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## G = C62.37C23order 288 = 25·32

### 32nd non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.37C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — Dic3⋊Dic3 — C62.37C23
 Lower central C32 — C62 — C62.37C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.37C23
G = < a,b,c,d,e | a6=b6=1, c2=e2=b3, d2=a3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b3c, ece-1=a3c, de=ed >

Subgroups: 394 in 125 conjugacy classes, 48 normal (44 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C42, C4⋊C4, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C42.C2, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, Dic3⋊C4, Dic3⋊C4, C4⋊Dic3, C4×C12, C3×C4⋊C4, C6×Dic3, C2×C3⋊Dic3, C6×C12, C12.6Q8, Dic3.Q8, Dic3⋊Dic3, C62.C22, Dic3×C12, C3×Dic3⋊C4, C6.Dic6, C62.37C23
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, Dic6, C22×S3, C42.C2, S32, C2×Dic6, C4○D12, D42S3, S3×Q8, C2×S32, C12.6Q8, Dic3.Q8, S3×Dic6, D6.D6, D6.3D6, C62.37C23

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

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

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

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

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 S3×Q8 C2×S32 S3×Dic6 D6.D6 D6.3D6 kernel C62.37C23 Dic3⋊Dic3 C62.C22 Dic3×C12 C3×Dic3⋊C4 C6.Dic6 C4×Dic3 Dic3⋊C4 C3×Dic3 C2×Dic3 C2×C12 C3×C6 Dic3 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 3 1 1 1 1 1 1 2 4 2 4 4 12 1 1 1 1 2 2 2

Matrix representation of C62.37C23 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 1 0 0 0 0 0 0 1
,
 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 1
,
 10 1 0 0 0 0 3 3 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 2 11 0 0 0 0 9 11
,
 8 0 0 0 0 0 9 5 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 11 9 0 0 0 0 4 2
,
 8 0 0 0 0 0 9 5 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 0 8

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

C62.37C23 in GAP, Magma, Sage, TeX

`C_6^2._{37}C_2^3`
`% in TeX`

`G:=Group("C6^2.37C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,515);`
`// by ID`

`G=gap.SmallGroup(288,515);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,64,254,100,1356,9414]);`
`// Polycyclic`

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

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