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G = D4×C31order 248 = 23·31

Direct product of C31 and D4

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

Aliases: D4×C31, C4⋊C62, C22⋊C62, C1243C2, C62.6C22, (C2×C62)⋊1C2, C2.1(C2×C62), SmallGroup(248,9)

Series: Derived Chief Lower central Upper central

C1C2 — D4×C31
C1C2C62C2×C62 — D4×C31
C1C2 — D4×C31
C1C62 — D4×C31

Generators and relations for D4×C31
 G = < a,b,c | a31=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

2C2
2C2
2C62
2C62

Smallest permutation representation of D4×C31
On 124 points
Generators in S124
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124)
(1 46 112 66)(2 47 113 67)(3 48 114 68)(4 49 115 69)(5 50 116 70)(6 51 117 71)(7 52 118 72)(8 53 119 73)(9 54 120 74)(10 55 121 75)(11 56 122 76)(12 57 123 77)(13 58 124 78)(14 59 94 79)(15 60 95 80)(16 61 96 81)(17 62 97 82)(18 32 98 83)(19 33 99 84)(20 34 100 85)(21 35 101 86)(22 36 102 87)(23 37 103 88)(24 38 104 89)(25 39 105 90)(26 40 106 91)(27 41 107 92)(28 42 108 93)(29 43 109 63)(30 44 110 64)(31 45 111 65)
(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)(61 81)(62 82)

G:=sub<Sym(124)| (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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,46,112,66)(2,47,113,67)(3,48,114,68)(4,49,115,69)(5,50,116,70)(6,51,117,71)(7,52,118,72)(8,53,119,73)(9,54,120,74)(10,55,121,75)(11,56,122,76)(12,57,123,77)(13,58,124,78)(14,59,94,79)(15,60,95,80)(16,61,96,81)(17,62,97,82)(18,32,98,83)(19,33,99,84)(20,34,100,85)(21,35,101,86)(22,36,102,87)(23,37,103,88)(24,38,104,89)(25,39,105,90)(26,40,106,91)(27,41,107,92)(28,42,108,93)(29,43,109,63)(30,44,110,64)(31,45,111,65), (32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82)>;

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)(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124), (1,46,112,66)(2,47,113,67)(3,48,114,68)(4,49,115,69)(5,50,116,70)(6,51,117,71)(7,52,118,72)(8,53,119,73)(9,54,120,74)(10,55,121,75)(11,56,122,76)(12,57,123,77)(13,58,124,78)(14,59,94,79)(15,60,95,80)(16,61,96,81)(17,62,97,82)(18,32,98,83)(19,33,99,84)(20,34,100,85)(21,35,101,86)(22,36,102,87)(23,37,103,88)(24,38,104,89)(25,39,105,90)(26,40,106,91)(27,41,107,92)(28,42,108,93)(29,43,109,63)(30,44,110,64)(31,45,111,65), (32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)(61,81)(62,82) );

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),(94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124)], [(1,46,112,66),(2,47,113,67),(3,48,114,68),(4,49,115,69),(5,50,116,70),(6,51,117,71),(7,52,118,72),(8,53,119,73),(9,54,120,74),(10,55,121,75),(11,56,122,76),(12,57,123,77),(13,58,124,78),(14,59,94,79),(15,60,95,80),(16,61,96,81),(17,62,97,82),(18,32,98,83),(19,33,99,84),(20,34,100,85),(21,35,101,86),(22,36,102,87),(23,37,103,88),(24,38,104,89),(25,39,105,90),(26,40,106,91),(27,41,107,92),(28,42,108,93),(29,43,109,63),(30,44,110,64),(31,45,111,65)], [(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,63),(44,64),(45,65),(46,66),(47,67),(48,68),(49,69),(50,70),(51,71),(52,72),(53,73),(54,74),(55,75),(56,76),(57,77),(58,78),(59,79),(60,80),(61,81),(62,82)])

D4×C31 is a maximal subgroup of   D4⋊D31  D4.D31  D42D31

155 conjugacy classes

class 1 2A2B2C 4 31A···31AD62A···62AD62AE···62CL124A···124AD
order1222431···3162···6262···62124···124
size112221···11···12···22···2

155 irreducible representations

dim11111122
type++++
imageC1C2C2C31C62C62D4D4×C31
kernelD4×C31C124C2×C62D4C4C22C31C1
# reps112303060130

Matrix representation of D4×C31 in GL2(𝔽373) generated by

2150
0215
,
328267
37145
,
1328
0372
G:=sub<GL(2,GF(373))| [215,0,0,215],[328,371,267,45],[1,0,328,372] >;

D4×C31 in GAP, Magma, Sage, TeX

D_4\times C_{31}
% in TeX

G:=Group("D4xC31");
// GroupNames label

G:=SmallGroup(248,9);
// by ID

G=gap.SmallGroup(248,9);
# by ID

G:=PCGroup([4,-2,-2,-31,-2,1009]);
// Polycyclic

G:=Group<a,b,c|a^31=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×C31 in TeX

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