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## G = C24.14D6order 192 = 26·3

### 3rd non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.14D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C2×C6.D4 — C24.14D6
 Lower central C3 — C2×C6 — C24.14D6
 Upper central C1 — C23 — C2×C22⋊C4

Generators and relations for C24.14D6
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=e5 >

Subgroups: 456 in 190 conjugacy classes, 69 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, 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, Dic3⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C24.C22, C6.C42, C2×C4×Dic3, C2×Dic3⋊C4, C2×C6.D4, C6×C22⋊C4, C24.14D6
Quotients:

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

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

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

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

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 2 2 4 4 type + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C4 S3 D4 D4 D6 D6 C4○D4 C3⋊D4 C4×S3 C4○D12 S3×D4 D4⋊2S3 kernel C24.14D6 C6.C42 C2×C4×Dic3 C2×Dic3⋊C4 C2×C6.D4 C6×C22⋊C4 C6.D4 C2×C22⋊C4 C2×Dic3 C2×C12 C22×C4 C24 C2×C6 C2×C4 C23 C22 C22 C22 # reps 1 2 1 1 2 1 8 1 2 2 2 1 8 4 4 4 1 3

Matrix representation of C24.14D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 7 1 0 0 0 0 0 0 1 0 0 0 0 0 3 12
,
 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 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 12 0 0 0 0 0 0 12 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 1 0 0 0 0 12 0 0 0 0 0 0 0 6 11 0 0 0 0 12 7 0 0 0 0 0 0 11 0 0 0 0 0 6 7
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 7 4 0 0 0 0 7 6

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

C24.14D6 in GAP, Magma, Sage, TeX

`C_2^4._{14}D_6`
`% in TeX`

`G:=Group("C2^4.14D6");`
`// GroupNames label`

`G:=SmallGroup(192,503);`
`// by ID`

`G=gap.SmallGroup(192,503);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,387,58,6278]);`
`// 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=e^5>;`
`// generators/relations`

׿
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