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G = D4×C23order 184 = 23·23

Direct product of C23 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C23, C4⋊C46, C923C2, C22⋊C46, C46.6C22, (C2×C46)⋊1C2, C2.1(C2×C46), SmallGroup(184,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C23
C1C2C46C2×C46 — D4×C23
C1C2 — D4×C23
C1C46 — D4×C23

Generators and relations for D4×C23
 G = < a,b,c | a23=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C46
2C46

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

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

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

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

D4×C23 is a maximal subgroup of   D4⋊D23  D4.D23  D42D23

115 conjugacy classes

class 1 2A2B2C 4 23A···23V46A···46V46W···46BN92A···92V
order1222423···2346···4646···4692···92
size112221···11···12···22···2

115 irreducible representations

dim11111122
type++++
imageC1C2C2C23C46C46D4D4×C23
kernelD4×C23C92C2×C46D4C4C22C23C1
# reps112222244122

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

270
027
,
030
360
,
017
360
G:=sub<GL(2,GF(47))| [27,0,0,27],[0,36,30,0],[0,36,17,0] >;

D4×C23 in GAP, Magma, Sage, TeX

D_4\times C_{23}
% in TeX

G:=Group("D4xC23");
// GroupNames label

G:=SmallGroup(184,9);
// by ID

G=gap.SmallGroup(184,9);
# by ID

G:=PCGroup([4,-2,-2,-23,-2,753]);
// Polycyclic

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

Export

Subgroup lattice of D4×C23 in TeX

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