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## G = C24.15D6order 192 = 26·3

### 4th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.15D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C24.15D6
 Lower central C3 — C2×C6 — C24.15D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.15D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=db=bd, ab=ba, eae-1=ac=ca, ad=da, faf-1=abcd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 456 in 190 conjugacy classes, 69 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×10], C22 [×7], C22 [×10], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×20], C23, C23 [×2], C23 [×6], Dic3 [×7], C12 [×3], C2×C6 [×7], C2×C6 [×10], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×6], C2×Dic3 [×9], C2×C12 [×2], C2×C12 [×5], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×2], C2×C42, C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×Dic3 [×2], Dic3⋊C4 [×2], C6.D4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C24.C22, C6.C42 [×2], C2×C4×Dic3, C2×Dic3⋊C4, C2×C6.D4 [×2], C6×C22⋊C4, C24.15D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C42⋊C2, C4×D4 [×2], C4⋊D4, C22.D4, C4.4D4, C422C2, S3×C2×C4, C4○D12, S3×D4, D42S3 [×3], C2×C3⋊D4, C24.C22, C23.16D6, C23.8D6, Dic34D4, C23.9D6, C4×C3⋊D4, C23.12D6, C23.14D6, C24.15D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 2 2 2 2 2 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.15D6 C6.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C6.D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 2 1 8 1 2 2 2 1 8 4 4 4 1 3

Matrix representation of C24.15D6 in GL6(𝔽13)

 1 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 1 0 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 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 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 6 0 0 0 0 0 3 2 0 0 0 0 0 0 2 0 0 0 0 0 0 6
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 10 4 0 0 0 0 11 3 0 0 0 0 0 0 0 12 0 0 0 0 1 0

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

C24.15D6 in GAP, Magma, Sage, TeX

`C_2^4._{15}D_6`
`% in TeX`

`G:=Group("C2^4.15D6");`
`// GroupNames label`

`G:=SmallGroup(192,504);`
`// by ID`

`G=gap.SmallGroup(192,504);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,64,926,219,6278]);`
`// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c*d,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=e^5>;`
`// generators/relations`

׿
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