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G = C23⋊D4order 184 = 23·23

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

metabelian, supersoluble, monomial, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C46 — C23⋊D4
C1C23C46D46 — C23⋊D4
C23C46 — C23⋊D4
C1C2C22

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

2C2
46C2
23C4
23C22
2D23
2C46
23D4

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 47 25 91)(2 69 26 90)(3 68 27 89)(4 67 28 88)(5 66 29 87)(6 65 30 86)(7 64 31 85)(8 63 32 84)(9 62 33 83)(10 61 34 82)(11 60 35 81)(12 59 36 80)(13 58 37 79)(14 57 38 78)(15 56 39 77)(16 55 40 76)(17 54 41 75)(18 53 42 74)(19 52 43 73)(20 51 44 72)(21 50 45 71)(22 49 46 70)(23 48 24 92)
(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(24 26)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)(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), (1,47,25,91)(2,69,26,90)(3,68,27,89)(4,67,28,88)(5,66,29,87)(6,65,30,86)(7,64,31,85)(8,63,32,84)(9,62,33,83)(10,61,34,82)(11,60,35,81)(12,59,36,80)(13,58,37,79)(14,57,38,78)(15,56,39,77)(16,55,40,76)(17,54,41,75)(18,53,42,74)(19,52,43,73)(20,51,44,72)(21,50,45,71)(22,49,46,70)(23,48,24,92), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(24,26)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(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), (1,47,25,91)(2,69,26,90)(3,68,27,89)(4,67,28,88)(5,66,29,87)(6,65,30,86)(7,64,31,85)(8,63,32,84)(9,62,33,83)(10,61,34,82)(11,60,35,81)(12,59,36,80)(13,58,37,79)(14,57,38,78)(15,56,39,77)(16,55,40,76)(17,54,41,75)(18,53,42,74)(19,52,43,73)(20,51,44,72)(21,50,45,71)(22,49,46,70)(23,48,24,92), (2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(24,26)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(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)], [(1,47,25,91),(2,69,26,90),(3,68,27,89),(4,67,28,88),(5,66,29,87),(6,65,30,86),(7,64,31,85),(8,63,32,84),(9,62,33,83),(10,61,34,82),(11,60,35,81),(12,59,36,80),(13,58,37,79),(14,57,38,78),(15,56,39,77),(16,55,40,76),(17,54,41,75),(18,53,42,74),(19,52,43,73),(20,51,44,72),(21,50,45,71),(22,49,46,70),(23,48,24,92)], [(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(24,26),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37),(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 2A2B2C 4 23A···23K46A···46AG
order1222423···2346···46
size11246462···22···2

49 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D23D46C23⋊D4
kernelC23⋊D4Dic23D46C2×C46C23C22C2C1
# reps11111111122

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

4115
1525
,
01
460
,
246
623
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

Export

Subgroup lattice of C23⋊D4 in TeX

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