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## G = C24.30D6order 192 = 26·3

### 19th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.30D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C24.30D6
 Lower central C3 — C2×C6 — C24.30D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C24.30D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=d, f2=db=bd, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 616 in 258 conjugacy classes, 91 normal (23 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3, C4 [×4], C4 [×6], C22 [×3], C22 [×4], C22 [×20], C6 [×3], C6 [×4], C6 [×4], C2×C4 [×6], C2×C4 [×14], D4 [×8], C23, C23 [×4], C23 [×12], Dic3 [×6], C12 [×4], C2×C6 [×3], C2×C6 [×4], C2×C6 [×20], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×Dic3 [×4], C2×Dic3 [×10], C2×C12 [×6], C3×D4 [×8], C22×C6, C22×C6 [×4], C22×C6 [×12], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C4×Dic3 [×2], C4⋊Dic3 [×2], C6.D4 [×8], C22×Dic3 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C24.3C22, C2×C4×Dic3, C2×C4⋊Dic3, C2×C6.D4 [×4], D4×C2×C6, C24.30D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, C6.D4 [×4], S3×D4 [×2], D42S3 [×2], C22×Dic3, C2×C3⋊D4 [×2], C24.3C22, D4×Dic3 [×2], C23.12D6, D63D4 [×2], C123D4, C2×C6.D4, C24.30D6

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

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

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

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

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 ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 4 4 4 4 2 2 2 2 2 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 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.30D6 C2×C4×Dic3 C2×C4⋊Dic3 C2×C6.D4 D4×C2×C6 C6×D4 C22×D4 C2×Dic3 C2×C12 C22×C4 C2×D4 C24 C2×C6 C2×C4 C22 C22 # reps 1 1 1 4 1 8 1 4 4 1 4 2 4 8 2 2

Matrix representation of C24.30D6 in GL5(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 12 1 0 0 0 0 1
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 12
,
 1 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 12 0 0 0 0 0 0 1 0 0 0 12 0 0 0 0 0 0 10 10 0 0 0 0 4
,
 5 0 0 0 0 0 0 5 0 0 0 8 0 0 0 0 0 0 10 6 0 0 0 7 3

`G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,1,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,10,4],[5,0,0,0,0,0,0,8,0,0,0,5,0,0,0,0,0,0,10,7,0,0,0,6,3] >;`

C24.30D6 in GAP, Magma, Sage, TeX

`C_2^4._{30}D_6`
`% in TeX`

`G:=Group("C2^4.30D6");`
`// GroupNames label`

`G:=SmallGroup(192,780);`
`// by ID`

`G=gap.SmallGroup(192,780);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,232,422,387,6278]);`
`// Polycyclic`

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

׿
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