Copied to
clipboard

## G = C24.60D6order 192 = 26·3

### 7th non-split extension by C24 of D6 acting via D6/S3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.60D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.60D6
 Lower central C3 — C2×C6 — C24.60D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.60D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, af=fa, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce5 >

Subgroups: 808 in 286 conjugacy classes, 83 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×6], C3, C4 [×8], C22 [×3], C22 [×8], C22 [×22], S3 [×2], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×26], D4 [×8], C23, C23 [×6], C23 [×12], Dic3 [×6], C12 [×2], D6 [×10], C2×C6 [×3], C2×C6 [×8], C2×C6 [×12], C22⋊C4 [×6], C22×C4 [×2], C22×C4 [×9], C2×D4 [×8], C24, C24, C2×Dic3 [×6], C2×Dic3 [×14], C3⋊D4 [×8], C2×C12 [×6], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×6], C22×C6 [×4], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4 [×2], C23×C4, C22×D4, D6⋊C4 [×4], C3×C22⋊C4 [×2], C22×Dic3, C22×Dic3 [×2], C22×Dic3 [×6], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C23.23D4, C6.C42 [×2], C2×D6⋊C4 [×2], C6×C22⋊C4, C23×Dic3, C22×C3⋊D4, C24.60D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, D6⋊C4 [×4], S3×C2×C4, C2×D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.23D4, Dic34D4 [×2], D6⋊D4, C23.21D6, C2×D6⋊C4, C23.14D6 [×2], C24.60D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C 4D 4E ··· 4L 4M 4N 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 2 2 2 2 12 12 2 4 4 4 4 6 ··· 6 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 C4×S3 D12 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.60D6 C6.C42 C2×D6⋊C4 C6×C22⋊C4 C23×Dic3 C22×C3⋊D4 C2×C3⋊D4 C2×C22⋊C4 C2×Dic3 C22×C6 C22×C4 C24 C2×C6 C23 C23 C23 C22 C22 # reps 1 2 2 1 1 1 8 1 4 4 2 1 4 4 4 4 2 2

Matrix representation of C24.60D6 in GL6(𝔽13)

 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 8 2 0 0 0 0 1 5
,
 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
,
 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 5 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 5 12
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 8 0 0 0 0 0 5 5 0 0 0 0 0 0 1 0 0 0 0 0 0 1

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

C24.60D6 in GAP, Magma, Sage, TeX

`C_2^4._{60}D_6`
`% in TeX`

`G:=Group("C2^4.60D6");`
`// GroupNames label`

`G:=SmallGroup(192,517);`
`// by ID`

`G=gap.SmallGroup(192,517);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,422,387,58,6278]);`
`// Polycyclic`

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

׿
×
𝔽