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G = D4×C29order 232 = 23·29

Direct product of C29 and D4

direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary

Aliases: D4×C29, C4⋊C58, C22⋊C58, C1163C2, C58.6C22, (C2×C58)⋊1C2, C2.1(C2×C58), SmallGroup(232,10)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C29
C1C2C58C2×C58 — D4×C29
C1C2 — D4×C29
C1C58 — D4×C29

Generators and relations for D4×C29
 G = < a,b,c | a29=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C58
2C58

Smallest permutation representation of D4×C29
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 32 106 62)(2 33 107 63)(3 34 108 64)(4 35 109 65)(5 36 110 66)(6 37 111 67)(7 38 112 68)(8 39 113 69)(9 40 114 70)(10 41 115 71)(11 42 116 72)(12 43 88 73)(13 44 89 74)(14 45 90 75)(15 46 91 76)(16 47 92 77)(17 48 93 78)(18 49 94 79)(19 50 95 80)(20 51 96 81)(21 52 97 82)(22 53 98 83)(23 54 99 84)(24 55 100 85)(25 56 101 86)(26 57 102 87)(27 58 103 59)(28 30 104 60)(29 31 105 61)
(30 60)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)(41 71)(42 72)(43 73)(44 74)(45 75)(46 76)(47 77)(48 78)(49 79)(50 80)(51 81)(52 82)(53 83)(54 84)(55 85)(56 86)(57 87)(58 59)

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

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

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

D4×C29 is a maximal subgroup of   D4⋊D29  D4.D29  D42D29

145 conjugacy classes

class 1 2A2B2C 4 29A···29AB58A···58AB58AC···58CF116A···116AB
order1222429···2958···5858···58116···116
size112221···11···12···22···2

145 irreducible representations

dim11111122
type++++
imageC1C2C2C29C58C58D4D4×C29
kernelD4×C29C116C2×C58D4C4C22C29C1
# reps112282856128

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

1420
0142
,
1272
90106
,
10
106232
G:=sub<GL(2,GF(233))| [142,0,0,142],[127,90,2,106],[1,106,0,232] >;

D4×C29 in GAP, Magma, Sage, TeX

D_4\times C_{29}
% in TeX

G:=Group("D4xC29");
// GroupNames label

G:=SmallGroup(232,10);
// by ID

G=gap.SmallGroup(232,10);
# by ID

G:=PCGroup([4,-2,-2,-29,-2,945]);
// Polycyclic

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

Export

Subgroup lattice of D4×C29 in TeX

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