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## G = C24.18D4order 192 = 26·3

### 18th non-split extension by C24 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C12 — C24.18D4
 Chief series C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C8○D12 — C24.18D4
 Lower central C3 — C6 — C2×C12 — C24.18D4
 Upper central C1 — C2 — C2×C4 — C8.C4

Generators and relations for C24.18D4
G = < a,b,c | a24=1, b4=c2=a12, bab-1=a7, cac-1=a-1, cbc-1=b3 >

Subgroups: 288 in 100 conjugacy classes, 37 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), SD16, Q16, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C4.10D4, C8.C4, C8○D4, C2×Q16, C8.C22, S3×C8, C8⋊S3, C24⋊C2, Dic12, C4.Dic3, C2×C24, C3×M4(2), C2×Dic6, C4○D12, D4.5D4, C12.47D4, C3×C8.C4, C8○D12, C2×Dic12, C8.D6, C24.18D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C2×D12, S3×D4, Q83S3, D4.5D4, C12⋊D4, C24.18D4

Character table of C24.18D4

 class 1 2A 2B 2C 3 4A 4B 4C 4D 4E 6A 6B 8A 8B 8C 8D 8E 8F 8G 12A 12B 12C 24A 24B 24C 24D 24E 24F 24G 24H size 1 1 2 12 2 2 2 12 24 24 2 4 2 2 4 8 8 12 12 2 2 4 4 4 4 4 8 8 8 8 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 -1 1 1 1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ4 1 1 1 -1 1 1 1 -1 -1 -1 1 1 1 1 1 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ5 1 1 1 -1 1 1 1 -1 1 -1 1 1 -1 -1 -1 -1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 -1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ9 2 2 2 0 -1 2 2 0 0 0 -1 -1 2 2 2 2 2 0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 2 -2 -2 2 2 -2 2 0 0 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ11 2 2 2 0 -1 2 2 0 0 0 -1 -1 -2 -2 -2 2 -2 0 0 -1 -1 -1 1 1 1 1 1 1 -1 -1 orthogonal lifted from D6 ρ12 2 2 -2 0 2 -2 2 0 0 0 2 -2 2 2 -2 0 0 0 0 -2 -2 2 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ13 2 2 2 0 -1 2 2 0 0 0 -1 -1 -2 -2 -2 -2 2 0 0 -1 -1 -1 1 1 1 1 -1 -1 1 1 orthogonal lifted from D6 ρ14 2 2 -2 2 2 2 -2 -2 0 0 2 -2 0 0 0 0 0 0 0 2 2 -2 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ15 2 2 2 0 -1 2 2 0 0 0 -1 -1 2 2 2 -2 -2 0 0 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 orthogonal lifted from D6 ρ16 2 2 -2 0 2 -2 2 0 0 0 2 -2 -2 -2 2 0 0 0 0 -2 -2 2 -2 -2 2 2 0 0 0 0 orthogonal lifted from D4 ρ17 2 2 -2 0 -1 -2 2 0 0 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 -1 1 1 √3 -√3 -√3 √3 orthogonal lifted from D12 ρ18 2 2 -2 0 -1 -2 2 0 0 0 -1 1 2 2 -2 0 0 0 0 1 1 -1 -1 -1 1 1 -√3 √3 √3 -√3 orthogonal lifted from D12 ρ19 2 2 -2 0 -1 -2 2 0 0 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 1 -1 -1 √3 -√3 √3 -√3 orthogonal lifted from D12 ρ20 2 2 -2 0 -1 -2 2 0 0 0 -1 1 -2 -2 2 0 0 0 0 1 1 -1 1 1 -1 -1 -√3 √3 -√3 √3 orthogonal lifted from D12 ρ21 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 0 -2i 2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ22 2 2 2 0 2 -2 -2 0 0 0 2 2 0 0 0 0 0 2i -2i -2 -2 -2 0 0 0 0 0 0 0 0 complex lifted from C4○D4 ρ23 4 4 4 0 -2 -4 -4 0 0 0 -2 -2 0 0 0 0 0 0 0 2 2 2 0 0 0 0 0 0 0 0 orthogonal lifted from Q8⋊3S3, Schur index 2 ρ24 4 4 -4 0 -2 4 -4 0 0 0 -2 2 0 0 0 0 0 0 0 -2 -2 2 0 0 0 0 0 0 0 0 orthogonal lifted from S3×D4 ρ25 4 -4 0 0 4 0 0 0 0 0 -4 0 -2√2 2√2 0 0 0 0 0 0 0 0 2√2 -2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ26 4 -4 0 0 4 0 0 0 0 0 -4 0 2√2 -2√2 0 0 0 0 0 0 0 0 -2√2 2√2 0 0 0 0 0 0 symplectic lifted from D4.5D4, Schur index 2 ρ27 4 -4 0 0 -2 0 0 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 -2√3 2√3 0 -√2 √2 -√6 √6 0 0 0 0 symplectic faithful, Schur index 2 ρ28 4 -4 0 0 -2 0 0 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 2√3 -2√3 0 √2 -√2 -√6 √6 0 0 0 0 symplectic faithful, Schur index 2 ρ29 4 -4 0 0 -2 0 0 0 0 0 2 0 2√2 -2√2 0 0 0 0 0 -2√3 2√3 0 √2 -√2 √6 -√6 0 0 0 0 symplectic faithful, Schur index 2 ρ30 4 -4 0 0 -2 0 0 0 0 0 2 0 -2√2 2√2 0 0 0 0 0 2√3 -2√3 0 -√2 √2 √6 -√6 0 0 0 0 symplectic faithful, Schur index 2

