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## G = C24.74D6order 192 = 26·3

### 3rd non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.74D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.74D6
 Lower central C3 — C2×C6 — C24.74D6
 Upper central C1 — C23 — C23×C4

Generators and relations for C24.74D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=bcd, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 472 in 218 conjugacy classes, 87 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C23×C4, C6.D4, C22×Dic3, C22×C12, C22×C12, C23×C6, C23.34D4, C6.C42, C2×C6.D4, C23×C12, C24.74D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, Dic3, D6, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×Dic3, C3⋊D4, C22×S3, C2×C22⋊C4, C42⋊C2, C22.D4, C6.D4, C4○D12, C22×Dic3, C2×C3⋊D4, C23.34D4, C23.26D6, C23.28D6, C2×C6.D4, C24.74D6

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

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

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

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

60 conjugacy classes

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

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + + image C1 C2 C2 C2 C4 S3 D4 Dic3 D6 D6 C4○D4 C3⋊D4 C4○D12 kernel C24.74D6 C6.C42 C2×C6.D4 C23×C12 C22×C12 C23×C4 C22×C6 C22×C4 C22×C4 C24 C2×C6 C23 C22 # reps 1 4 2 1 8 1 4 4 2 1 8 8 16

Matrix representation of C24.74D6 in GL5(𝔽13)

 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 5 0 0 0 0 0 8 0 0 0 0 0 9 0 0 0 0 0 10
,
 8 0 0 0 0 0 0 5 0 0 0 5 0 0 0 0 0 0 0 3 0 0 0 4 0

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

C24.74D6 in GAP, Magma, Sage, TeX

`C_2^4._{74}D_6`
`% in TeX`

`G:=Group("C2^4.74D6");`
`// GroupNames label`

`G:=SmallGroup(192,770);`
`// by ID`

`G=gap.SmallGroup(192,770);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,422,184,6278]);`
`// Polycyclic`

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

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