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## G = C24.75D6order 192 = 26·3

### 4th non-split extension by C24 of D6 acting via D6/C6=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.75D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4⋊Dic3 — C24.75D6
 Lower central C3 — C2×C6 — C24.75D6
 Upper central C1 — C23 — C23×C4

Generators and relations for C24.75D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=b, ab=ba, faf-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de5 >

Subgroups: 504 in 234 conjugacy classes, 103 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×8], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×8], C2×C4 [×22], C23, C23 [×6], C23 [×4], Dic3 [×4], C12 [×4], C12 [×2], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×8], C24, C2×Dic3 [×12], C2×C12 [×8], C2×C12 [×10], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4⋊Dic3 [×4], C6.D4 [×4], C22×Dic3 [×4], C22×C12 [×2], C22×C12 [×4], C22×C12 [×4], C23×C6, C23.7Q8, C6.C42 [×2], C2×C4⋊Dic3 [×2], C2×C6.D4 [×2], C23×C12, C24.75D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×2], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic6 [×2], D12 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C4⋊Dic3 [×4], C6.D4 [×4], C2×Dic6, C2×D12, C4○D12 [×2], C22×Dic3, C2×C3⋊D4 [×2], C23.7Q8, C12.48D4 [×2], C2×C4⋊Dic3, C23.26D6, C127D4 [×2], C2×C6.D4, C24.75D6

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - - + + - + image C1 C2 C2 C2 C2 C4 S3 D4 D4 Q8 Dic3 D6 D6 C4○D4 C3⋊D4 Dic6 D12 C4○D12 kernel C24.75D6 C6.C42 C2×C4⋊Dic3 C2×C6.D4 C23×C12 C22×C12 C23×C4 C2×C12 C22×C6 C22×C6 C22×C4 C22×C4 C24 C2×C6 C2×C4 C23 C23 C22 # reps 1 2 2 2 1 8 1 4 2 2 4 2 1 4 8 4 4 8

Matrix representation of C24.75D6 in GL6(𝔽13)

 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 2 12
,
 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 1 0 0 0 0 0 0 1 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 3 10 0 0 0 0 3 6 0 0 0 0 0 0 2 0 0 0 0 0 6 7 0 0 0 0 0 0 2 0 0 0 0 0 8 7
,
 1 0 0 0 0 0 12 12 0 0 0 0 0 0 3 9 0 0 0 0 9 10 0 0 0 0 0 0 12 1 0 0 0 0 0 1

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

C24.75D6 in GAP, Magma, Sage, TeX

`C_2^4._{75}D_6`
`% in TeX`

`G:=Group("C2^4.75D6");`
`// GroupNames label`

`G:=SmallGroup(192,771);`
`// by ID`

`G=gap.SmallGroup(192,771);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,232,422,6278]);`
`// Polycyclic`

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

׿
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