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## G = C24.29D6order 192 = 26·3

### 18th non-split extension by C24 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C24.29D6
 Chief series C1 — C3 — C6 — C2×C6 — C22×C6 — C22×Dic3 — C23×Dic3 — C24.29D6
 Lower central C3 — C2×C6 — C24.29D6
 Upper central C1 — C23 — C22×D4

Generators and relations for C24.29D6
G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=1, f2=b, ab=ba, ac=ca, eae-1=ad=da, faf-1=acd, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 680 in 286 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C2.C42, C2×C22⋊C4, C23×C4, C22×D4, C6.D4, C22×Dic3, C22×Dic3, C22×C12, C6×D4, C6×D4, C23×C6, C23.23D4, C6.C42, C2×C6.D4, C2×C6.D4, C23×Dic3, D4×C2×C6, C24.29D6
Quotients:

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

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

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

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

48 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 2L 2M 3 4A 4B 4C ··· 4J 4K 4L 4M 4N 6A ··· 6G 6H ··· 6O 12A 12B 12C 12D order 1 2 ··· 2 2 2 2 2 2 2 3 4 4 4 ··· 4 4 4 4 4 6 ··· 6 6 ··· 6 12 12 12 12 size 1 1 ··· 1 2 2 2 2 4 4 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 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C4 S3 D4 D4 D6 Dic3 D6 C4○D4 C3⋊D4 S3×D4 D4⋊2S3 kernel C24.29D6 C6.C42 C2×C6.D4 C23×Dic3 D4×C2×C6 C6×D4 C22×D4 C2×Dic3 C22×C6 C22×C4 C2×D4 C24 C2×C6 C23 C22 C22 # reps 1 2 3 1 1 8 1 4 4 1 4 2 4 8 2 2

Matrix representation of C24.29D6 in GL6(𝔽13)

 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 1 0 0 0 0 0 10 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 12 0 0 0 0 0 0 12 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
,
 4 0 0 0 0 0 0 10 0 0 0 0 0 0 4 0 0 0 0 0 0 10 0 0 0 0 0 0 10 11 0 0 0 0 4 3
,
 0 10 0 0 0 0 9 0 0 0 0 0 0 0 0 10 0 0 0 0 4 0 0 0 0 0 0 0 3 2 0 0 0 0 9 10

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

C24.29D6 in GAP, Magma, Sage, TeX

`C_2^4._{29}D_6`
`% in TeX`

`G:=Group("C2^4.29D6");`
`// GroupNames label`

`G:=SmallGroup(192,779);`
`// by ID`

`G=gap.SmallGroup(192,779);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,422,387,6278]);`
`// Polycyclic`

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

׿
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