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## G = C24.24D6order 192 = 26·3

### 13rd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.24D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — S3×C23 — C22×C3⋊D4 — C24.24D6
 Lower central C3 — C2×C6 — C24.24D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.24D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, 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=de5 >

Subgroups: 744 in 258 conjugacy classes, 77 normal (51 characteristic)
C1, C2 [×7], C2 [×4], C3, C4 [×10], C22 [×7], C22 [×20], S3 [×2], C6 [×7], C6 [×2], C2×C4 [×2], C2×C4 [×18], D4 [×8], C23, C23 [×2], C23 [×14], Dic3 [×4], Dic3 [×3], C12 [×3], D6 [×10], C2×C6 [×7], C2×C6 [×10], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×8], C24, C24, C2×Dic3 [×8], C2×Dic3 [×5], 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], Dic3⋊C4 [×2], D6⋊C4 [×4], C6.D4 [×2], C3×C22⋊C4 [×2], C22×Dic3 [×3], C2×C3⋊D4 [×4], C2×C3⋊D4 [×4], C22×C12 [×2], S3×C23, C23×C6, C24.3C22, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4 [×2], C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.24D6
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], C3⋊D4 [×2], C22×S3, C2×C22⋊C4, C4×D4 [×2], C4⋊D4 [×2], C4.4D4, C41D4, S3×C2×C4, C4○D12, S3×D4 [×3], D42S3, C2×C3⋊D4, C24.3C22, S3×C22⋊C4, Dic34D4, Dic3⋊D4, C23.11D6, C4×C3⋊D4, C23.14D6, C123D4, C24.24D6

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 12 12 2 2 2 2 2 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 1 2 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.24D6 C2×C4×Dic3 C2×Dic3⋊C4 C2×D6⋊C4 C2×C6.D4 C6×C22⋊C4 C22×C3⋊D4 C2×C3⋊D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 1 1 2 1 1 1 8 1 6 2 2 1 4 4 4 4 3 1

Matrix representation of C24.24D6 in GL6(𝔽13)

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

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

C24.24D6 in GAP, Magma, Sage, TeX

`C_2^4._{24}D_6`
`% in TeX`

`G:=Group("C2^4.24D6");`
`// GroupNames label`

`G:=SmallGroup(192,516);`
`// by ID`

`G=gap.SmallGroup(192,516);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,758,387,58,6278]);`
`// Polycyclic`

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

׿
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