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## G = C29⋊D4order 232 = 23·29

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

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C58 — C29⋊D4
 Chief series C1 — C29 — C58 — D58 — C29⋊D4
 Lower central C29 — C58 — C29⋊D4
 Upper central C1 — C2 — C22

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 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 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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