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## G = C24.76D6order 192 = 26·3

### 5th non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.76D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.76D6
 Lower central C3 — C2×C6 — C24.76D6
 Upper central C1 — C23 — C23×C4

Generators and relations for C24.76D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=db=bd, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 760 in 286 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, D6⋊C4, C6.D4, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C22×C12, S3×C23, C23×C6, C23.23D4, C6.C42, C2×D6⋊C4, C2×C6.D4, C22×C3⋊D4, C23×C12, C24.76D6
Quotients:

Smallest permutation representation of C24.76D6
On 96 points
Generators in S96
```(1 72)(2 61)(3 62)(4 63)(5 64)(6 65)(7 66)(8 67)(9 68)(10 69)(11 70)(12 71)(13 92)(14 93)(15 94)(16 95)(17 96)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 79)(26 80)(27 81)(28 82)(29 83)(30 84)(31 73)(32 74)(33 75)(34 76)(35 77)(36 78)(37 58)(38 59)(39 60)(40 49)(41 50)(42 51)(43 52)(44 53)(45 54)(46 55)(47 56)(48 57)
(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 82)(2 83)(3 84)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 50)(14 51)(15 52)(16 53)(17 54)(18 55)(19 56)(20 57)(21 58)(22 59)(23 60)(24 49)(25 69)(26 70)(27 71)(28 72)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 85)(47 86)(48 87)
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 49)(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 89)(26 90)(27 91)(28 92)(29 93)(30 94)(31 95)(32 96)(33 85)(34 86)(35 87)(36 88)(37 68)(38 69)(39 70)(40 71)(41 72)(42 61)(43 62)(44 63)(45 64)(46 65)(47 66)(48 67)
(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 6 56 49)(2 60 57 5)(3 4 58 59)(7 12 50 55)(8 54 51 11)(9 10 52 53)(13 18 76 81)(14 80 77 17)(15 16 78 79)(19 24 82 75)(20 74 83 23)(21 22 84 73)(25 43 95 68)(26 67 96 42)(27 41 85 66)(28 65 86 40)(29 39 87 64)(30 63 88 38)(31 37 89 62)(32 61 90 48)(33 47 91 72)(34 71 92 46)(35 45 93 70)(36 69 94 44)```

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

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

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

60 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A ··· 4H 4I ··· 4N 6A ··· 6O 12A ··· 12P order 1 2 ··· 2 2 2 2 2 2 2 3 4 ··· 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 12 12 2 2 ··· 2 12 ··· 12 2 ··· 2 2 ··· 2

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 D12 C3⋊D4 C4○D12 kernel C24.76D6 C6.C42 C2×D6⋊C4 C2×C6.D4 C22×C3⋊D4 C23×C12 C2×C3⋊D4 C23×C4 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C23 C23 C22 # reps 1 2 2 1 1 1 8 1 4 4 2 1 4 8 4 4 4 8

Matrix representation of C24.76D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 11 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 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 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 12 0 0 0 0 0 0 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 5 5 0 0 0 0 8 0 0 0 0 0 0 0 8 3 0 0 0 0 0 5 0 0 0 0 0 0 11 2 0 0 0 0 11 9
,
 5 5 0 0 0 0 0 8 0 0 0 0 0 0 5 10 0 0 0 0 8 8 0 0 0 0 0 0 2 11 0 0 0 0 9 11

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

C24.76D6 in GAP, Magma, Sage, TeX

`C_2^4._{76}D_6`
`% in TeX`

`G:=Group("C2^4.76D6");`
`// GroupNames label`

`G:=SmallGroup(192,772);`
`// by ID`

`G=gap.SmallGroup(192,772);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,58,6278]);`
`// Polycyclic`

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

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