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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, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C24, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C22×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, S3×C23, C23×C6, C24.3C22, C2×C4×Dic3, C2×Dic3⋊C4, C2×D6⋊C4, C2×C6.D4, C6×C22⋊C4, C22×C3⋊D4, C24.24D6
Quotients:

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

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

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

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

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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