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## G = C24.28D4order 192 = 26·3

### 28th non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.28D4
 Chief series C1 — C3 — C6 — C2×C6 — C2×C12 — C2×D12 — C2×D24 — C24.28D4
 Lower central C3 — C6 — C2×C12 — C24.28D4
 Upper central C1 — C22 — C2×C4 — C2×Q16

Generators and relations for C24.28D4
G = < a,b,c | a24=b4=c2=1, bab-1=a17, cac=a-1, cbc=a12b-1 >

Subgroups: 440 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C8 [×2], C8 [×2], C2×C4, C2×C4 [×4], D4 [×4], Q8 [×4], C23 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×C8, C2×C8, D8 [×2], SD16 [×4], Q16 [×2], C2×D4 [×2], C2×Q8 [×2], C3⋊C8 [×2], C24 [×2], D12 [×4], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×4], C22×S3 [×2], C4×C8, C4.4D4 [×2], C2×D8, C2×SD16 [×2], C2×Q16, D24 [×2], C2×C3⋊C8, C4×Dic3, D6⋊C4 [×4], Q82S3 [×4], C2×C24, C3×Q16 [×2], C2×D12 [×2], C6×Q8 [×2], C8.12D4, C8×Dic3, C2×D24, C2×Q82S3 [×2], C12.23D4 [×2], C6×Q16, C24.28D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×6], C23, D6 [×3], C2×D4 [×3], C3⋊D4 [×2], C22×S3, C41D4, C4○D8 [×2], S3×D4 [×2], C2×C3⋊D4, C8.12D4, D24⋊C2 [×2], C123D4, C24.28D4

Smallest permutation representation of C24.28D4
On 96 points
Generators in S96
```(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 49 91 33)(2 66 92 26)(3 59 93 43)(4 52 94 36)(5 69 95 29)(6 62 96 46)(7 55 73 39)(8 72 74 32)(9 65 75 25)(10 58 76 42)(11 51 77 35)(12 68 78 28)(13 61 79 45)(14 54 80 38)(15 71 81 31)(16 64 82 48)(17 57 83 41)(18 50 84 34)(19 67 85 27)(20 60 86 44)(21 53 87 37)(22 70 88 30)(23 63 89 47)(24 56 90 40)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 69)(26 68)(27 67)(28 66)(29 65)(30 64)(31 63)(32 62)(33 61)(34 60)(35 59)(36 58)(37 57)(38 56)(39 55)(40 54)(41 53)(42 52)(43 51)(44 50)(45 49)(46 72)(47 71)(48 70)(73 85)(74 84)(75 83)(76 82)(77 81)(78 80)(86 96)(87 95)(88 94)(89 93)(90 92)```

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

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

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

36 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 12A 12B 12C 12D 12E 12F 24A 24B 24C 24D order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 6 6 8 8 8 8 8 8 8 8 12 12 12 12 12 12 24 24 24 24 size 1 1 1 1 24 24 2 2 2 6 6 6 6 8 8 2 2 2 2 2 2 2 6 6 6 6 4 4 8 8 8 8 4 4 4 4

36 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 S3 D4 D4 D4 D6 D6 C3⋊D4 C4○D8 S3×D4 S3×D4 D24⋊C2 kernel C24.28D4 C8×Dic3 C2×D24 C2×Q8⋊2S3 C12.23D4 C6×Q16 C2×Q16 C3⋊C8 C24 C2×Dic3 C2×C8 C2×Q8 C8 C6 C4 C22 C2 # reps 1 1 1 2 2 1 1 2 2 2 1 2 4 8 1 1 4

Matrix representation of C24.28D4 in GL6(𝔽73)

 11 3 0 0 0 0 8 62 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 16 57 0 0 0 0 16 16
,
 62 70 0 0 0 0 65 11 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 27 0 0 0 0 0 0 27
,
 1 0 0 0 0 0 17 72 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 72

`G:=sub<GL(6,GF(73))| [11,8,0,0,0,0,3,62,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,16,16,0,0,0,0,57,16],[62,65,0,0,0,0,70,11,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,27,0,0,0,0,0,0,27],[1,17,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,72] >;`

C24.28D4 in GAP, Magma, Sage, TeX

`C_{24}._{28}D_4`
`% in TeX`

`G:=Group("C24.28D4");`
`// GroupNames label`

`G:=SmallGroup(192,750);`
`// by ID`

`G=gap.SmallGroup(192,750);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,555,184,1684,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^17,c*a*c=a^-1,c*b*c=a^12*b^-1>;`
`// generators/relations`

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