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## G = Dic23order 92 = 22·23

### Dicyclic group

Aliases: Dic23, C23⋊C4, C46.C2, C2.D23, SmallGroup(92,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — Dic23
 Chief series C1 — C23 — C46 — Dic23
 Lower central C23 — Dic23
 Upper central C1 — C2

Generators and relations for Dic23
G = < a,b | a46=1, b2=a23, bab-1=a-1 >

Character table of Dic23

 class 1 2 4A 4B 23A 23B 23C 23D 23E 23F 23G 23H 23I 23J 23K 46A 46B 46C 46D 46E 46F 46G 46H 46I 46J 46K size 1 1 23 23 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 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 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 linear of order 2 ρ3 1 -1 i -i 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 4 ρ4 1 -1 -i i 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 4 ρ5 2 2 0 0 ζ2315+ζ238 ζ2313+ζ2310 ζ2312+ζ2311 ζ2314+ζ239 ζ2316+ζ237 ζ2318+ζ235 ζ2320+ζ233 ζ2322+ζ23 ζ2321+ζ232 ζ2319+ζ234 ζ2317+ζ236 ζ2317+ζ236 ζ2315+ζ238 ζ2313+ζ2310 ζ2312+ζ2311 ζ2314+ζ239 ζ2316+ζ237 ζ2318+ζ235 ζ2320+ζ233 ζ2322+ζ23 ζ2321+ζ232 ζ2319+ζ234 orthogonal lifted from D23 ρ6 2 2 0 0 ζ2319+ζ234 ζ2318+ζ235 ζ2317+ζ236 ζ2316+ζ237 ζ2315+ζ238 ζ2314+ζ239 ζ2313+ζ2310 ζ2312+ζ2311 ζ2322+ζ23 ζ2321+ζ232 ζ2320+ζ233 ζ2320+ζ233 ζ2319+ζ234 ζ2318+ζ235 ζ2317+ζ236 ζ2316+ζ237 ζ2315+ζ238 ζ2314+ζ239 ζ2313+ζ2310 ζ2312+ζ2311 ζ2322+ζ23 ζ2321+ζ232 orthogonal lifted from D23 ρ7 2 2 0 0 ζ2316+ζ237 ζ2320+ζ233 ζ2322+ζ23 ζ2318+ζ235 ζ2314+ζ239 ζ2313+ζ2310 ζ2317+ζ236 ζ2321+ζ232 ζ2319+ζ234 ζ2315+ζ238 ζ2312+ζ2311 ζ2312+ζ2311 ζ2316+ζ237 ζ2320+ζ233 ζ2322+ζ23 ζ2318+ζ235 ζ2314+ζ239 ζ2313+ζ2310 ζ2317+ζ236 ζ2321+ζ232 ζ2319+ζ234 ζ2315+ζ238 orthogonal lifted from D23 ρ8 2 2 0 0 ζ2317+ζ236 ζ2319+ζ234 ζ2314+ζ239 ζ2322+ζ23 ζ2312+ζ2311 ζ2321+ζ232 ζ2315+ζ238 ζ2318+ζ235 ζ2313+ζ2310 ζ2320+ζ233 ζ2316+ζ237 ζ2316+ζ237 ζ2317+ζ236 ζ2319+ζ234 ζ2314+ζ239 ζ2322+ζ23 ζ2312+ζ2311 ζ2321+ζ232 ζ2315+ζ238 ζ2318+ζ235 ζ2313+ζ2310 ζ2320+ζ233 orthogonal lifted from D23 ρ9 2 2 0 0 ζ2313+ζ2310 ζ2322+ζ23 ζ2315+ζ238 ζ2317+ζ236 ζ2320+ζ233 ζ2312+ζ2311 ζ2321+ζ232 ζ2316+ζ237 ζ2314+ζ239 ζ2318+ζ235 ζ2319+ζ234 ζ2319+ζ234 ζ2313+ζ2310 ζ2322+ζ23 ζ2315+ζ238 ζ2317+ζ236 ζ2320+ζ233 ζ2312+ζ2311 ζ2321+ζ232 ζ2316+ζ237 ζ2314+ζ239 ζ2318+ζ235 orthogonal lifted from D23 ρ10 2 2 0 0 ζ2320+ζ233 ζ2321+ζ232 ζ2316+ζ237 ζ2312+ζ2311 ζ2317+ζ236 ζ2322+ζ23 ζ2319+ζ234 ζ2314+ζ239 ζ2318+ζ235 ζ2313+ζ2310 ζ2315+ζ238 ζ2315+ζ238 ζ2320+ζ233 ζ2321+ζ232 ζ2316+ζ237 