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## G = C22×D29order 232 = 23·29

### Direct product of C22 and D29

Aliases: C22×D29, C29⋊C23, C58⋊C22, (C2×C58)⋊3C2, SmallGroup(232,13)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — C22×D29
 Chief series C1 — C29 — D29 — D58 — C22×D29
 Lower central C29 — C22×D29
 Upper central C1 — C22

Generators and relations for C22×D29
G = < a,b,c,d | a2=b2=c29=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Smallest permutation representation of C22×D29
On 116 points
Generators in S116
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 111)(11 112)(12 113)(13 114)(14 115)(15 116)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 97)(26 98)(27 99)(28 100)(29 101)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 59)(45 60)(46 61)(47 62)(48 63)(49 64)(50 65)(51 66)(52 67)(53 68)(54 69)(55 70)(56 71)(57 72)(58 73)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 49)(8 50)(9 51)(10 52)(11 53)(12 54)(13 55)(14 56)(15 57)(16 58)(17 30)(18 31)(19 32)(20 33)(21 34)(22 35)(23 36)(24 37)(25 38)(26 39)(27 40)(28 41)(29 42)(59 103)(60 104)(61 105)(62 106)(63 107)(64 108)(65 109)(66 110)(67 111)(68 112)(69 113)(70 114)(71 115)(72 116)(73 88)(74 89)(75 90)(76 91)(77 92)(78 93)(79 94)(80 95)(81 96)(82 97)(83 98)(84 99)(85 100)(86 101)(87 102)
(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)
(1 101)(2 100)(3 99)(4 98)(5 97)(6 96)(7 95)(8 94)(9 93)(10 92)(11 91)(12 90)(13 89)(14 88)(15 116)(16 115)(17 114)(18 113)(19 112)(20 111)(21 110)(22 109)(23 108)(24 107)(25 106)(26 105)(27 104)(28 103)(29 102)(30 70)(31 69)(32 68)(33 67)(34 66)(35 65)(36 64)(37 63)(38 62)(39 61)(40 60)(41 59)(42 87)(43 86)(44 85)(45 84)(46 83)(47 82)(48 81)(49 80)(50 79)(51 78)(52 77)(53 76)(54 75)(55 74)(56 73)(57 72)(58 71)

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

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

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

C22×D29 is a maximal subgroup of   D58⋊C4  D29.D4
C22×D29 is a maximal quotient of   D1165C2  D42D29  Q82D29

64 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 29A ··· 29N 58A ··· 58AP order 1 2 2 2 2 2 2 2 29 ··· 29 58 ··· 58 size 1 1 1 1 29 29 29 29 2 ··· 2 2 ··· 2

64 irreducible representations

 dim 1 1 1 2 2 type + + + + + image C1 C2 C2 D29 D58 kernel C22×D29 D58 C2×C58 C22 C2 # reps 1 6 1 14 42

Matrix representation of C22×D29 in GL4(𝔽59) generated by

 1 0 0 0 0 58 0 0 0 0 1 0 0 0 0 1
,
 58 0 0 0 0 58 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 58 14
,
 58 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(59))| [1,0,0,0,0,58,0,0,0,0,1,0,0,0,0,1],[58,0,0,0,0,58,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,58,0,0,1,14],[58,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;

C22×D29 in GAP, Magma, Sage, TeX

C_2^2\times D_{29}
% in TeX

G:=Group("C2^2xD29");
// GroupNames label

G:=SmallGroup(232,13);
// by ID

G=gap.SmallGroup(232,13);
# by ID

G:=PCGroup([4,-2,-2,-2,-29,3587]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^29=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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