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G = Dic23  order 92 = 22·23

Dicyclic group

metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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 >

C1
C2
C23
23C4
C46
Dic23

Character table of Dic23

 class 124A4B23A23B23C23D23E23F23G23H23I23J23K46A46B46C46D46E46F46G46H46I46J46K
 size 1123232222222222222222222222
ρ111111111111111111111111111    trivial
ρ211-1-11111111111111111111111    linear of order 2
ρ31-1i-i11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ41-1-ii11111111111-1-1-1-1-1-1-1-1-1-1-1    linear of order 4
ρ52200ζ2315+ζ238ζ2313+ζ2310ζ2312+ζ2311ζ2314+ζ239ζ2316+ζ237ζ2318+ζ235ζ2320+ζ233ζ2322+ζ23ζ2321+ζ232ζ2319+ζ234ζ2317+ζ236ζ2317+ζ236ζ2315+ζ238ζ2313+ζ2310ζ2312+ζ2311ζ2314+ζ239ζ2316+ζ237ζ2318+ζ235ζ2320+ζ233ζ2322+ζ23ζ2321+ζ232ζ2319+ζ234    orthogonal lifted from D23
ρ62200ζ2319+ζ234ζ2318+ζ235ζ2317+ζ236ζ2316+ζ237ζ2315+ζ238ζ2314+ζ239ζ2313+ζ2310ζ2312+ζ2311ζ2322+ζ23ζ2321+ζ232ζ2320+ζ233ζ2320+ζ233ζ2319+ζ234ζ2318+ζ235ζ2317+ζ236ζ2316+ζ237ζ2315+ζ238ζ2314+ζ239ζ2313+ζ2310ζ2312+ζ2311ζ2322+ζ23ζ2321+ζ232    orthogonal lifted from D23
ρ72200ζ2316+ζ237ζ2320+ζ233ζ2322+ζ23ζ2318+ζ235ζ2314+ζ239ζ2313+ζ2310ζ2317+ζ236ζ2321+ζ232ζ2319+ζ234ζ2315+ζ238ζ2312+ζ2311ζ2312+ζ2311ζ2316+ζ237ζ2320+ζ233ζ2322+ζ23ζ2318+ζ235ζ2314+ζ239ζ2313+ζ2310ζ2317+ζ236ζ2321+ζ232ζ2319+ζ234ζ2315+ζ238    orthogonal lifted from D23
ρ82200ζ2317+ζ236ζ2319+ζ234ζ2314+ζ239ζ2322+ζ23ζ2312+ζ2311ζ2321+ζ232ζ2315+ζ238ζ2318+ζ235ζ2313+ζ2310ζ2320+ζ233ζ2316+ζ237ζ2316+ζ237ζ2317+ζ236ζ2319+ζ234ζ2314+ζ239ζ2322+ζ23ζ2312+ζ2311ζ2321+ζ232ζ2315+ζ238ζ2318+ζ235ζ2313+ζ2310ζ2320+ζ233    orthogonal lifted from D23
ρ92200ζ2313+ζ2310ζ2322+ζ23ζ2315+ζ238ζ2317+ζ236ζ2320+ζ233ζ2312+ζ2311ζ2321+ζ232ζ2316+ζ237ζ2314+ζ239ζ2318+ζ235ζ2319+ζ234ζ2319+ζ234ζ2313+ζ2310ζ2322+ζ23ζ2315+ζ238ζ2317+ζ236ζ2320+ζ233ζ2312+ζ2311ζ2321+ζ232ζ2316+ζ237ζ2314+ζ239ζ2318+ζ235    orthogonal lifted from D23
ρ102200ζ2320+ζ233ζ2321+ζ232ζ2316+ζ237ζ2312+ζ2311ζ2317+ζ236ζ2322+ζ23ζ2319+ζ234ζ2314+ζ239ζ2318+ζ235ζ2313+ζ2310ζ2315+ζ238ζ2315+ζ238ζ2320+ζ233ζ2321+ζ232ζ2316+ζ237ζ2312+ζ2311ζ2317+ζ236ζ2322+ζ23ζ2319+ζ234ζ2314+ζ239ζ2318+ζ235ζ2313+ζ2310    orthogonal lifted from D23
ρ112200ζ2318+ζ235ζ2312+ζ2311ζ2319+ζ234ζ2320+ζ233ζ2313+ζ2310ζ2317+ζ236ζ2322+ζ23ζ2315+ζ238ζ2316+ζ237ζ2314+ζ239ζ2321+ζ232ζ2321+ζ232ζ2318+ζ235ζ2312+ζ2311ζ2319+ζ234ζ2320+ζ233ζ2313+ζ2310ζ2317+ζ236ζ2322+ζ23ζ2315+ζ238ζ2316+ζ237ζ2314+ζ239    orthogonal lifted from D23
ρ122200ζ2314+ζ239ζ2317+ζ236ζ2321+ζ232ζ2313+ζ2310ζ2318+ζ235ζ2320+ζ233ζ2312+ζ2311ζ2319+ζ234ζ2315+ζ238ζ2316+ζ237ζ2322+ζ23ζ2322+ζ23ζ2314+ζ239ζ2317+ζ236ζ2321+ζ232ζ2313+ζ2310ζ2318+ζ235ζ2320+ζ233ζ2312+ζ2311ζ2319+ζ234ζ2315+ζ238ζ2316+ζ237    orthogonal lifted from D23
