direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C23, C4⋊C46, C92⋊3C2, C22⋊C46, C46.6C22, (C2×C46)⋊1C2, C2.1(C2×C46), SmallGroup(184,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C23
G = < a,b,c | a23=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
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 57 87 33)(2 58 88 34)(3 59 89 35)(4 60 90 36)(5 61 91 37)(6 62 92 38)(7 63 70 39)(8 64 71 40)(9 65 72 41)(10 66 73 42)(11 67 74 43)(12 68 75 44)(13 69 76 45)(14 47 77 46)(15 48 78 24)(16 49 79 25)(17 50 80 26)(18 51 81 27)(19 52 82 28)(20 53 83 29)(21 54 84 30)(22 55 85 31)(23 56 86 32)
(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)(41 65)(42 66)(43 67)(44 68)(45 69)(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), (1,57,87,33)(2,58,88,34)(3,59,89,35)(4,60,90,36)(5,61,91,37)(6,62,92,38)(7,63,70,39)(8,64,71,40)(9,65,72,41)(10,66,73,42)(11,67,74,43)(12,68,75,44)(13,69,76,45)(14,47,77,46)(15,48,78,24)(16,49,79,25)(17,50,80,26)(18,51,81,27)(19,52,82,28)(20,53,83,29)(21,54,84,30)(22,55,85,31)(23,56,86,32), (24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(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), (1,57,87,33)(2,58,88,34)(3,59,89,35)(4,60,90,36)(5,61,91,37)(6,62,92,38)(7,63,70,39)(8,64,71,40)(9,65,72,41)(10,66,73,42)(11,67,74,43)(12,68,75,44)(13,69,76,45)(14,47,77,46)(15,48,78,24)(16,49,79,25)(17,50,80,26)(18,51,81,27)(19,52,82,28)(20,53,83,29)(21,54,84,30)(22,55,85,31)(23,56,86,32), (24,48)(25,49)(26,50)(27,51)(28,52)(29,53)(30,54)(31,55)(32,56)(33,57)(34,58)(35,59)(36,60)(37,61)(38,62)(39,63)(40,64)(41,65)(42,66)(43,67)(44,68)(45,69)(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)], [(1,57,87,33),(2,58,88,34),(3,59,89,35),(4,60,90,36),(5,61,91,37),(6,62,92,38),(7,63,70,39),(8,64,71,40),(9,65,72,41),(10,66,73,42),(11,67,74,43),(12,68,75,44),(13,69,76,45),(14,47,77,46),(15,48,78,24),(16,49,79,25),(17,50,80,26),(18,51,81,27),(19,52,82,28),(20,53,83,29),(21,54,84,30),(22,55,85,31),(23,56,86,32)], [(24,48),(25,49),(26,50),(27,51),(28,52),(29,53),(30,54),(31,55),(32,56),(33,57),(34,58),(35,59),(36,60),(37,61),(38,62),(39,63),(40,64),(41,65),(42,66),(43,67),(44,68),(45,69),(46,47)]])
D4×C23 is a maximal subgroup of
D4⋊D23 D4.D23 D4⋊2D23
115 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46BN | 92A | ··· | 92V |
| order | 1 | 2 | 2 | 2 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C23 | C46 | C46 | D4 | D4×C23 |
| kernel | D4×C23 | C92 | C2×C46 | D4 | C4 | C22 | C23 | C1 |
| # reps | 1 | 1 | 2 | 22 | 22 | 44 | 1 | 22 |
Matrix representation of D4×C23 ►in GL2(𝔽47) generated by
| 27 | 0 |
| 0 | 27 |
| 0 | 30 |
| 36 | 0 |
| 0 | 17 |
| 36 | 0 |
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