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## G = C22×C29⋊C4order 464 = 24·29

### Direct product of C22 and C29⋊C4

Aliases: C22×C29⋊C4, D583C4, D29.C23, D58.7C22, C58⋊(C2×C4), D29⋊(C2×C4), C29⋊(C22×C4), (C2×C58)⋊2C4, (C22×D29).3C2, SmallGroup(464,49)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C29 — C22×C29⋊C4
 Chief series C1 — C29 — D29 — C29⋊C4 — C2×C29⋊C4 — C22×C29⋊C4
 Lower central C29 — C22×C29⋊C4
 Upper central C1 — C22

Generators and relations for C22×C29⋊C4
G = < a,b,c,d | a2=b2=c29=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c17 >

Subgroups: 670 in 54 conjugacy classes, 32 normal (7 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C23, C22×C4, C29, D29, D29, C58, C29⋊C4, D58, C2×C58, C2×C29⋊C4, C22×D29, C22×C29⋊C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C29⋊C4, C2×C29⋊C4, C22×C29⋊C4

Smallest permutation representation of C22×C29⋊C4
On 116 points
Generators in S116
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 101)(15 102)(16 103)(17 104)(18 105)(19 106)(20 107)(21 108)(22 109)(23 110)(24 111)(25 112)(26 113)(27 114)(28 115)(29 116)(30 59)(31 60)(32 61)(33 62)(34 63)(35 64)(36 65)(37 66)(38 67)(39 68)(40 69)(41 70)(42 71)(43 72)(44 73)(45 74)(46 75)(47 76)(48 77)(49 78)(50 79)(51 80)(52 81)(53 82)(54 83)(55 84)(56 85)(57 86)(58 87)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(59 88)(60 89)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 101)(73 102)(74 103)(75 104)(76 105)(77 106)(78 107)(79 108)(80 109)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)
(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 88)(2 100 29 105)(3 112 28 93)(4 95 27 110)(5 107 26 98)(6 90 25 115)(7 102 24 103)(8 114 23 91)(9 97 22 108)(10 109 21 96)(11 92 20 113)(12 104 19 101)(13 116 18 89)(14 99 17 106)(15 111 16 94)(30 59)(31 71 58 76)(32 83 57 64)(33 66 56 81)(34 78 55 69)(35 61 54 86)(36 73 53 74)(37 85 52 62)(38 68 51 79)(39 80 50 67)(40 63 49 84)(41 75 48 72)(42 87 47 60)(43 70 46 77)(44 82 45 65)

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

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

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

44 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A ··· 4H 29A ··· 29G 58A ··· 58U order 1 2 2 2 2 2 2 2 4 ··· 4 29 ··· 29 58 ··· 58 size 1 1 1 1 29 29 29 29 29 ··· 29 4 ··· 4 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 4 4 type + + + + + image C1 C2 C2 C4 C4 C29⋊C4 C2×C29⋊C4 kernel C22×C29⋊C4 C2×C29⋊C4 C22×D29 D58 C2×C58 C22 C2 # reps 1 6 1 6 2 7 21

Matrix representation of C22×C29⋊C4 in GL5(𝔽233)

 232 0 0 0 0 0 232 0 0 0 0 0 232 0 0 0 0 0 232 0 0 0 0 0 232
,
 232 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 186 185 194 232 0 181 164 75 217 0 29 219 148 189 0 105 18 103 146
,
 144 0 0 0 0 0 109 118 77 141 0 217 0 129 117 0 231 38 140 135 0 215 112 79 217

G:=sub<GL(5,GF(233))| [232,0,0,0,0,0,232,0,0,0,0,0,232,0,0,0,0,0,232,0,0,0,0,0,232],[232,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,186,181,29,105,0,185,164,219,18,0,194,75,148,103,0,232,217,189,146],[144,0,0,0,0,0,109,217,231,215,0,118,0,38,112,0,77,129,140,79,0,141,117,135,217] >;

C22×C29⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_{29}\rtimes C_4
% in TeX

G:=Group("C2^2xC29:C4");
// GroupNames label

G:=SmallGroup(464,49);
// by ID

G=gap.SmallGroup(464,49);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-29,40,4804,1419]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^29=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^17>;
// generators/relations

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