metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D58⋊2C4, D29.2D4, D58.6C22, (C2×C58)⋊1C4, C29⋊(C22⋊C4), C22⋊(C29⋊C4), C58.7(C2×C4), (C22×D29).2C2, (C2×C29⋊C4)⋊C2, C2.7(C2×C29⋊C4), SmallGroup(464,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D29.D4
G = < a,b,c,d | a29=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a17, cbc-1=dbd-1=a16b, dcd-1=a-1bc-1 >
2C2
29C2
29C2
58C2
29C22
29C22
58C4
58C4
58C22
58C22
2D29
2C58
29C23
29C2×C4
29C2×C4
2D58
2D58
29C22⋊C4
Smallest permutation representation of D29.D4
►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 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 57)(31 56)(32 55)(33 54)(34 53)(35 52)(36 51)(37 50)(38 49)(39 48)(40 47)(41 46)(42 45)(43 44)(59 65)(60 64)(61 63)(66 87)(67 86)(68 85)(69 84)(70 83)(71 82)(72 81)(73 80)(74 79)(75 78)(76 77)(88 92)(89 91)(93 116)(94 115)(95 114)(96 113)(97 112)(98 111)(99 110)(100 109)(101 108)(102 107)(103 106)(104 105)
(1 77 44 105)(2 60 43 93)(3 72 42 110)(4 84 41 98)(5 67 40 115)(6 79 39 103)(7 62 38 91)(8 74 37 108)(9 86 36 96)(10 69 35 113)(11 81 34 101)(12 64 33 89)(13 76 32 106)(14 59 31 94)(15 71 30 111)(16 83 58 99)(17 66 57 116)(18 78 56 104)(19 61 55 92)(20 73 54 109)(21 85 53 97)(22 68 52 114)(23 80 51 102)(24 63 50 90)(25 75 49 107)(26 87 48 95)(27 70 47 112)(28 82 46 100)(29 65 45 88)
(2 13 29 18)(3 25 28 6)(4 8 27 23)(5 20 26 11)(7 15 24 16)(9 10 22 21)(12 17 19 14)(30 50 58 38)(31 33 57 55)(32 45 56 43)(34 40 54 48)(35 52 53 36)(37 47 51 41)(39 42 49 46)(59 92 66 89)(60 104 65 106)(61 116 64 94)(62 99 63 111)(67 101 87 109)(68 113 86 97)(69 96 85 114)(70 108 84 102)(71 91 83 90)(72 103 82 107)(73 115 81 95)(74 98 80 112)(75 110 79 100)(76 93 78 88)(77 105)
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), (1,29)(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)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(59,65)(60,64)(61,63)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(88,92)(89,91)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,77,44,105)(2,60,43,93)(3,72,42,110)(4,84,41,98)(5,67,40,115)(6,79,39,103)(7,62,38,91)(8,74,37,108)(9,86,36,96)(10,69,35,113)(11,81,34,101)(12,64,33,89)(13,76,32,106)(14,59,31,94)(15,71,30,111)(16,83,58,99)(17,66,57,116)(18,78,56,104)(19,61,55,92)(20,73,54,109)(21,85,53,97)(22,68,52,114)(23,80,51,102)(24,63,50,90)(25,75,49,107)(26,87,48,95)(27,70,47,112)(28,82,46,100)(29,65,45,88), (2,13,29,18)(3,25,28,6)(4,8,27,23)(5,20,26,11)(7,15,24,16)(9,10,22,21)(12,17,19,14)(30,50,58,38)(31,33,57,55)(32,45,56,43)(34,40,54,48)(35,52,53,36)(37,47,51,41)(39,42,49,46)(59,92,66,89)(60,104,65,106)(61,116,64,94)(62,99,63,111)(67,101,87,109)(68,113,86,97)(69,96,85,114)(70,108,84,102)(71,91,83,90)(72,103,82,107)(73,115,81,95)(74,98,80,112)(75,110,79,100)(76,93,78,88)(77,105)>;
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), (1,29)(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)(30,57)(31,56)(32,55)(33,54)(34,53)(35,52)(36,51)(37,50)(38,49)(39,48)(40,47)(41,46)(42,45)(43,44)(59,65)(60,64)(61,63)(66,87)(67,86)(68,85)(69,84)(70,83)(71,82)(72,81)(73,80)(74,79)(75,78)(76,77)(88,92)(89,91)(93,116)(94,115)(95,114)(96,113)(97,112)(98,111)(99,110)(100,109)(101,108)(102,107)(103,106)(104,105), (1,77,44,105)(2,60,43,93)(3,72,42,110)(4,84,41,98)(5,67,40,115)(6,79,39,103)(7,62,38,91)(8,74,37,108)(9,86,36,96)(10,69,35,113)(11,81,34,101)(12,64,33,89)(13,76,32,106)(14,59,31,94)(15,71,30,111)(16,83,58,99)(17,66,57,116)(18,78,56,104)(19,61,55,92)(20,73,54,109)(21,85,53,97)(22,68,52,114)(23,80,51,102)(24,63,50,90)(25,75,49,107)(26,87,48,95)(27,70,47,112)(28,82,46,100)(29,65,45,88), (2,13,29,18)(3,25,28,6)(4,8,27,23)(5,20,26,11)(7,15,24,16)(9,10,22,21)(12,17,19,14)(30,50,58,38)(31,33,57,55)(32,45,56,43)(34,40,54,48)(35,52,53,36)(37,47,51,41)(39,42,49,46)(59,92,66,89)(60,104,65,106)(61,116,64,94)(62,99,63,111)(67,101,87,109)(68,113,86,97)(69,96,85,114)(70,108,84,102)(71,91,83,90)(72,103,82,107)(73,115,81,95)(74,98,80,112)(75,110,79,100)(76,93,78,88)(77,105) );
