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## G = C24.20D6order 192 = 26·3

### 9th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.20D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.20D6
 Lower central C3 — C22×C6 — C24.20D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.20D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=bcd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >

Subgroups: 440 in 170 conjugacy classes, 57 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C3, C4 [×7], C22 [×7], C22 [×10], C6 [×7], C6 [×2], C2×C4 [×19], C23, C23 [×2], C23 [×6], Dic3 [×5], C12 [×2], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×6], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×4], C24, C2×Dic3 [×2], C2×Dic3 [×11], C2×C12 [×6], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42 [×3], C2×C22⋊C4, C2×C22⋊C4 [×2], C2×C4⋊C4, Dic3⋊C4 [×2], C6.D4 [×4], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.11D4, C6.C42 [×3], C2×Dic3⋊C4, C2×C6.D4 [×2], C6×C22⋊C4, C24.20D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×4], C23, D6 [×3], C2×D4 [×2], C4○D4 [×5], C3⋊D4 [×2], C22×S3, C4⋊D4, C22.D4 [×3], C4.4D4, C422C2 [×2], C4○D12 [×2], S3×D4, D42S3 [×3], C2×C3⋊D4, C23.11D4, C23.8D6 [×2], C23.9D6, C23.11D6, C23.28D6, C23.23D6, C23.14D6, C24.20D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4○D12 S3×D4 D4⋊2S3 kernel C24.20D6 C6.C42 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C2×C22⋊C4 C2×Dic3 C22×C6 C22×C4 C24 C2×C6 C23 C22 C22 C22 # reps 1 3 1 2 1 1 2 2 2 1 10 4 8 1 3

Matrix representation of C24.20D6 in GL6(𝔽13)

 1 0 0 0 0 0 1 12 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 1 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 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
,
 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
,
 6 0 0 0 0 0 4 11 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 11 4 0 0 0 0 9 2 0 0 0 0 0 0 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 5 0 0 0 0 8 0

`G:=sub<GL(6,GF(13))| [1,1,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[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],[6,4,0,0,0,0,0,11,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[11,9,0,0,0,0,4,2,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,8,0,0,0,0,5,0] >;`

C24.20D6 in GAP, Magma, Sage, TeX

`C_2^4._{20}D_6`
`% in TeX`

`G:=Group("C2^4.20D6");`
`// GroupNames label`

`G:=SmallGroup(192,511);`
`// by ID`

`G=gap.SmallGroup(192,511);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,254,387,100,6278]);`
`// Polycyclic`

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

׿
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