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## G = C62.40C23order 288 = 25·32

### 35th non-split extension by C62 of C23 acting via C23/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — C62.40C23
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C62.C22 — C62.40C23
 Lower central C32 — C62 — C62.40C23
 Upper central C1 — C22 — C2×C4

Generators and relations for C62.40C23
G = < a,b,c,d,e | a6=b6=1, c2=d2=b3, e2=a3b3, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=a3c, ece-1=b3c, ede-1=b3d >

Subgroups: 394 in 127 conjugacy classes, 46 normal (18 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C2×C4, C32, 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, C3×C4⋊C4, C6×Dic3, C2×C3⋊Dic3, C6×C12, Dic3.Q8, Dic3⋊Dic3, C62.C22, C3×Dic3⋊C4, C4×C3⋊Dic3, C62.40C23
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C4○D4, C22×S3, C42.C2, S32, C4○D12, D42S3, S3×Q8, C2×S32, Dic3.Q8, Dic3.D6, D6.D6, D6.4D6, C62.40C23

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

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

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

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

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 2 2 12 12 12 12 18 18 18 18 2 ··· 2 4 4 4 4 ··· 4 12 ··· 12

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + - + + + - - + - image C1 C2 C2 C2 C2 S3 Q8 D6 D6 C4○D4 C4○D12 S32 D4⋊2S3 S3×Q8 C2×S32 Dic3.D6 D6.D6 D6.4D6 kernel C62.40C23 Dic3⋊Dic3 C62.C22 C3×Dic3⋊C4 C4×C3⋊Dic3 Dic3⋊C4 C3⋊Dic3 C2×Dic3 C2×C12 C3×C6 C6 C2×C4 C6 C6 C22 C2 C2 C2 # reps 1 2 2 2 1 2 2 4 2 4 8 1 2 2 1 2 2 2

Matrix representation of C62.40C23 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 9 0 0 0 0 0 3 3
,
 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 4 5 0 0 0 0 10 9
,
 2 1 0 0 0 0 8 11 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 12 0 0 0 0 0 12 1
,
 12 7 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 8 0 0 0 0 0 0 8

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

C62.40C23 in GAP, Magma, Sage, TeX

`C_6^2._{40}C_2^3`
`% in TeX`

`G:=Group("C6^2.40C2^3");`
`// GroupNames label`

`G:=SmallGroup(288,518);`
`// by ID`

`G=gap.SmallGroup(288,518);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,120,422,219,142,1356,9414]);`
`// Polycyclic`

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

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