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## G = C24.25D6order 192 = 26·3

### 14th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C6 — C24.25D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.25D6
 Lower central C3 — C22×C6 — C24.25D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.25D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=c, f2=cb=bc, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=be5 >

Subgroups: 728 in 238 conjugacy classes, 61 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C3, C4 [×7], C22 [×7], C22 [×20], S3 [×2], C6 [×7], C6 [×2], C2×C4 [×17], D4 [×8], C23, C23 [×2], C23 [×14], Dic3 [×5], C12 [×2], 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 [×4], C2×Dic3 [×7], C3⋊D4 [×8], C2×C12 [×6], 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, Dic3⋊C4 [×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×Dic3⋊C4, C2×D6⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.25D6
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×8], C23, D6 [×3], C2×D4 [×4], C4○D4 [×3], C3⋊D4 [×2], C22×S3, C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C4○D12 [×2], S3×D4 [×3], D42S3, C2×C3⋊D4, C23.10D4, C23.9D6, Dic3⋊D4 [×2], C23.11D6, C23.28D6, C232D6, C23.14D6, C24.25D6

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

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

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

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

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 4 4 type + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C4○D4 C3⋊D4 C4○D12 S3×D4 D4⋊2S3 kernel C24.25D6 C6.C42 C2×Dic3⋊C4 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 C22×C3⋊D4 C2×C22⋊C4 C2×Dic3 C22×S3 C22×C6 C22×C4 C24 C2×C6 C23 C22 C22 C22 # reps 1 1 1 2 1 1 1 1 4 2 2 2 1 6 4 8 3 1

Matrix representation of C24.25D6 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 2 9 0 0 0 0 4 11
,
 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 12 0 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
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 7 3 0 0 0 0 10 10
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 1 2 0 0 0 0 12 12 0 0 0 0 0 0 3 3 0 0 0 0 6 10

`G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,2,4,0,0,0,0,9,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[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],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,7,10,0,0,0,0,3,10],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,1,12,0,0,0,0,2,12,0,0,0,0,0,0,3,6,0,0,0,0,3,10] >;`

C24.25D6 in GAP, Magma, Sage, TeX

`C_2^4._{25}D_6`
`% in TeX`

`G:=Group("C2^4.25D6");`
`// GroupNames label`

`G:=SmallGroup(192,518);`
`// by ID`

`G=gap.SmallGroup(192,518);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,387,100,6278]);`
`// Polycyclic`

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