metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C29⋊2D4, C22⋊D29, D58⋊2C2, Dic29⋊C2, C2.5D58, C58.5C22, (C2×C58)⋊2C2, SmallGroup(232,8)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C29⋊D4
G = < a,b,c | a29=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >
Smallest permutation representation of C29⋊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 72 41 91)(2 71 42 90)(3 70 43 89)(4 69 44 88)(5 68 45 116)(6 67 46 115)(7 66 47 114)(8 65 48 113)(9 64 49 112)(10 63 50 111)(11 62 51 110)(12 61 52 109)(13 60 53 108)(14 59 54 107)(15 87 55 106)(16 86 56 105)(17 85 57 104)(18 84 58 103)(19 83 30 102)(20 82 31 101)(21 81 32 100)(22 80 33 99)(23 79 34 98)(24 78 35 97)(25 77 36 96)(26 76 37 95)(27 75 38 94)(28 74 39 93)(29 73 40 92)
(2 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 20)(12 19)(13 18)(14 17)(15 16)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(53 58)(54 57)(55 56)(59 104)(60 103)(61 102)(62 101)(63 100)(64 99)(65 98)(66 97)(67 96)(68 95)(69 94)(70 93)(71 92)(72 91)(73 90)(74 89)(75 88)(76 116)(77 115)(78 114)(79 113)(80 112)(81 111)(82 110)(83 109)(84 108)(85 107)(86 106)(87 105)
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), (1,72,41,91)(2,71,42,90)(3,70,43,89)(4,69,44,88)(5,68,45,116)(6,67,46,115)(7,66,47,114)(8,65,48,113)(9,64,49,112)(10,63,50,111)(11,62,51,110)(12,61,52,109)(13,60,53,108)(14,59,54,107)(15,87,55,106)(16,86,56,105)(17,85,57,104)(18,84,58,103)(19,83,30,102)(20,82,31,101)(21,81,32,100)(22,80,33,99)(23,79,34,98)(24,78,35,97)(25,77,36,96)(26,76,37,95)(27,75,38,94)(28,74,39,93)(29,73,40,92), (2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(53,58)(54,57)(55,56)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,116)(77,115)(78,114)(79,113)(80,112)(81,111)(82,110)(83,109)(84,108)(85,107)(86,106)(87,105)>;
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), (1,72,41,91)(2,71,42,90)(3,70,43,89)(4,69,44,88)(5,68,45,116)(6,67,46,115)(7,66,47,114)(8,65,48,113)(9,64,49,112)(10,63,50,111)(11,62,51,110)(12,61,52,109)(13,60,53,108)(14,59,54,107)(15,87,55,106)(16,86,56,105)(17,85,57,104)(18,84,58,103)(19,83,30,102)(20,82,31,101)(21,81,32,100)(22,80,33,99)(23,79,34,98)(24,78,35,97)(25,77,36,96)(26,76,37,95)(27,75,38,94)(28,74,39,93)(29,73,40,92), (2,29)(3,28)(4,27)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(53,58)(54,57)(55,56)(59,104)(60,103)(61,102)(62,101)(63,100)(64,99)(65,98)(66,97)(67,96)(68,95)(69,94)(70,93)(71,92)(72,91)(73,90)(74,89)(75,88)(76,116)(77,115)(78,114)(79,113)(80,112)(81,111)(82,110)(83,109)(84,108)(85,107)(86,106)(87,105) );
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)], [(1,72,41,91),(2,71,42,90),(3,70,43,89),(4,69,44,88),(5,68,45,116),(6,67,46,115),(7,66,47,114),(8,65,48,113),(9,64,49,112),(10,63,50,111),(11,62,51,110),(12,61,52,109),(13,60,53,108),(14,59,54,107),(15,87,55,106),(16,86,56,105),(17,85,57,104),(18,84,58,103),(19,83,30,102),(20,82,31,101),(21,81,32,100),(22,80,33,99),(23,79,34,98),(24,78,35,97),(25,77,36,96),(26,76,37,95),(27,75,38,94),(28,74,39,93),(29,73,40,92)], [(2,29),(3,28),(4,27),(5,26),(6,25),(7,24),(8,23),(9,22),(10,21),(11,20),(12,19),(13,18),(14,17),(15,16),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(53,58),(54,57),(55,56),(59,104),(60,103),(61,102),(62,101),(63,100),(64,99),(65,98),(66,97),(67,96),(68,95),(69,94),(70,93),(71,92),(72,91),(73,90),(74,89),(75,88),(76,116),(77,115),(78,114),(79,113),(80,112),(81,111),(82,110),(83,109),(84,108),(85,107),(86,106),(87,105)]])
C29⋊D4 is a maximal subgroup of
D116⋊5C2 D4×D29 D4⋊2D29
C29⋊D4 is a maximal quotient of C58.D4 D58⋊C4 D4⋊D29 D4.D29 Q8⋊D29 C29⋊Q16 C23.D29
61 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 29A | ··· | 29N | 58A | ··· | 58AP |
| order | 1 | 2 | 2 | 2 | 4 | 29 | ··· | 29 | 58 | ··· | 58 |
| size | 1 | 1 | 2 | 58 | 58 | 2 | ··· | 2 | 2 | ··· | 2 |
61 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | D4 | D29 | D58 | C29⋊D4 |
| kernel | C29⋊D4 | Dic29 | D58 | C2×C58 | C29 | C22 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 14 | 14 | 28 |
Matrix representation of C29⋊D4 ►in GL2(𝔽233) generated by
| 0 | 1 |
| 232 | 175 |
| 219 | 112 |
| 225 | 14 |
| 1 | 0 |
| 175 | 232 |
G:=sub<GL(2,GF(233))| [0,232,1,175],[219,225,112,14],[1,175,0,232] >;
C29⋊D4 in GAP, Magma, Sage, TeX
C_{29}\rtimes D_4 % in TeX
G:=Group("C29:D4"); // GroupNames label
G:=SmallGroup(232,8);
// by ID
G=gap.SmallGroup(232,8);
# by ID
G:=PCGroup([4,-2,-2,-2,-29,49,3587]);
// Polycyclic
G:=Group<a,b,c|a^29=b^4=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations
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