direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C29, C4⋊C58, C22⋊C58, C116⋊3C2, C58.6C22, (C2×C58)⋊1C2, C2.1(C2×C58), SmallGroup(232,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C29
G = < a,b,c | a29=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C29
►On 116 points
Generators in S116
(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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116)
(1 64 110 32)(2 65 111 33)(3 66 112 34)(4 67 113 35)(5 68 114 36)(6 69 115 37)(7 70 116 38)(8 71 88 39)(9 72 89 40)(10 73 90 41)(11 74 91 42)(12 75 92 43)(13 76 93 44)(14 77 94 45)(15 78 95 46)(16 79 96 47)(17 80 97 48)(18 81 98 49)(19 82 99 50)(20 83 100 51)(21 84 101 52)(22 85 102 53)(23 86 103 54)(24 87 104 55)(25 59 105 56)(26 60 106 57)(27 61 107 58)(28 62 108 30)(29 63 109 31)
(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 69)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 81)(50 82)(51 83)(52 84)(53 85)(54 86)(55 87)(56 59)(57 60)(58 61)
G:=sub<Sym(116)| (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)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,64,110,32)(2,65,111,33)(3,66,112,34)(4,67,113,35)(5,68,114,36)(6,69,115,37)(7,70,116,38)(8,71,88,39)(9,72,89,40)(10,73,90,41)(11,74,91,42)(12,75,92,43)(13,76,93,44)(14,77,94,45)(15,78,95,46)(16,79,96,47)(17,80,97,48)(18,81,98,49)(19,82,99,50)(20,83,100,51)(21,84,101,52)(22,85,102,53)(23,86,103,54)(24,87,104,55)(25,59,105,56)(26,60,106,57)(27,61,107,58)(28,62,108,30)(29,63,109,31), (30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,59)(57,60)(58,61)>;
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)(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116), (1,64,110,32)(2,65,111,33)(3,66,112,34)(4,67,113,35)(5,68,114,36)(6,69,115,37)(7,70,116,38)(8,71,88,39)(9,72,89,40)(10,73,90,41)(11,74,91,42)(12,75,92,43)(13,76,93,44)(14,77,94,45)(15,78,95,46)(16,79,96,47)(17,80,97,48)(18,81,98,49)(19,82,99,50)(20,83,100,51)(21,84,101,52)(22,85,102,53)(23,86,103,54)(24,87,104,55)(25,59,105,56)(26,60,106,57)(27,61,107,58)(28,62,108,30)(29,63,109,31), (30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,69)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(47,79)(48,80)(49,81)(50,82)(51,83)(52,84)(53,85)(54,86)(55,87)(56,59)(57,60)(58,61) );
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),(88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116)], [(1,64,110,32),(2,65,111,33),(3,66,112,34),(4,67,113,35),(5,68,114,36),(6,69,115,37),(7,70,116,38),(8,71,88,39),(9,72,89,40),(10,73,90,41),(11,74,91,42),(12,75,92,43),(13,76,93,44),(14,77,94,45),(15,78,95,46),(16,79,96,47),(17,80,97,48),(18,81,98,49),(19,82,99,50),(20,83,100,51),(21,84,101,52),(22,85,102,53),(23,86,103,54),(24,87,104,55),(25,59,105,56),(26,60,106,57),(27,61,107,58),(28,62,108,30),(29,63,109,31)], [(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,69),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(47,79),(48,80),(49,81),(50,82),(51,83),(52,84),(53,85),(54,86),(55,87),(56,59),(57,60),(58,61)]])
D4×C29 is a maximal subgroup of
D4⋊D29 D4.D29 D4⋊2D29
145 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 29A | ··· | 29AB | 58A | ··· | 58AB | 58AC | ··· | 58CF | 116A | ··· | 116AB |
| order | 1 | 2 | 2 | 2 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
145 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C29 | C58 | C58 | D4 | D4×C29 |
| kernel | D4×C29 | C116 | C2×C58 | D4 | C4 | C22 | C29 | C1 |
| # reps | 1 | 1 | 2 | 28 | 28 | 56 | 1 | 28 |
Matrix representation of D4×C29 ►in GL2(𝔽233) generated by
| 142 | 0 |
| 0 | 142 |
| 127 | 2 |
| 90 | 106 |
| 1 | 0 |
| 106 | 232 |
G:=sub<GL(2,GF(233))| [142,0,0,142],[127,90,2,106],[1,106,0,232] >;
D4×C29 in GAP, Magma, Sage, TeX
D_4\times C_{29} % in TeX
G:=Group("D4xC29"); // GroupNames label
G:=SmallGroup(232,10);
// by ID
G=gap.SmallGroup(232,10);
# by ID
G:=PCGroup([4,-2,-2,-29,-2,945]);
// Polycyclic
G:=Group<a,b,c|a^29=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export