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## G = C24.19D6order 192 = 26·3

### 8th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.19D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C4×Dic3 — C24.19D6
 Lower central C3 — C2×C6 — C24.19D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.19D6
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=e5 >

Subgroups: 456 in 190 conjugacy classes, 75 normal (51 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C22×C6, C22×C6, C2.C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C4×Dic3, C4⋊Dic3, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C24.C22, C6.C42, C2×C4×Dic3, C2×C4⋊Dic3, C2×C6.D4, C6×C22⋊C4, C24.19D6
Quotients:

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 3 4A 4B 4C 4D 4E 4F 4G ··· 4N 4O 4P 4Q 4R 6A ··· 6G 6H 6I 6J 6K 12A ··· 12H order 1 2 ··· 2 2 2 3 4 4 4 4 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 6 6 6 12 ··· 12 size 1 1 ··· 1 4 4 2 2 2 2 2 4 4 6 ··· 6 12 12 12 12 2 ··· 2 4 4 4 4 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 type + + + + + + + + - + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 Dic3 D6 D6 C4○D4 C4○D12 S3×D4 D4⋊2S3 kernel C24.19D6 C6.C42 C2×C4×Dic3 C2×C4⋊Dic3 C2×C6.D4 C6×C22⋊C4 C3×C22⋊C4 C2×C22⋊C4 C2×Dic3 C22⋊C4 C22×C4 C24 C2×C6 C22 C22 C22 # reps 1 2 1 1 2 1 8 1 4 4 2 1 8 8 2 2

Matrix representation of C24.19D6 in GL5(𝔽13)

 12 0 0 0 0 0 1 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 4 12
,
 12 0 0 0 0 0 12 0 0 0 0 0 12 0 0 0 0 0 1 0 0 0 0 0 1
,
 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
,
 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 6 0 0 0 0 0 2 0 0 0 0 0 11 1 0 0 0 10 2
,
 8 0 0 0 0 0 0 2 0 0 0 7 0 0 0 0 0 0 2 12 0 0 0 3 11

`G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,4,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[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],[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],[1,0,0,0,0,0,6,0,0,0,0,0,2,0,0,0,0,0,11,10,0,0,0,1,2],[8,0,0,0,0,0,0,7,0,0,0,2,0,0,0,0,0,0,2,3,0,0,0,12,11] >;`

C24.19D6 in GAP, Magma, Sage, TeX

`C_2^4._{19}D_6`
`% in TeX`

`G:=Group("C2^4.19D6");`
`// GroupNames label`

`G:=SmallGroup(192,510);`
`// by ID`

`G=gap.SmallGroup(192,510);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,120,422,387,6278]);`
`// 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=e^5>;`
`// generators/relations`

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