ρ132200ζ2312+ζ2311ζ2315+ζ238ζ2318+ζ235ζ2321+ζ232ζ2322+ζ23ζ2319+ζ234ζ2316+ζ237ζ2313+ζ2310ζ2320+ζ233ζ2317+ζ236ζ2314+ζ239ζ2314+ζ239ζ2312+ζ2311ζ2315+ζ238ζ2318+ζ235ζ2321+ζ232ζ2322+ζ23ζ2319+ζ234ζ2316+ζ237ζ2313+ζ2310ζ2320+ζ233ζ2317+ζ236    orthogonal lifted from D23
ρ142200ζ2321+ζ232ζ2314+ζ239ζ2320+ζ233ζ2315+ζ238ζ2319+ζ234ζ2316+ζ237ζ2318+ζ235ζ2317+ζ236ζ2312+ζ2311ζ2322+ζ23ζ2313+ζ2310ζ2313+ζ2310ζ2321+ζ232ζ2314+ζ239ζ2320+ζ233ζ2315+ζ238ζ2319+ζ234ζ2316+ζ237ζ2318+ζ235ζ2317+ζ236ζ2312+ζ2311ζ2322+ζ23    orthogonal lifted from D23
ρ152200ζ2322+ζ23ζ2316+ζ237ζ2313+ζ2310ζ2319+ζ234ζ2321+ζ232ζ2315+ζ238ζ2314+ζ239ζ2320+ζ233ζ2317+ζ236ζ2312+ζ2311ζ2318+ζ235ζ2318+ζ235ζ2322+ζ23ζ2316+ζ237ζ2313+ζ2310ζ2319+ζ234ζ2321+ζ232ζ2315+ζ238ζ2314+ζ239ζ2320+ζ233ζ2317+ζ236ζ2312+ζ2311    orthogonal lifted from D23
ρ162-200ζ2321+ζ232ζ2314+ζ239ζ2320+ζ233ζ2315+ζ238ζ2319+ζ234ζ2316+ζ237ζ2318+ζ235ζ2317+ζ236ζ2312+ζ2311ζ2322+ζ23ζ2313+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
ρ172-200ζ2320+ζ233ζ2321+ζ232ζ2316+ζ237ζ2312+ζ2311ζ2317+ζ236ζ2322+ζ23ζ2319+ζ234ζ2314+ζ239ζ2318+ζ235ζ2313+ζ2310ζ2315+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
ρ182-200ζ2315+ζ238ζ2313+ζ2310ζ2312+ζ2311ζ2314+ζ239ζ2316+ζ237ζ2318+ζ235ζ2320+ζ233ζ2322+ζ23ζ2321+ζ232ζ2319+ζ234ζ2317+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
ρ192-200ζ2318+ζ235ζ2312+ζ2311ζ2319+ζ234ζ2320+ζ233ζ2313+ζ2310ζ2317+ζ236ζ2322+ζ23ζ2315+ζ238ζ2316+ζ237ζ2314+ζ239ζ2321+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
ρ202-200ζ2319+ζ234ζ2318+ζ235ζ2317+ζ236ζ2316+ζ237ζ2315+ζ238ζ2314+ζ239ζ2313+ζ2310ζ2312+ζ2311ζ2322+ζ23ζ2321+ζ232ζ2320+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
ρ212-200ζ2314+ζ239ζ2317+ζ236ζ2321+ζ232ζ2313+ζ2310ζ2318+ζ235ζ2320+ζ233ζ2312+ζ2311ζ2319+ζ234ζ2315+ζ238ζ2316+ζ237ζ2322+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
ρ222-200ζ2322+ζ23ζ2316+ζ237ζ2313+ζ2310ζ2319+ζ234ζ2321+ζ232ζ2315+ζ238ζ2314+ζ239ζ2320+ζ233ζ2317+ζ236ζ2312+ζ2311ζ2318+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
ρ232-200ζ2312+ζ2311ζ2315+ζ238ζ2318+ζ235ζ2321+ζ232ζ2322+ζ23ζ2319+ζ234ζ2316+ζ237ζ2313+ζ2310ζ2320+ζ233ζ2317+ζ236ζ2314+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
ρ242-200ζ2313+ζ2310ζ2322+ζ23ζ2315+ζ238ζ2317+ζ236ζ2320+ζ233ζ2312+ζ2311ζ2321+ζ232ζ2316+ζ237ζ2314+ζ239ζ2318+ζ235ζ2319+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
ρ252-200ζ2317+ζ236ζ2319+ζ234ζ2314+ζ239ζ2322+ζ23ζ2312+ζ2311ζ2321+ζ232ζ2315+ζ238ζ2318+ζ235ζ2313+ζ2310ζ2320+ζ233ζ2316+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
ρ262-200ζ2316+ζ237ζ2320+ζ233ζ2322+ζ23ζ2318+ζ235ζ2314+ζ239ζ2313+ζ2310ζ2317+ζ236ζ2321+ζ232ζ2319+ζ234ζ2315+ζ238ζ2312+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

4130
171
,
3117
716
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
 

Export

Subgroup lattice of Dic23 in TeX
Character table of Dic23 in TeX

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