Smallest permutation representation of C24.18D4
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 53 81 45 13 65 93 33)(2 60 82 28 14 72 94 40)(3 67 83 35 15 55 95 47)(4 50 84 42 16 62 96 30)(5 57 85 25 17 69 73 37)(6 64 86 32 18 52 74 44)(7 71 87 39 19 59 75 27)(8 54 88 46 20 66 76 34)(9 61 89 29 21 49 77 41)(10 68 90 36 22 56 78 48)(11 51 91 43 23 63 79 31)(12 58 92 26 24 70 80 38)
(1 45 13 33)(2 44 14 32)(3 43 15 31)(4 42 16 30)(5 41 17 29)(6 40 18 28)(7 39 19 27)(8 38 20 26)(9 37 21 25)(10 36 22 48)(11 35 23 47)(12 34 24 46)(49 73 61 85)(50 96 62 84)(51 95 63 83)(52 94 64 82)(53 93 65 81)(54 92 66 80)(55 91 67 79)(56 90 68 78)(57 89 69 77)(58 88 70 76)(59 87 71 75)(60 86 72 74)```

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

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

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

Matrix representation of C24.18D4 in GL4(𝔽73) generated by

 0 16 0 57 57 16 16 57 0 16 0 16 57 16 57 16
,
 14 12 65 14 61 26 59 6 65 14 59 61 59 6 12 47
,
 12 14 14 65 26 61 6 59 14 65 61 59 6 59 47 12
`G:=sub<GL(4,GF(73))| [0,57,0,57,16,16,16,16,0,16,0,57,57,57,16,16],[14,61,65,59,12,26,14,6,65,59,59,12,14,6,61,47],[12,26,14,6,14,61,65,59,14,6,61,47,65,59,59,12] >;`

C24.18D4 in GAP, Magma, Sage, TeX

`C_{24}._{18}D_4`
`% in TeX`

`G:=Group("C24.18D4");`
`// GroupNames label`

`G:=SmallGroup(192,455);`
`// by ID`

`G=gap.SmallGroup(192,455);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,254,219,58,1123,136,438,102,6278]);`
`// Polycyclic`

`G:=Group<a,b,c|a^24=1,b^4=c^2=a^12,b*a*b^-1=a^7,c*a*c^-1=a^-1,c*b*c^-1=b^3>;`
`// generators/relations`

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