ζ2312+ζ2311 ζ2317+ζ236 ζ2322+ζ23 ζ2319+ζ234 ζ2314+ζ239 ζ2318+ζ235 ζ2313+ζ2310 orthogonal lifted from D23 ρ11 2 2 0 0 ζ2318+ζ235 ζ2312+ζ2311 ζ2319+ζ234 ζ2320+ζ233 ζ2313+ζ2310 ζ2317+ζ236 ζ2322+ζ23 ζ2315+ζ238 ζ2316+ζ237 ζ2314+ζ239 ζ2321+ζ232 ζ2321+ζ232 ζ2318+ζ235 ζ2312+ζ2311 ζ2319+ζ234 ζ2320+ζ233 ζ2313+ζ2310 ζ2317+ζ236 ζ2322+ζ23 ζ2315+ζ238 ζ2316+ζ237 ζ2314+ζ239 orthogonal lifted from D23 ρ12 2 2 0 0 ζ2314+ζ239 ζ2317+ζ236 ζ2321+ζ232 ζ2313+ζ2310 ζ2318+ζ235 ζ2320+ζ233 ζ2312+ζ2311 ζ2319+ζ234 ζ2315+ζ238 ζ2316+ζ237 ζ2322+ζ23 ζ2322+ζ23 ζ2314+ζ239 ζ2317+ζ236 ζ2321+ζ232 ζ2313+ζ2310 ζ2318+ζ235 ζ2320+ζ233 ζ2312+ζ2311 ζ2319+ζ234 ζ2315+ζ238 ζ2316+ζ237 orthogonal lifted from D23 ρ13 2 2 0 0 ζ2312+ζ2311 ζ2315+ζ238 ζ2318+ζ235 ζ2321+ζ232 ζ2322+ζ23 ζ2319+ζ234 ζ2316+ζ237 ζ2313+ζ2310 ζ2320+ζ233 ζ2317+ζ236 ζ2314+ζ239 ζ2314+ζ239 ζ2312+ζ2311 ζ2315+ζ238 ζ2318+ζ235 ζ2321+ζ232 ζ2322+ζ23 ζ2319+ζ234 ζ2316+ζ237 ζ2313+ζ2310 ζ2320+ζ233 ζ2317+ζ236 orthogonal lifted from D23 ρ14 2 2 0 0 ζ2321+ζ232 ζ2314+ζ239 ζ2320+ζ233 ζ2315+ζ238 ζ2319+ζ234 ζ2316+ζ237 ζ2318+ζ235 ζ2317+ζ236 ζ2312+ζ2311 ζ2322+ζ23 ζ2313+ζ2310 ζ2313+ζ2310 ζ2321+ζ232 ζ2314+ζ239 ζ2320+ζ233 ζ2315+ζ238 ζ2319+ζ234 ζ2316+ζ237 ζ2318+ζ235 ζ2317+ζ236 ζ2312+ζ2311 ζ2322+ζ23 orthogonal lifted from D23 ρ15 2 2 0 0 ζ2322+ζ23 ζ2316+ζ237 ζ2313+ζ2310 ζ2319+ζ234 ζ2321+ζ232 ζ2315+ζ238 ζ2314+ζ239 ζ2320+ζ233 ζ2317+ζ236 ζ2312+ζ2311 ζ2318+ζ235 ζ2318+ζ235 ζ2322+ζ23 ζ2316+ζ237 ζ2313+ζ2310 ζ2319+ζ234 ζ2321+ζ232 ζ2315+ζ238 ζ2314+ζ239 ζ2320+ζ233 ζ2317+ζ236 ζ2312+ζ2311 orthogonal lifted from D23 ρ16 2 -2 0 0 ζ2321+ζ232 ζ2314+ζ239 ζ2320+ζ233 ζ2315+ζ238 ζ2319+ζ234 ζ2316+ζ237 ζ2318+ζ235 ζ2317+ζ236 ζ2312+ζ2311 ζ2322+ζ23 ζ2313+ζ2310 -ζ2313-ζ2310 -ζ2321-ζ232 -ζ2314-ζ239 -ζ2320-ζ233 -ζ2315-ζ238 -ζ2319-ζ234 -ζ2316-ζ237 -ζ2318-ζ235 -ζ2317-ζ236 -ζ2312-ζ2311 -ζ2322-ζ23 symplectic faithful, Schur index 2 ρ17 2 -2 0 0 ζ2320+ζ233 ζ2321+ζ232 ζ2316+ζ237 ζ2312+ζ2311 ζ2317+ζ236 ζ2322+ζ23 ζ2319+ζ234 ζ2314+ζ239 ζ2318+ζ235 ζ2313+ζ2310 ζ2315+ζ238 -ζ2315-ζ238 -ζ2320-ζ233 -ζ2321-ζ232 -ζ2316-ζ237 -ζ2312-ζ2311 -ζ2317-ζ236 -ζ2322-ζ23 -ζ2319-ζ234 -ζ2314-ζ239 -ζ2318-ζ235 -ζ2313-ζ2310 symplectic faithful, Schur index 2 ρ18 2 -2 0 0 ζ2315+ζ238 ζ2313+ζ2310 ζ2312+ζ2311 ζ2314+ζ239 ζ2316+ζ237 ζ2318+ζ235 ζ2320+ζ233 ζ2322+ζ23 ζ2321+ζ232 ζ2319+ζ234 ζ2317+ζ236 -ζ2317-ζ236 -ζ2315-ζ238 -ζ2313-ζ2310 -ζ2312-ζ2311 -ζ2314-ζ239 -ζ2316-ζ237 -ζ2318-ζ235 -ζ2320-ζ233 -ζ2322-ζ23 -ζ2321-ζ232 -ζ2319-ζ234 symplectic faithful, Schur index 