direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D23, C92⋊2C2, D46.C2, C2.1D46, Dic23⋊2C2, C46.2C22, C23⋊1(C2×C4), SmallGroup(184,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C23 — C4×D23 |
Generators and relations for C4×D23
G = < a,b,c | a4=b23=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D23
►On 92 points
Generators in S92
(1 75 26 68)(2 76 27 69)(3 77 28 47)(4 78 29 48)(5 79 30 49)(6 80 31 50)(7 81 32 51)(8 82 33 52)(9 83 34 53)(10 84 35 54)(11 85 36 55)(12 86 37 56)(13 87 38 57)(14 88 39 58)(15 89 40 59)(16 90 41 60)(17 91 42 61)(18 92 43 62)(19 70 44 63)(20 71 45 64)(21 72 46 65)(22 73 24 66)(23 74 25 67)
(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 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 27)(25 26)(28 46)(29 45)(30 44)(31 43)(32 42)(33 41)(34 40)(35 39)(36 38)(47 65)(48 64)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(66 69)(67 68)(70 79)(71 78)(72 77)(73 76)(74 75)(80 92)(81 91)(82 90)(83 89)(84 88)(85 87)
G:=sub<Sym(92)| (1,75,26,68)(2,76,27,69)(3,77,28,47)(4,78,29,48)(5,79,30,49)(6,80,31,50)(7,81,32,51)(8,82,33,52)(9,83,34,53)(10,84,35,54)(11,85,36,55)(12,86,37,56)(13,87,38,57)(14,88,39,58)(15,89,40,59)(16,90,41,60)(17,91,42,61)(18,92,43,62)(19,70,44,63)(20,71,45,64)(21,72,46,65)(22,73,24,66)(23,74,25,67), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,27)(25,26)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(66,69)(67,68)(70,79)(71,78)(72,77)(73,76)(74,75)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87)>;
G:=Group( (1,75,26,68)(2,76,27,69)(3,77,28,47)(4,78,29,48)(5,79,30,49)(6,80,31,50)(7,81,32,51)(8,82,33,52)(9,83,34,53)(10,84,35,54)(11,85,36,55)(12,86,37,56)(13,87,38,57)(14,88,39,58)(15,89,40,59)(16,90,41,60)(17,91,42,61)(18,92,43,62)(19,70,44,63)(20,71,45,64)(21,72,46,65)(22,73,24,66)(23,74,25,67), (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), (1,23)(2,22)(3,21)(4,20)(5,19)(6,18)(7,17)(8,16)(9,15)(10,14)(11,13)(24,27)(25,26)(28,46)(29,45)(30,44)(31,43)(32,42)(33,41)(34,40)(35,39)(36,38)(47,65)(48,64)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(66,69)(67,68)(70,79)(71,78)(72,77)(73,76)(74,75)(80,92)(81,91)(82,90)(83,89)(84,88)(85,87) );
G=PermutationGroup([[(1,75,26,68),(2,76,27,69),(3,77,28,47),(4,78,29,48),(5,79,30,49),(6,80,31,50),(7,81,32,51),(8,82,33,52),(9,83,34,53),(10,84,35,54),(11,85,36,55),(12,86,37,56),(13,87,38,57),(14,88,39,58),(15,89,40,59),(16,90,41,60),(17,91,42,61),(18,92,43,62),(19,70,44,63),(20,71,45,64),(21,72,46,65),(22,73,24,66),(23,74,25,67)], [(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)], [(1,23),(2,22),(3,21),(4,20),(5,19),(6,18),(7,17),(8,16),(9,15),(10,14),(11,13),(24,27),(25,26),(28,46),(29,45),(30,44),(31,43),(32,42),(33,41),(34,40),(35,39),(36,38),(47,65),(48,64),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(66,69),(67,68),(70,79),(71,78),(72,77),(73,76),(74,75),(80,92),(81,91),(82,90),(83,89),(84,88),(85,87)]])
C4×D23 is a maximal subgroup of
C8⋊D23 D92⋊5C2 D4⋊2D23 D92⋊C2
C4×D23 is a maximal quotient of C8⋊D23 Dic23⋊C4 D46⋊C4
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 23A | ··· | 23K | 46A | ··· | 46K | 92A | ··· | 92V |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 23 | ··· | 23 | 46 | ··· | 46 | 92 | ··· | 92 |
| size | 1 | 1 | 23 | 23 | 1 | 1 | 23 | 23 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D23 | D46 | C4×D23 |
| kernel | C4×D23 | Dic23 | C92 | D46 | D23 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 11 | 11 | 22 |
Matrix representation of C4×D23 ►in GL2(𝔽277) generated by
| 217 | 0 |
| 0 | 217 |
| 0 | 1 |
| 276 | 57 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(277))| [217,0,0,217],[0,276,1,57],[0,1,1,0] >;
C4×D23 in GAP, Magma, Sage, TeX
C_4\times D_{23} % in TeX
G:=Group("C4xD23"); // GroupNames label
G:=SmallGroup(184,4);
// by ID
G=gap.SmallGroup(184,4);
# by ID
G:=PCGroup([4,-2,-2,-2,-23,21,2819]);
// Polycyclic
G:=Group<a,b,c|a^4=b^23=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export