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## G = C62.39C23order 288 = 25·32

### 34th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.39C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.39C23
 Lower central C32 — C62 — C62.39C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.39C23
G = < a,b,c,d,e | a6=b6=1, c2=a3, d2=a3b3, e2=b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, de=ed >

Subgroups: 394 in 125 conjugacy classes, 52 normal (34 characteristic)
C1, C2 [×3], C3 [×2], C3, C4 [×2], C4 [×6], C22, C6 [×6], C6 [×3], C2×C4, C2×C4 [×6], C32, Dic3 [×10], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6, C42, C4⋊C4 [×6], C3×C6 [×3], C2×Dic3 [×2], C2×Dic3 [×2], C2×Dic3 [×6], C2×C12 [×2], C2×C12 [×5], C42.C2, C3×Dic3 [×4], C3⋊Dic3 [×2], C3×C12 [×2], C62, C4×Dic3, Dic3⋊C4 [×6], C4⋊Dic3, C4⋊Dic3 [×5], C4×C12, C3×C4⋊C4, C6×Dic3 [×2], C6×Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12, C12.6Q8, C4.Dic6, Dic3⋊Dic3 [×2], C62.C22 [×2], Dic3×C12, C3×C4⋊Dic3, C12⋊Dic3, C62.39C23
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], Q8 [×2], C23, D6 [×6], C2×Q8, C4○D4 [×2], Dic6 [×4], C22×S3 [×2], C42.C2, S32, C2×Dic6 [×2], C4○D12 [×2], D42S3, Q83S3, C322Q8 [×2], C2×S32, C12.6Q8, C4.Dic6, D125S3, D6.6D6, C2×C322Q8, C62.39C23

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

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

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

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

48 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 12B 12C 12D 12E ··· 12J 12K ··· 12R 12S 12T 12U 12V 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 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 1 1 2 2 4 2 2 6 6 6 6 12 12 36 36 2 ··· 2 4 4 4 2 2 2 2 4 ··· 4 6 ··· 6 12 12 12 12

48 irreducible representations

 dim 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 S3 S3 Q8 D6 D6 C4○D4 Dic6 C4○D12 S32 D4⋊2S3 Q8⋊3S3 C32⋊2Q8 C2×S32 D12⋊5S3 D6.6D6 kernel C62.39C23 Dic3⋊Dic3 C62.C22 Dic3×C12 C3×C4⋊Dic3 C12⋊Dic3 C4×Dic3 C4⋊Dic3 C3×C12 C2×Dic3 C2×C12 C3×C6 C12 C6 C2×C4 C6 C6 C4 C22 C2 C2 # reps 1 2 2 1 1 1 1 1 2 4 2 4 8 8 1 1 1 2 1 2 2

Matrix representation of C62.39C23 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 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 0 1 0 0 0 0 12 12
,
 6 1 0 0 0 0 2 7 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 8 0 0 0 0 0 5 5
,
 7 11 0 0 0 0 11 6 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 10 7 0 0 0 0 6 3
,
 9 3 0 0 0 0 3 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 0 12

`G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12],[6,2,0,0,0,0,1,7,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,8,5,0,0,0,0,0,5],[7,11,0,0,0,0,11,6,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,10,6,0,0,0,0,7,3],[9,3,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12] >;`

C62.39C23 in GAP, Magma, Sage, TeX

`C_6^2._{39}C_2^3`
`% in TeX`

`G:=Group("C6^2.39C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,517);`
`// by ID`

`G=gap.SmallGroup(288,517);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,176,422,100,1356,9414]);`
`// Polycyclic`

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

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