Copied to
clipboard

G = C29⋊D4order 232 = 23·29

The semidirect product of C29 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C292D4, C22⋊D29, D582C2, Dic29⋊C2, C2.5D58, C58.5C22, (C2×C58)⋊2C2, SmallGroup(232,8)

Series: Derived Chief Lower central Upper central

C1C58 — C29⋊D4
C1C29C58D58 — C29⋊D4
C29C58 — C29⋊D4
C1C2C22

Generators and relations for C29⋊D4
 G = < a,b,c | a29=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
58C2
29C4
29C22
2D29
2C58
29D4

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 81 54 109)(2 80 55 108)(3 79 56 107)(4 78 57 106)(5 77 58 105)(6 76 30 104)(7 75 31 103)(8 74 32 102)(9 73 33 101)(10 72 34 100)(11 71 35 99)(12 70 36 98)(13 69 37 97)(14 68 38 96)(15 67 39 95)(16 66 40 94)(17 65 41 93)(18 64 42 92)(19 63 43 91)(20 62 44 90)(21 61 45 89)(22 60 46 88)(23 59 47 116)(24 87 48 115)(25 86 49 114)(26 85 50 113)(27 84 51 112)(28 83 52 111)(29 82 53 110)
(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 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(50 58)(51 57)(52 56)(53 55)(59 102)(60 101)(61 100)(62 99)(63 98)(64 97)(65 96)(66 95)(67 94)(68 93)(69 92)(70 91)(71 90)(72 89)(73 88)(74 116)(75 115)(76 114)(77 113)(78 112)(79 111)(80 110)(81 109)(82 108)(83 107)(84 106)(85 105)(86 104)(87 103)

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), (1,81,54,109)(2,80,55,108)(3,79,56,107)(4,78,57,106)(5,77,58,105)(6,76,30,104)(7,75,31,103)(8,74,32,102)(9,73,33,101)(10,72,34,100)(11,71,35,99)(12,70,36,98)(13,69,37,97)(14,68,38,96)(15,67,39,95)(16,66,40,94)(17,65,41,93)(18,64,42,92)(19,63,43,91)(20,62,44,90)(21,61,45,89)(22,60,46,88)(23,59,47,116)(24,87,48,115)(25,86,49,114)(26,85,50,113)(27,84,51,112)(28,83,52,111)(29,82,53,110), (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)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(50,58)(51,57)(52,56)(53,55)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103)>;

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), (1,81,54,109)(2,80,55,108)(3,79,56,107)(4,78,57,106)(5,77,58,105)(6,76,30,104)(7,75,31,103)(8,74,32,102)(9,73,33,101)(10,72,34,100)(11,71,35,99)(12,70,36,98)(13,69,37,97)(14,68,38,96)(15,67,39,95)(16,66,40,94)(17,65,41,93)(18,64,42,92)(19,63,43,91)(20,62,44,90)(21,61,45,89)(22,60,46,88)(23,59,47,116)(24,87,48,115)(25,86,49,114)(26,85,50,113)(27,84,51,112)(28,83,52,111)(29,82,53,110), (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)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)(50,58)(51,57)(52,56)(53,55)(59,102)(60,101)(61,100)(62,99)(63,98)(64,97)(65,96)(66,95)(67,94)(68,93)(69,92)(70,91)(71,90)(72,89)(73,88)(74,116)(75,115)(76,114)(77,113)(78,112)(79,111)(80,110)(81,109)(82,108)(83,107)(84,106)(85,105)(86,104)(87,103) );

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)], [(1,81,54,109),(2,80,55,108),(3,79,56,107),(4,78,57,106),(5,77,58,105),(6,76,30,104),(7,75,31,103),(8,74,32,102),(9,73,33,101),(10,72,34,100),(11,71,35,99),(12,70,36,98),(13,69,37,97),(14,68,38,96),(15,67,39,95),(16,66,40,94),(17,65,41,93),(18,64,42,92),(19,63,43,91),(20,62,44,90),(21,61,45,89),(22,60,46,88),(23,59,47,116),(24,87,48,115),(25,86,49,114),(26,85,50,113),(27,84,51,112),(28,83,52,111),(29,82,53,110)], [(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),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40),(50,58),(51,57),(52,56),(53,55),(59,102),(60,101),(61,100),(62,99),(63,98),(64,97),(65,96),(66,95),(67,94),(68,93),(69,92),(70,91),(71,90),(72,89),(73,88),(74,116),(75,115),(76,114),(77,113),(78,112),(79,111),(80,110),(81,109),(82,108),(83,107),(84,106),(85,105),(86,104),(87,103)])

C29⋊D4 is a maximal subgroup of   D1165C2  D4×D29  D42D29
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 2A2B2C 4 29A···29N58A···58AP
order1222429···2958···58
size11258582···22···2

61 irreducible representations

dim11112222
type+++++++
imageC1C2C2C2D4D29D58C29⋊D4
kernelC29⋊D4Dic29D58C2×C58C29C22C2C1
# reps11111141428

Matrix representation of C29⋊D4 in GL2(𝔽233) generated by

01
232175
,
219112
22514
,
10
175232
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

Export

Subgroup lattice of C29⋊D4 in TeX

׿
×
𝔽