2 ρ19 2 -2 0 0 ζ2318+ζ235 ζ2312+ζ2311 ζ2319+ζ234 ζ2320+ζ233 ζ2313+ζ2310 ζ2317+ζ236 ζ2322+ζ23 ζ2315+ζ238 ζ2316+ζ237 ζ2314+ζ239 ζ2321+ζ232 -ζ2321-ζ232 -ζ2318-ζ235 -ζ2312-ζ2311 -ζ2319-ζ234 -ζ2320-ζ233 -ζ2313-ζ2310 -ζ2317-ζ236 -ζ2322-ζ23 -ζ2315-ζ238 -ζ2316-ζ237 -ζ2314-ζ239 symplectic faithful, Schur index 2 ρ20 2 -2 0 0 ζ2319+ζ234 ζ2318+ζ235 ζ2317+ζ236 ζ2316+ζ237 ζ2315+ζ238 ζ2314+ζ239 ζ2313+ζ2310 ζ2312+ζ2311 ζ2322+ζ23 ζ2321+ζ232 ζ2320+ζ233 -ζ2320-ζ233 -ζ2319-ζ234 -ζ2318-ζ235 -ζ2317-ζ236 -ζ2316-ζ237 -ζ2315-ζ238 -ζ2314-ζ239 -ζ2313-ζ2310 -ζ2312-ζ2311 -ζ2322-ζ23 -ζ2321-ζ232 symplectic faithful, Schur index 2 ρ21 2 -2 0 0 ζ2314+ζ239 ζ2317+ζ236 ζ2321+ζ232 ζ2313+ζ2310 ζ2318+ζ235 ζ2320+ζ233 ζ2312+ζ2311 ζ2319+ζ234 ζ2315+ζ238 ζ2316+ζ237 ζ2322+ζ23 -ζ2322-ζ23 -ζ2314-ζ239 -ζ2317-ζ236 -ζ2321-ζ232 -ζ2313-ζ2310 -ζ2318-ζ235 -ζ2320-ζ233 -ζ2312-ζ2311 -ζ2319-ζ234 -ζ2315-ζ238 -ζ2316-ζ237 symplectic faithful, Schur index 2 ρ22 2 -2 0 0 ζ2322+ζ23 ζ2316+ζ237 ζ2313+ζ2310 ζ2319+ζ234 ζ2321+ζ232 ζ2315+ζ238 ζ2314+ζ239 ζ2320+ζ233 ζ2317+ζ236 ζ2312+ζ2311 ζ2318+ζ235 -ζ2318-ζ235 -ζ2322-ζ23 -ζ2316-ζ237 -ζ2313-ζ2310 -ζ2319-ζ234 -ζ2321-ζ232 -ζ2315-ζ238 -ζ2314-ζ239 -ζ2320-ζ233 -ζ2317-ζ236 -ζ2312-ζ2311 symplectic faithful, Schur index 2 ρ23 2 -2 0 0 ζ2312+ζ2311 ζ2315+ζ238 ζ2318+ζ235 ζ2321+ζ232 ζ2322+ζ23 ζ2319+ζ234 ζ2316+ζ237 ζ2313+ζ2310 ζ2320+ζ233 ζ2317+ζ236 ζ2314+ζ239 -ζ2314-ζ239 -ζ2312-ζ2311 -ζ2315-ζ238 -ζ2318-ζ235 -ζ2321-ζ232 -ζ2322-ζ23 -ζ2319-ζ234 -ζ2316-ζ237 -ζ2313-ζ2310 -ζ2320-ζ233 -ζ2317-ζ236 symplectic faithful, Schur index 2 ρ24 2 -2 0 0 ζ2313+ζ2310 ζ2322+ζ23 ζ2315+ζ238 ζ2317+ζ236 ζ2320+ζ233 ζ2312+ζ2311 ζ2321+ζ232 ζ2316+ζ237 ζ2314+ζ239 ζ2318+ζ235 ζ2319+ζ234 -ζ2319-ζ234 -ζ2313-ζ2310 -ζ2322-ζ23 -ζ2315-ζ238 -ζ2317-ζ236 -ζ2320-ζ233 -ζ2312-ζ2311 -ζ2321-ζ232 -ζ2316-ζ237 -ζ2314-ζ239 -ζ2318-ζ235 symplectic faithful, Schur index 2 ρ25 2 -2 0 0 ζ2317+ζ236 ζ2319+ζ234 ζ2314+ζ239 ζ2322+ζ23 ζ2312+ζ2311 ζ2321+ζ232 ζ2315+ζ238 ζ2318+ζ235 ζ2313+ζ2310 ζ2320+ζ233 ζ2316+ζ237 -ζ2316-ζ237 -ζ2317-ζ236 -ζ2319-ζ234 -ζ2314-ζ239 -ζ2322-ζ23 -ζ2312-ζ2311 -ζ2321-ζ232 -ζ2315-ζ238 -ζ2318-ζ235 -ζ2313-ζ2310 -ζ2320-ζ233 symplectic faithful, Schur index 2 ρ26 2 -2 0 0 ζ2316+ζ237 ζ2320+ζ233 ζ2322+ζ23 ζ2318+ζ235 ζ2314+ζ239 ζ2313+ζ2310 ζ2317+ζ236 ζ2321+ζ232 ζ2319+ζ234 ζ2315+ζ238 ζ2312+ζ2311 -ζ2312-ζ2311 -ζ2316-ζ237 -ζ2320-ζ233 -ζ2322-ζ23 -ζ2318-ζ235 -ζ2314-ζ239 -ζ2313-ζ2310 -ζ2317-ζ236 -ζ2321-ζ232 -ζ2319-ζ234 -ζ2315-ζ238 symplectic faithful, Schur index 2

