direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D29, C4⋊1D58, C116⋊C22, D116⋊3C2, C22⋊1D58, D58⋊2C22, C58.5C23, Dic29⋊1C22, C29⋊2(C2×D4), (C2×C58)⋊C22, (C4×D29)⋊1C2, (D4×C29)⋊2C2, C29⋊D4⋊1C2, (C22×D29)⋊2C2, C2.6(C22×D29), SmallGroup(464,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D29
G = < a,b,c,d | a4=b2=c29=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 770 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C29, D29, D29, C58, C58, Dic29, C116, D58, D58, D58, C2×C58, C4×D29, D116, C29⋊D4, D4×C29, C22×D29, D4×D29
Quotients: C1, C2, C22, D4, C23, C2×D4, D29, D58, C22×D29, D4×D29
►On 116 points
Generators in S116
(1 108 53 61)(2 109 54 62)(3 110 55 63)(4 111 56 64)(5 112 57 65)(6 113 58 66)(7 114 30 67)(8 115 31 68)(9 116 32 69)(10 88 33 70)(11 89 34 71)(12 90 35 72)(13 91 36 73)(14 92 37 74)(15 93 38 75)(16 94 39 76)(17 95 40 77)(18 96 41 78)(19 97 42 79)(20 98 43 80)(21 99 44 81)(22 100 45 82)(23 101 46 83)(24 102 47 84)(25 103 48 85)(26 104 49 86)(27 105 50 87)(28 106 51 59)(29 107 52 60)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 59)(29 60)(30 114)(31 115)(32 116)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(59 62)(60 61)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(88 98)(89 97)(90 96)(91 95)(92 94)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)
G:=sub<Sym(116)| (1,108,53,61)(2,109,54,62)(3,110,55,63)(4,111,56,64)(5,112,57,65)(6,113,58,66)(7,114,30,67)(8,115,31,68)(9,116,32,69)(10,88,33,70)(11,89,34,71)(12,90,35,72)(13,91,36,73)(14,92,37,74)(15,93,38,75)(16,94,39,76)(17,95,40,77)(18,96,41,78)(19,97,42,79)(20,98,43,80)(21,99,44,81)(22,100,45,82)(23,101,46,83)(24,102,47,84)(25,103,48,85)(26,104,49,86)(27,105,50,87)(28,106,51,59)(29,107,52,60), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,59)(29,60)(30,114)(31,115)(32,116)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(59,62)(60,61)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(88,98)(89,97)(90,96)(91,95)(92,94)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108)>;
G:=Group( (1,108,53,61)(2,109,54,62)(3,110,55,63)(4,111,56,64)(5,112,57,65)(6,113,58,66)(7,114,30,67)(8,115,31,68)(9,116,32,69)(10,88,33,70)(11,89,34,71)(12,90,35,72)(13,91,36,73)(14,92,37,74)(15,93,38,75)(16,94,39,76)(17,95,40,77)(18,96,41,78)(19,97,42,79)(20,98,43,80)(21,99,44,81)(22,100,45,82)(23,101,46,83)(24,102,47,84)(25,103,48,85)(26,104,49,86)(27,105,50,87)(28,106,51,59)(29,107,52,60), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,81)(22,82)(23,83)(24,84)(25,85)(26,86)(27,87)(28,59)(29,60)(30,114)(31,115)(32,116)(33,88)(34,89)(35,90)(36,91)(37,92)(38,93)(39,94)(40,95)(41,96)(42,97)(43,98)(44,99)(45,100)(46,101)(47,102)(48,103)(49,104)(50,105)(51,106)(52,107)(53,108)(54,109)(55,110)(56,111)(57,112)(58,113), (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,24)(7,23)(8,22)(9,21)(10,20)(11,19)(12,18)(13,17)(14,16)(30,46)(31,45)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)(59,62)(60,61)(63,87)(64,86)(65,85)(66,84)(67,83)(68,82)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)(88,98)(89,97)(90,96)(91,95)(92,94)(99,116)(100,115)(101,114)(102,113)(103,112)(104,111)(105,110)(106,109)(107,108) );
G=PermutationGroup([[(1,108,53,61),(2,109,54,62),(3,110,55,63),(4,111,56,64),(5,112,57,65),(6,113,58,66),(7,114,30,67),(8,115,31,68),(9,116,32,69),(10,88,33,70),(11,89,34,71),(12,90,35,72),(13,91,36,73),(14,92,37,74),(15,93,38,75),(16,94,39,76),(17,95,40,77),(18,96,41,78),(19,97,42,79),(20,98,43,80),(21,99,44,81),(22,100,45,82),(23,101,46,83),(24,102,47,84),(25,103,48,85),(26,104,49,86),(27,105,50,87),(28,106,51,59),(29,107,52,60)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,81),(22,82),(23,83),(24,84),(25,85),(26,86),(27,87),(28,59),(29,60),(30,114),(31,115),(32,116),(33,88),(34,89),(35,90),(36,91),(37,92),(38,93),(39,94),(40,95),(41,96),(42,97),(43,98),(44,99),(45,100),(46,101),(47,102),(48,103),(49,104),(50,105),(51,106),(52,107),(53,108),(54,109),(55,110),(56,111),(57,112),(58,113)], [(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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,24),(7,23),(8,22),(9,21),(10,20),(11,19),(12,18),(13,17),(14,16),(30,46),(31,45),(32,44),(33,43),(34,42),(35,41),(36,40),(37,39),(47,58),(48,57),(49,56),(50,55),(51,54),(52,53),(59,62),(60,61),(63,87),(64,86),(65,85),(66,84),(67,83),(68,82),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76),(88,98),(89,97),(90,96),(91,95),(92,94),(99,116),(100,115),(101,114),(102,113),(103,112),(104,111),(105,110),(106,109),(107,108)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 29A | ··· | 29N | 58A | ··· | 58N | 58O | ··· | 58AP | 116A | ··· | 116N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 2 | 2 | 29 | 29 | 58 | 58 | 2 | 58 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D29 | D58 | D58 | D4×D29 |
| kernel | D4×D29 | C4×D29 | D116 | C29⋊D4 | D4×C29 | C22×D29 | D29 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 14 | 14 | 28 | 14 |
Matrix representation of D4×D29 ►in GL4(𝔽233) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 220 | 232 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 36 |
| 0 | 0 | 0 | 232 |
| 0 | 1 | 0 | 0 |
| 232 | 122 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(233))| [1,0,0,0,0,1,0,0,0,0,1,220,0,0,36,232],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,36,232],[0,232,0,0,1,122,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;
D4×D29 in GAP, Magma, Sage, TeX
D_4\times D_{29} % in TeX
G:=Group("D4xD29"); // GroupNames label
G:=SmallGroup(464,39);
// by ID
G=gap.SmallGroup(464,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,97,11204]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^29=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations