metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: D92, C4⋊D23, C23⋊1D4, C92⋊1C2, D46⋊1C2, C2.4D46, C46.3C22, sometimes denoted D184 or Dih92 or Dih184, SmallGroup(184,5)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D92
G = < a,b | a92=b2=1, bab=a-1 >
Smallest permutation representation of D92
►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 92)(2 91)(3 90)(4 89)(5 88)(6 87)(7 86)(8 85)(9 84)(10 83)(11 82)(12 81)(13 80)(14 79)(15 78)(16 77)(17 76)(18 75)(19 74)(20 73)(21 72)(22 71)(23 70)(24 69)(25 68)(26 67)(27 66)(28 65)(29 64)(30 63)(31 62)(32 61)(33 60)(34 59)(35 58)(36 57)(37 56)(38 55)(39 54)(40 53)(41 52)(42 51)(43 50)(44 49)(45 48)(46 47)
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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47)>;
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), (1,92)(2,91)(3,90)(4,89)(5,88)(6,87)(7,86)(8,85)(9,84)(10,83)(11,82)(12,81)(13,80)(14,79)(15,78)(16,77)(17,76)(18,75)(19,74)(20,73)(21,72)(22,71)(23,70)(24,69)(25,68)(26,67)(27,66)(28,65)(29,64)(30,63)(31,62)(32,61)(33,60)(34,59)(35,58)(36,57)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(43,50)(44,49)(45,48)(46,47) );
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)], [(1,92),(2,91),(3,90),(4,89),(5,88),(6,87),(7,86),(8,85),(9,84),(10,83),(11,82),(12,81),(13,80),(14,79),(15,78),(16,77),(17,76),(18,75),(19,74),(20,73),(21,72),(22,71),(23,70),(24,69),(25,68),(26,67),(27,66),(28,65),(29,64),(30,63),(31,62),(32,61),(33,60),(34,59),(35,58),(36,57),(37,56),(38,55),(39,54),(40,53),(41,52),(42,51),(43,50),(44,49),(45,48),(46,47)]])
D92 is a maximal subgroup of
C184⋊C2 D184 D4⋊D23 Q8⋊D23 D92⋊5C2 D4×D23 D92⋊C2
D92 is a maximal quotient of C184⋊C2 D184 Dic92 C92⋊C4 D46⋊C4
49 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V |
| order | 1 | 2 | 2 | 2 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 46 | 46 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
49 irreducible representations
| dim | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | D4 | D23 | D46 | D92 |
| kernel | D92 | C92 | D46 | C23 | C4 | C2 | C1 |
| # reps | 1 | 1 | 2 | 1 | 11 | 11 | 22 |
Matrix representation of D92 ►in GL2(𝔽277) generated by
| 248 | 197 |
| 127 | 16 |
| 145 | 139 |
| 56 | 132 |
G:=sub<GL(2,GF(277))| [248,127,197,16],[145,56,139,132] >;
D92 in GAP, Magma, Sage, TeX
D_{92} % in TeX
G:=Group("D92"); // GroupNames label
G:=SmallGroup(184,5);
// by ID
G=gap.SmallGroup(184,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-23,49,21,2819]);
// Polycyclic
G:=Group<a,b|a^92=b^2=1,b*a*b=a^-1>;
// generators/relations
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