Copied to
clipboard

## G = C24.27D6order 192 = 26·3

### 16th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.27D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C2×D6⋊C4 — C24.27D6
 Lower central C3 — C22×C6 — C24.27D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.27D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=dc=cd, f2=d, 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=ce5 >

Subgroups: 728 in 238 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C3, C4 [×7], C22 [×7], C22 [×20], S3 [×2], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×15], D4 [×8], C23, C23 [×2], C23 [×14], Dic3 [×4], C12 [×3], D6 [×10], C2×C6 [×7], C2×C6 [×10], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×6], C24, C24, C2×Dic3 [×2], C2×Dic3 [×8], C3⋊D4 [×8], C2×C12 [×2], C2×C12 [×5], C22×S3 [×2], C22×S3 [×6], C22×C6, C22×C6 [×2], C22×C6 [×6], C2.C42, C2×C22⋊C4, C2×C22⋊C4 [×3], C2×C4⋊C4, C22×D4, C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×3], C2×C3⋊D4 [×6], C22×C12 [×2], S3×C23, C23×C6, C23.10D4, C6.C42, C2×C4⋊Dic3, C2×D6⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.27D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×8], C23, D6 [×3], C2×D4 [×4], C4○D4 [×3], D12 [×2], C3⋊D4 [×2], C22×S3, C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×D12, C4○D12, S3×D4 [×2], D42S3 [×2], C2×C3⋊D4, C23.10D4, D6⋊D4, C23.9D6, C23.11D6, C23.21D6, C127D4, D63D4, C23.14D6, C24.27D6

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 D12 C4○D12 S3×D4 D4⋊2S3 kernel C24.27D6 C6.C42 C2×C4⋊Dic3 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 C22×C3⋊D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×S3 C22×C6 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 1 1 2 1 1 1 1 2 2 2 2 2 1 6 4 4 4 2 2

Matrix representation of C24.27D6 in GL6(𝔽13)

 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 8 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 12 0 0 0 0 0 0 12 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 5 2 0 0 0 0 1 8 0 0 0 0 0 0 3 10 0 0 0 0 3 6
,
 8 0 0 0 0 0 0 8 0 0 0 0 0 0 8 11 0 0 0 0 0 5 0 0 0 0 0 0 10 3 0 0 0 0 6 3

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

C24.27D6 in GAP, Magma, Sage, TeX

`C_2^4._{27}D_6`
`% in TeX`

`G:=Group("C2^4.27D6");`
`// GroupNames label`

`G:=SmallGroup(192,520);`
`// by ID`

`G=gap.SmallGroup(192,520);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,6278]);`
`// Polycyclic`

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

׿
×
𝔽