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G = C22xD29order 232 = 23·29

Direct product of C22 and D29

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: C22xD29, C29:C23, C58:C22, (C2xC58):3C2, SmallGroup(232,13)

Series: Derived Chief Lower central Upper central

C1C29 — C22xD29
C1C29D29D58 — C22xD29
C29 — C22xD29
C1C22

Generators and relations for C22xD29
 G = < a,b,c,d | a2=b2=c29=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 340 in 32 conjugacy classes, 21 normal (5 characteristic)
Quotients: C1, C2, C22, C23, D29, D58, C22xD29
29C2
29C2
29C2
29C2
29C22
29C22
29C22
29C22
29C22
29C22
29C23

Smallest permutation representation of C22xD29
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)]])

C22xD29 is a maximal subgroup of   D58:C4  D29.D4
C22xD29 is a maximal quotient of   D116:5C2  D4:2D29  Q8:2D29

64 conjugacy classes

class 1 2A2B2C2D2E2F2G29A···29N58A···58AP
order1222222229···2958···58
size1111292929292···22···2

64 irreducible representations

dim11122
type+++++
imageC1C2C2D29D58
kernelC22xD29D58C2xC58C22C2
# reps1611442

Matrix representation of C22xD29 in GL4(F59) generated by

1000
05800
0010
0001
,
58000
05800
0010
0001
,
1000
0100
0001
005814
,
58000
0100
0001
0010
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] >;

C22xD29 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

Export

Subgroup lattice of C22xD29 in TeX

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