C23 Dic32 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C6.Dic6 C2×C32⋊2Q8 C6×Dic6 Dic3⋊C4 C2×Dic6 C3×Dic3 C3×Dic3 C3⋊Dic3 C2×Dic3 C2×C12 Dic3 Dic3 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 2 2 4 2 4 4 1 1 3 1 2 2 2

Matrix representation of C62.9C23 in GL6(𝔽13)

 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
,
 1 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 12 0 0 0 0 1 0
,
 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 8 5 0 0 0 0 0 5
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 0 0 0 0 0 12 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 11 4 0 0 0 0 9 2

`G:=sub<GL(6,GF(13))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,5,5],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,11,9,0,0,0,0,4,2] >;`

C62.9C23 in GAP, Magma, Sage, TeX

`C_6^2._9C_2^3`
`% in TeX`

`G:=Group("C6^2.9C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,487);`
`// by ID`

`G=gap.SmallGroup(288,487);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,64,422,135,58,1356,9414]);`
`// Polycyclic`

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

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