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)], [(1,29),(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),(30,57),(31,56),(32,55),(33,54),(34,53),(35,52),(36,51),(37,50),(38,49),(39,48),(40,47),(41,46),(42,45),(43,44),(59,65),(60,64),(61,63),(66,87),(67,86),(68,85),(69,84),(70,83),(71,82),(72,81),(73,80),(74,79),(75,78),(76,77),(88,92),(89,91),(93,116),(94,115),(95,114),(96,113),(97,112),(98,111),(99,110),(100,109),(101,108),(102,107),(103,106),(104,105)], [(1,77,44,105),(2,60,43,93),(3,72,42,110),(4,84,41,98),(5,67,40,115),(6,79,39,103),(7,62,38,91),(8,74,37,108),(9,86,36,96),(10,69,35,113),(11,81,34,101),(12,64,33,89),(13,76,32,106),(14,59,31,94),(15,71,30,111),(16,83,58,99),(17,66,57,116),(18,78,56,104),(19,61,55,92),(20,73,54,109),(21,85,53,97),(22,68,52,114),(23,80,51,102),(24,63,50,90),(25,75,49,107),(26,87,48,95),(27,70,47,112),(28,82,46,100),(29,65,45,88)], [(2,13,29,18),(3,25,28,6),(4,8,27,23),(5,20,26,11),(7,15,24,16),(9,10,22,21),(12,17,19,14),(30,50,58,38),(31,33,57,55),(32,45,56,43),(34,40,54,48),(35,52,53,36),(37,47,51,41),(39,42,49,46),(59,92,66,89),(60,104,65,106),(61,116,64,94),(62,99,63,111),(67,101,87,109),(68,113,86,97),(69,96,85,114),(70,108,84,102),(71,91,83,90),(72,103,82,107),(73,115,81,95),(74,98,80,112),(75,110,79,100),(76,93,78,88),(77,105)]])
38 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 29A | ··· | 29G | 58A | ··· | 58U |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | 2 | 29 | 29 | 58 | 58 | 58 | 58 | 58 | 4 | ··· | 4 | 4 | ··· | 4 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C4 | C4 | D4 | C29⋊C4 | C2×C29⋊C4 | D29.D4 |
| kernel | D29.D4 | C2×C29⋊C4 | C22×D29 | D58 | C2×C58 | D29 | C22 | C2 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 7 | 7 | 14 |
Matrix representation of D29.D4 ►in GL6(𝔽233)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 29 | 1 | 0 | 0 |
| 0 | 0 | 189 | 0 | 1 | 0 |
| 0 | 0 | 82 | 0 | 0 | 1 |
| 0 | 0 | 214 | 99 | 132 | 118 |
| 232 | 0 | 0 | 0 | 0 | 0 |
| 0 | 232 | 0 | 0 | 0 | 0 |
| 0 | 0 | 45 | 159 | 231 | 84 |
| 0 | 0 | 226 | 90 | 71 | 19 |
| 0 | 0 | 75 | 213 | 102 | 87 |
| 0 | 0 | 27 | 124 | 177 | 229 |
| 104 | 177 | 0 | 0 | 0 | 0 |
| 164 | 129 | 0 | 0 | 0 | 0 |
| 0 | 0 | 194 | 72 | 140 | 4 |
| 0 | 0 | 33 | 168 | 8 | 216 |
| 0 | 0 | 175 | 30 | 9 | 33 |
| 0 | 0 | 157 | 75 | 152 | 95 |
| 144 | 0 | 0 | 0 | 0 | 0 |
| 202 | 89 | 0 | 0 | 0 | 0 |
| 0 | 0 | 194 | 72 | 140 | 4 |
| 0 | 0 | 33 | 168 | 8 | 216 |
| 0 | 0 | 175 | 30 | 9 | 33 |
| 0 | 0 | 157 | 75 | 152 | 95 |
G:=sub<GL(6,GF(233))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,29,189,82,214,0,0,1,0,0,99,0,0,0,1,0,132,0,0,0,0,1,118],[232,0,0,0,0,0,0,232,0,0,0,0,0,0,45,226,75,27,0,0,159,90,213,124,0,0,231,71,102,177,0,0,84,19,87,229],[104,164,0,0,0,0,177,129,0,0,0,0,0,0,194,33,175,157,0,0,72,168,30,75,0,0,140,8,9,152,0,0,4,216,33,95],[144,202,0,0,0,0,0,89,0,0,0,0,0,0,194,33,175,157,0,0,72,168,30,75,0,0,140,8,9,152,0,0,4,216,33,95] >;
D29.D4 in GAP, Magma, Sage, TeX
D_{29}.D_4 % in TeX
G:=Group("D29.D4"); // GroupNames label
G:=SmallGroup(464,34);
// by ID
G=gap.SmallGroup(464,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-29,20,101,4804,2814]);
// Polycyclic
G:=Group<a,b,c,d|a^29=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^17,c*b*c^-1=d*b*d^-1=a^16*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations
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