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## G = C23⋊D4order 184 = 23·23

### The semidirect product of C23 and D4 acting via D4/C22=C2

Aliases: C232D4, C22⋊D23, D462C2, Dic23⋊C2, C2.5D46, C46.5C22, (C2×C46)⋊2C2, SmallGroup(184,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C46 — C23⋊D4
 Chief series C1 — C23 — C46 — D46 — C23⋊D4
 Lower central C23 — C46 — C23⋊D4
 Upper central C1 — C2 — C22

Generators and relations for C23⋊D4
G = < a,b,c | a23=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

Smallest permutation representation of C23⋊D4
On 92 points
Generators in S92
```(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)
(1 60 46 78)(2 59 24 77)(3 58 25 76)(4 57 26 75)(5 56 27 74)(6 55 28 73)(7 54 29 72)(8 53 30 71)(9 52 31 70)(10 51 32 92)(11 50 33 91)(12 49 34 90)(13 48 35 89)(14 47 36 88)(15 69 37 87)(16 68 38 86)(17 67 39 85)(18 66 40 84)(19 65 41 83)(20 64 42 82)(21 63 43 81)(22 62 44 80)(23 61 45 79)
(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(24 45)(25 44)(26 43)(27 42)(28 41)(29 40)(30 39)(31 38)(32 37)(33 36)(34 35)(47 91)(48 90)(49 89)(50 88)(51 87)(52 86)(53 85)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(69 92)```

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

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

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

C23⋊D4 is a maximal subgroup of   D925C2  D4×D23  D42D23
C23⋊D4 is a maximal quotient of   Dic23⋊C4  D46⋊C4  D4⋊D23  D4.D23  Q8⋊D23  C23⋊Q16  C23.D23

49 conjugacy classes

 class 1 2A 2B 2C 4 23A ··· 23K 46A ··· 46AG order 1 2 2 2 4 23 ··· 23 46 ··· 46 size 1 1 2 46 46 2 ··· 2 2 ··· 2

49 irreducible representations

 dim 1 1 1 1 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 D4 D23 D46 C23⋊D4 kernel C23⋊D4 Dic23 D46 C2×C46 C23 C22 C2 C1 # reps 1 1 1 1 1 11 11 22

Matrix representation of C23⋊D4 in GL2(𝔽47) generated by

 41 15 15 25
,
 0 1 46 0
,
 24 6 6 23
`G:=sub<GL(2,GF(47))| [41,15,15,25],[0,46,1,0],[24,6,6,23] >;`

C23⋊D4 in GAP, Magma, Sage, TeX

`C_{23}\rtimes D_4`
`% in TeX`

`G:=Group("C23:D4");`
`// GroupNames label`

`G:=SmallGroup(184,7);`
`// by ID`

`G=gap.SmallGroup(184,7);`
`# by ID`

`G:=PCGroup([4,-2,-2,-2,-23,49,2819]);`
`// Polycyclic`

`G:=Group<a,b,c|a^23=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;`
`// generators/relations`

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