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## G = C24.57D6order 192 = 26·3

### 4th non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.57D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C24.57D6
 Lower central C3 — C2×C6 — C24.57D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.57D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=b, ab=ba, eae-1=faf-1=ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >

Subgroups: 552 in 234 conjugacy classes, 83 normal (27 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×10], C22 [×3], C22 [×8], C22 [×12], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×30], C23, C23 [×6], C23 [×4], Dic3 [×8], C12 [×2], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×6], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×10], C24, C2×Dic3 [×8], C2×Dic3 [×16], C2×C12 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C23×C4, Dic3⋊C4 [×4], C6.D4 [×4], C3×C22⋊C4 [×2], C22×Dic3 [×4], C22×Dic3 [×6], C22×C12 [×2], C23×C6, C23.8Q8, C6.C42 [×2], C2×Dic3⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C23×Dic3, C24.57D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×6], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], Dic6 [×2], C4×S3 [×2], C3⋊D4 [×2], C22×S3, C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, Dic3⋊C4 [×4], C2×Dic6, S3×C2×C4, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.8Q8, Dic3.D4 [×2], Dic34D4 [×2], C2×Dic3⋊C4, C23.23D6, C232D6, C24.57D6

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

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

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

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

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 1 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 Q8 D6 D6 C4○D4 Dic6 C4×S3 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.57D6 C6.C42 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C23×Dic3 C6.D4 C2×C22⋊C4 C2×Dic3 C22×C6 C22×C6 C22×C4 C24 C2×C6 C23 C23 C23 C22 C22 # reps 1 2 2 1 1 1 8 1 4 2 2 2 1 4 4 4 4 2 2

Matrix representation of C24.57D6 in GL6(𝔽13)

 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 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 1 0 0 0 0 0 0 1
,
 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 12 0 0 0 0 0 1 0 0 0 0 0 0 5 5 0 0 0 0 8 0
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 7 11 0 0 0 0 12 6 0 0 0 0 0 0 10 7 0 0 0 0 10 3

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,0,0,0,0,5,8,0,0,0,0,5,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,7,12,0,0,0,0,11,6,0,0,0,0,0,0,10,10,0,0,0,0,7,3] >;`

C24.57D6 in GAP, Magma, Sage, TeX

`C_2^4._{57}D_6`
`% in TeX`

`G:=Group("C2^4.57D6");`
`// GroupNames label`

`G:=SmallGroup(192,505);`
`// by ID`

`G=gap.SmallGroup(192,505);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,254,219,184,6278]);`
`// Polycyclic`

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

׿
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