direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C4×D29, C116⋊2C2, C2.1D58, D58.2C2, Dic29⋊2C2, C58.2C22, C29⋊2(C2×C4), SmallGroup(232,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C29 — C4×D29 |
Generators and relations for C4×D29
G = < a,b,c | a4=b29=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of C4×D29
►On 116 points
Generators in S116
(1 108 36 83)(2 109 37 84)(3 110 38 85)(4 111 39 86)(5 112 40 87)(6 113 41 59)(7 114 42 60)(8 115 43 61)(9 116 44 62)(10 88 45 63)(11 89 46 64)(12 90 47 65)(13 91 48 66)(14 92 49 67)(15 93 50 68)(16 94 51 69)(17 95 52 70)(18 96 53 71)(19 97 54 72)(20 98 55 73)(21 99 56 74)(22 100 57 75)(23 101 58 76)(24 102 30 77)(25 103 31 78)(26 104 32 79)(27 105 33 80)(28 106 34 81)(29 107 35 82)
(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 41)(31 40)(32 39)(33 38)(34 37)(35 36)(42 58)(43 57)(44 56)(45 55)(46 54)(47 53)(48 52)(49 51)(59 77)(60 76)(61 75)(62 74)(63 73)(64 72)(65 71)(66 70)(67 69)(78 87)(79 86)(80 85)(81 84)(82 83)(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)| (1,108,36,83)(2,109,37,84)(3,110,38,85)(4,111,39,86)(5,112,40,87)(6,113,41,59)(7,114,42,60)(8,115,43,61)(9,116,44,62)(10,88,45,63)(11,89,46,64)(12,90,47,65)(13,91,48,66)(14,92,49,67)(15,93,50,68)(16,94,51,69)(17,95,52,70)(18,96,53,71)(19,97,54,72)(20,98,55,73)(21,99,56,74)(22,100,57,75)(23,101,58,76)(24,102,30,77)(25,103,31,78)(26,104,32,79)(27,105,33,80)(28,106,34,81)(29,107,35,82), (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)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(78,87)(79,86)(80,85)(81,84)(82,83)(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( (1,108,36,83)(2,109,37,84)(3,110,38,85)(4,111,39,86)(5,112,40,87)(6,113,41,59)(7,114,42,60)(8,115,43,61)(9,116,44,62)(10,88,45,63)(11,89,46,64)(12,90,47,65)(13,91,48,66)(14,92,49,67)(15,93,50,68)(16,94,51,69)(17,95,52,70)(18,96,53,71)(19,97,54,72)(20,98,55,73)(21,99,56,74)(22,100,57,75)(23,101,58,76)(24,102,30,77)(25,103,31,78)(26,104,32,79)(27,105,33,80)(28,106,34,81)(29,107,35,82), (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)(30,41)(31,40)(32,39)(33,38)(34,37)(35,36)(42,58)(43,57)(44,56)(45,55)(46,54)(47,53)(48,52)(49,51)(59,77)(60,76)(61,75)(62,74)(63,73)(64,72)(65,71)(66,70)(67,69)(78,87)(79,86)(80,85)(81,84)(82,83)(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([[(1,108,36,83),(2,109,37,84),(3,110,38,85),(4,111,39,86),(5,112,40,87),(6,113,41,59),(7,114,42,60),(8,115,43,61),(9,116,44,62),(10,88,45,63),(11,89,46,64),(12,90,47,65),(13,91,48,66),(14,92,49,67),(15,93,50,68),(16,94,51,69),(17,95,52,70),(18,96,53,71),(19,97,54,72),(20,98,55,73),(21,99,56,74),(22,100,57,75),(23,101,58,76),(24,102,30,77),(25,103,31,78),(26,104,32,79),(27,105,33,80),(28,106,34,81),(29,107,35,82)], [(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),(30,41),(31,40),(32,39),(33,38),(34,37),(35,36),(42,58),(43,57),(44,56),(45,55),(46,54),(47,53),(48,52),(49,51),(59,77),(60,76),(61,75),(62,74),(63,73),(64,72),(65,71),(66,70),(67,69),(78,87),(79,86),(80,85),(81,84),(82,83),(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)]])
C4×D29 is a maximal subgroup of
C8⋊D29 D29⋊C8 C116.C4 C116⋊C4 D116⋊5C2 D4⋊2D29 Q8⋊2D29
C4×D29 is a maximal quotient of C8⋊D29 C58.D4 D58⋊C4
64 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 29A | ··· | 29N | 58A | ··· | 58N | 116A | ··· | 116AB |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 29 | ··· | 29 | 58 | ··· | 58 | 116 | ··· | 116 |
| size | 1 | 1 | 29 | 29 | 1 | 1 | 29 | 29 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
64 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C4 | D29 | D58 | C4×D29 |
| kernel | C4×D29 | Dic29 | C116 | D58 | D29 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 4 | 14 | 14 | 28 |
Matrix representation of C4×D29 ►in GL2(𝔽233) generated by
| 144 | 0 |
| 0 | 144 |
| 0 | 1 |
| 232 | 179 |
| 0 | 1 |
| 1 | 0 |
G:=sub<GL(2,GF(233))| [144,0,0,144],[0,232,1,179],[0,1,1,0] >;
C4×D29 in GAP, Magma, Sage, TeX
C_4\times D_{29} % in TeX
G:=Group("C4xD29"); // GroupNames label
G:=SmallGroup(232,5);
// by ID
G=gap.SmallGroup(232,5);
# by ID
G:=PCGroup([4,-2,-2,-2,-29,21,3587]);
// Polycyclic
G:=Group<a,b,c|a^4=b^29=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export