Smallest permutation representation of Dic23
Regular action 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 89 24 66)(2 88 25 65)(3 87 26 64)(4 86 27 63)(5 85 28 62)(6 84 29 61)(7 83 30 60)(8 82 31 59)(9 81 32 58)(10 80 33 57)(11 79 34 56)(12 78 35 55)(13 77 36 54)(14 76 37 53)(15 75 38 52)(16 74 39 51)(17 73 40 50)(18 72 41 49)(19 71 42 48)(20 70 43 47)(21 69 44 92)(22 68 45 91)(23 67 46 90)

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

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

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

Dic23 is a maximal subgroup of   Dic46  C4×D23  C23⋊D4  Dic69  Dic115  C23⋊F5
Dic23 is a maximal quotient of   C23⋊C8  Dic69  Dic115  C23⋊F5

Matrix representation of Dic23 in GL2(𝔽47) generated by

 41 30 17 1
,
 31 17 7 16
G:=sub<GL(2,GF(47))| [41,17,30,1],[31,7,17,16] >;

Dic23 in GAP, Magma, Sage, TeX

{\rm Dic}_{23}
% in TeX

G:=Group("Dic23");
// GroupNames label

G:=SmallGroup(92,1);
// by ID

G=gap.SmallGroup(92,1);
# by ID

G:=PCGroup([3,-2,-2,-23,6,794]);
// Polycyclic

G:=Group<a,b|a^46=1,b^2=a^23,b*a*b^-1=a^-1>;
// generators/relations

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