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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 [×3], C2 [×4], C2 [×6], C3, C4 [×8], C22 [×3], C22 [×8], C22 [×22], C6 [×3], C6 [×4], C6 [×6], C2×C4 [×2], C2×C4 [×24], D4 [×8], C23, C23 [×8], C23 [×10], Dic3 [×6], C12 [×2], C2×C6 [×3], C2×C6 [×8], C2×C6 [×22], C22⋊C4 [×6], C22×C4, C22×C4 [×10], C2×D4 [×4], C2×D4 [×4], C24 [×2], C2×Dic3 [×4], C2×Dic3 [×18], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×C6, C22×C6 [×8], C22×C6 [×10], C2.C42 [×2], C2×C22⋊C4 [×3], C23×C4, C22×D4, C6.D4 [×6], C22×Dic3 [×4], C22×Dic3 [×6], C22×C12, C6×D4 [×4], C6×D4 [×4], C23×C6 [×2], C23.23D4, C6.C42 [×2], C2×C6.D4, C2×C6.D4 [×2], C23×Dic3, D4×C2×C6, C24.29D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×8], C23, Dic3 [×4], D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×Dic3 [×6], C3⋊D4 [×4], C22×S3, C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C6.D4 [×4], S3×D4 [×2], D42S3 [×2], C22×Dic3, C2×C3⋊D4 [×2], C23.23D4, D4×Dic3 [×2], C23.23D6, C232D6, C23.14D6 [×2], C2×C6.D4, C24.29D6

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