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## G = C22.D24order 192 = 26·3

### 3rd non-split extension by C22 of D24 acting via D24/D12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C22.D24
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C2×D12 — C12⋊7D4 — C22.D24
 Lower central C3 — C6 — C2×C12 — C22.D24
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for C22.D24
G = < a,b,c,d | a2=b2=c24=1, d2=b, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=bc-1 >

Subgroups: 384 in 114 conjugacy classes, 43 normal (25 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C22⋊C8, D4⋊C4, C2.D8, C2×C4⋊C4, C4⋊D4, C4⋊Dic3, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C2×C24, C2×D12, C22×Dic3, C2×C3⋊D4, C22×C12, C22.D8, C241C4, C2.D24, C3×C22⋊C8, C2×C4⋊Dic3, C127D4, C22.D24
Quotients: C1, C2, C22, S3, D4, C23, D6, D8, C2×D4, C4○D4, D12, C22×S3, C22.D4, C2×D8, C8.C22, D24, C2×D12, D42S3, C22.D8, C23.21D6, C2×D24, C8.D6, C22.D24

Smallest permutation representation of C22.D24
On 96 points
Generators in S96
```(1 96)(2 26)(3 74)(4 28)(5 76)(6 30)(7 78)(8 32)(9 80)(10 34)(11 82)(12 36)(13 84)(14 38)(15 86)(16 40)(17 88)(18 42)(19 90)(20 44)(21 92)(22 46)(23 94)(24 48)(25 71)(27 49)(29 51)(31 53)(33 55)(35 57)(37 59)(39 61)(41 63)(43 65)(45 67)(47 69)(50 75)(52 77)(54 79)(56 81)(58 83)(60 85)(62 87)(64 89)(66 91)(68 93)(70 95)(72 73)
(1 71)(2 72)(3 49)(4 50)(5 51)(6 52)(7 53)(8 54)(9 55)(10 56)(11 57)(12 58)(13 59)(14 60)(15 61)(16 62)(17 63)(18 64)(19 65)(20 66)(21 67)(22 68)(23 69)(24 70)(25 96)(26 73)(27 74)(28 75)(29 76)(30 77)(31 78)(32 79)(33 80)(34 81)(35 82)(36 83)(37 84)(38 85)(39 86)(40 87)(41 88)(42 89)(43 90)(44 91)(45 92)(46 93)(47 94)(48 95)
(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 70 71 24)(2 23 72 69)(3 68 49 22)(4 21 50 67)(5 66 51 20)(6 19 52 65)(7 64 53 18)(8 17 54 63)(9 62 55 16)(10 15 56 61)(11 60 57 14)(12 13 58 59)(25 48 96 95)(26 94 73 47)(27 46 74 93)(28 92 75 45)(29 44 76 91)(30 90 77 43)(31 42 78 89)(32 88 79 41)(33 40 80 87)(34 86 81 39)(35 38 82 85)(36 84 83 37)```

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

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

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

39 conjugacy classes

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

39 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - - - image C1 C2 C2 C2 C2 C2 S3 D4 D4 D6 D6 C4○D4 D8 D12 D12 D24 C8.C22 D4⋊2S3 C8.D6 kernel C22.D24 C24⋊1C4 C2.D24 C3×C22⋊C8 C2×C4⋊Dic3 C12⋊7D4 C22⋊C8 C2×C12 C22×C6 C2×C8 C22×C4 C12 C2×C6 C2×C4 C23 C22 C6 C4 C2 # reps 1 2 2 1 1 1 1 1 1 2 1 4 4 2 2 8 1 2 2

Matrix representation of C22.D24 in GL4(𝔽73) generated by

 72 0 0 0 0 72 0 0 0 0 17 8 0 0 37 56
,
 1 0 0 0 0 1 0 0 0 0 72 0 0 0 0 72
,
 18 50 0 0 23 68 0 0 0 0 52 3 0 0 48 21
,
 23 68 0 0 18 50 0 0 0 0 52 3 0 0 23 21
`G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,17,37,0,0,8,56],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[18,23,0,0,50,68,0,0,0,0,52,48,0,0,3,21],[23,18,0,0,68,50,0,0,0,0,52,23,0,0,3,21] >;`

C22.D24 in GAP, Magma, Sage, TeX

`C_2^2.D_{24}`
`% in TeX`

`G:=Group("C2^2.D24");`
`// GroupNames label`

`G:=SmallGroup(192,295);`
`// by ID`

`G=gap.SmallGroup(192,295);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,254,219,310,1123,136,6278]);`
`// Polycyclic`

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

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