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## G = C24.56D6order 192 = 26·3

### 3rd non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.56D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C24.56D6
 Lower central C3 — C2×C6 — C24.56D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.56D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=bcd, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 520 in 218 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2 [×6], C2 [×4], C3, C4 [×8], C22 [×3], C22 [×8], C22 [×12], C6, C6 [×6], C6 [×4], C2×C4 [×28], C23, C23 [×6], C23 [×4], Dic3 [×6], C12 [×2], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, C2×Dic3 [×4], C2×Dic3 [×18], C2×C12 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×4], C2×C22⋊C4, C2×C22⋊C4, C23×C4, C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×8], C22×Dic3 [×4], C22×C12 [×2], C23×C6, C23.34D4, C6.C42 [×4], C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.56D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4○D4 [×4], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C42⋊C2 [×2], C22.D4 [×4], D6⋊C4 [×4], S3×C2×C4, C2×D12, D42S3 [×4], C2×C3⋊D4, C23.34D4, C23.16D6 [×2], C23.21D6 [×2], C2×D6⋊C4, C23.23D6 [×2], C24.56D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 4O 4P 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 2 4 4 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 2 2 2 2 2 2 2 2 4 type + + + + + + + + + + - image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4○D4 C4×S3 D12 C3⋊D4 D4⋊2S3 kernel C24.56D6 C6.C42 C2×C6.D4 C6×C22⋊C4 C23×Dic3 C22×Dic3 C2×C22⋊C4 C22×C6 C22×C4 C24 C2×C6 C23 C23 C23 C22 # reps 1 4 1 1 1 8 1 4 2 1 8 4 4 4 4

Matrix representation of C24.56D6 in GL8(𝔽13)

 1 0 0 0 0 0 0 0 8 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1
,
 8 11 0 0 0 0 0 0 12 5 0 0 0 0 0 0 0 0 8 1 0 0 0 0 0 0 2 5 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 12 1
,
 12 10 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 12 6 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 12 1 0 0 0 0 0 0 0 1

`G:=sub<GL(8,GF(13))| [1,8,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[8,12,0,0,0,0,0,0,11,5,0,0,0,0,0,0,0,0,8,2,0,0,0,0,0,0,1,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[12,0,0,0,0,0,0,0,10,1,0,0,0,0,0,0,0,0,12,4,0,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,1] >;`

C24.56D6 in GAP, Magma, Sage, TeX

`C_2^4._{56}D_6`
`% in TeX`

`G:=Group("C2^4.56D6");`
`// GroupNames label`

`G:=SmallGroup(192,502);`
`// by ID`

`G=gap.SmallGroup(192,502);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,422,387,58,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=b*c*d,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*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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