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G = C24.31D6order 192 = 26·3

20th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.31D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.31D6
 Lower central C3 — C22×C6 — C24.31D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C24.31D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=dc=cd, ab=ba, ac=ca, eae-1=ad=da, faf-1=abd, bc=cb, bd=db, be=eb, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >

Subgroups: 600 in 238 conjugacy classes, 65 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×7], C22 [×3], C22 [×4], C22 [×20], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×2], C2×C4 [×15], D4 [×8], C23, C23 [×4], C23 [×12], Dic3 [×5], C12 [×2], C2×C6 [×3], C2×C6 [×4], C2×C6 [×20], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C2×D4 [×6], C24 [×2], C2×Dic3 [×2], C2×Dic3 [×11], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×C6, C22×C6 [×4], C22×C6 [×12], C2.C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, Dic3⋊C4 [×2], C6.D4 [×8], C22×Dic3 [×2], C22×Dic3 [×2], C22×C12, C6×D4 [×6], C23×C6 [×2], C23.10D4, C6.C42, C2×Dic3⋊C4, C2×C6.D4 [×4], D4×C2×C6, C24.31D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×8], C23, D6 [×3], C2×D4 [×4], C4○D4 [×3], C3⋊D4 [×6], C22×S3, C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, S3×D4, D42S3 [×3], C2×C3⋊D4 [×3], C23.10D4, C23.23D6 [×2], C23.12D6, D63D4, C23.14D6 [×2], C244S3, C24.31D6

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

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

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

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

42 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C ··· 4J 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 3 4 4 4 ··· 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 2 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 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.31D6 C6.C42 C2×Dic3⋊C4 C2×C6.D4 D4×C2×C6 C22×D4 C2×Dic3 C2×C12 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 # reps 1 1 1 4 1 1 2 2 4 1 2 6 4 8 1 3

Matrix representation of C24.31D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 5 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 5 12
,
 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 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 1 0 0 0 0 0 0 1
,
 11 2 0 0 0 0 11 9 0 0 0 0 0 0 12 0 0 0 0 0 3 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 9 11 0 0 0 0 2 4 0 0 0 0 0 0 8 0 0 0 0 0 2 5 0 0 0 0 0 0 3 4 0 0 0 0 11 10

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,12,0,0,0,0,0,0,1,5,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[11,11,0,0,0,0,2,9,0,0,0,0,0,0,12,3,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,2,0,0,0,0,11,4,0,0,0,0,0,0,8,2,0,0,0,0,0,5,0,0,0,0,0,0,3,11,0,0,0,0,4,10] >;`

C24.31D6 in GAP, Magma, Sage, TeX

`C_2^4._{31}D_6`
`% in TeX`

`G:=Group("C2^4.31D6");`
`// GroupNames label`

`G:=SmallGroup(192,781);`
`// by ID`

`G=gap.SmallGroup(192,781);`
`# by ID`

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

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

׿
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