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## G = C6.C42order 96 = 25·3

### 5th non-split extension by C6 of C42 acting via C42/C2×C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — C6.C42
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C6.C42
 Lower central C3 — C6 — C6.C42
 Upper central C1 — C23 — C22×C4

Generators and relations for C6.C42
G = < a,b,c,d,e | a2=b2=c2=1, d6=b, e2=abc, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=cd5 >

Subgroups: 146 in 76 conjugacy classes, 45 normal (19 characteristic)
C1, C2 [×3], C2 [×4], C3, C4 [×6], C22 [×3], C22 [×4], C6 [×3], C6 [×4], C2×C4 [×2], C2×C4 [×10], C23, Dic3 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×4], C22×C4, C22×C4 [×2], C2×Dic3 [×4], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C2.C42, C22×Dic3 [×2], C22×C12, C6.C42
Quotients: C1, C2 [×3], C4 [×6], C22, S3, C2×C4 [×3], D4 [×3], Q8, Dic3 [×2], D6, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic6, C4×S3 [×2], D12, C2×Dic3, C3⋊D4 [×2], C2.C42, C4×Dic3, Dic3⋊C4 [×2], C4⋊Dic3, D6⋊C4 [×2], C6.D4, C6.C42

Smallest permutation representation of C6.C42
Regular action on 96 points
Generators in S96
```(1 79)(2 80)(3 81)(4 82)(5 83)(6 84)(7 73)(8 74)(9 75)(10 76)(11 77)(12 78)(13 52)(14 53)(15 54)(16 55)(17 56)(18 57)(19 58)(20 59)(21 60)(22 49)(23 50)(24 51)(25 68)(26 69)(27 70)(28 71)(29 72)(30 61)(31 62)(32 63)(33 64)(34 65)(35 66)(36 67)(37 86)(38 87)(39 88)(40 89)(41 90)(42 91)(43 92)(44 93)(45 94)(46 95)(47 96)(48 85)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(73 79)(74 80)(75 81)(76 82)(77 83)(78 84)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)
(1 49)(2 50)(3 51)(4 52)(5 53)(6 54)(7 55)(8 56)(9 57)(10 58)(11 59)(12 60)(13 82)(14 83)(15 84)(16 73)(17 74)(18 75)(19 76)(20 77)(21 78)(22 79)(23 80)(24 81)(25 85)(26 86)(27 87)(28 88)(29 89)(30 90)(31 91)(32 92)(33 93)(34 94)(35 95)(36 96)(37 69)(38 70)(39 71)(40 72)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)
(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 25 16 42)(2 90 17 67)(3 35 18 40)(4 88 19 65)(5 33 20 38)(6 86 21 63)(7 31 22 48)(8 96 23 61)(9 29 24 46)(10 94 13 71)(11 27 14 44)(12 92 15 69)(26 78 43 54)(28 76 45 52)(30 74 47 50)(32 84 37 60)(34 82 39 58)(36 80 41 56)(49 85 73 62)(51 95 75 72)(53 93 77 70)(55 91 79 68)(57 89 81 66)(59 87 83 64)```

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

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

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

36 conjugacy classes

 class 1 2A ··· 2G 3 4A 4B 4C 4D 4E ··· 4L 6A ··· 6G 12A ··· 12H order 1 2 ··· 2 3 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 6 ··· 6 2 ··· 2 2 ··· 2

36 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 type + + + + + - - + - + image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 D6 Dic6 C4×S3 D12 C3⋊D4 kernel C6.C42 C22×Dic3 C22×C12 C2×Dic3 C2×C12 C22×C4 C2×C6 C2×C6 C2×C4 C23 C22 C22 C22 C22 # reps 1 2 1 8 4 1 3 1 2 1 2 4 2 4

Matrix representation of C6.C42 in GL4(𝔽13) generated by

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

C6.C42 in GAP, Magma, Sage, TeX

`C_6.C_4^2`
`% in TeX`

`G:=Group("C6.C4^2");`
`// GroupNames label`

`G:=SmallGroup(96,38);`
`// by ID`

`G=gap.SmallGroup(96,38);`
`# by ID`

`G:=PCGroup([6,-2,-2,-2,-2,-2,-3,24,217,55,2309]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^6=b,e^2=a*b*c,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=c*d^5>;`
`// generators/relations`

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