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## G = D4×D31order 496 = 24·31

### Direct product of D4 and D31

Aliases: D4×D31, C41D62, C124⋊C22, D1243C2, C221D62, D622C22, C62.5C23, Dic311C22, C312(C2×D4), (C2×C62)⋊C22, (C4×D31)⋊1C2, (D4×C31)⋊2C2, C31⋊D41C2, (C22×D31)⋊2C2, C2.6(C22×D31), SmallGroup(496,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C62 — D4×D31
 Chief series C1 — C31 — C62 — D62 — C22×D31 — D4×D31
 Lower central C31 — C62 — D4×D31
 Upper central C1 — C2 — D4

Generators and relations for D4×D31
G = < a,b,c,d | a4=b2=c31=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 820 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C31, D31, D31, C62, C62, Dic31, C124, D62, D62, D62, C2×C62, C4×D31, D124, C31⋊D4, D4×C31, C22×D31, D4×D31
Quotients: C1, C2, C22, D4, C23, C2×D4, D31, D62, C22×D31, D4×D31

Smallest permutation representation of D4×D31
On 124 points
Generators in S124
(1 106 46 68)(2 107 47 69)(3 108 48 70)(4 109 49 71)(5 110 50 72)(6 111 51 73)(7 112 52 74)(8 113 53 75)(9 114 54 76)(10 115 55 77)(11 116 56 78)(12 117 57 79)(13 118 58 80)(14 119 59 81)(15 120 60 82)(16 121 61 83)(17 122 62 84)(18 123 32 85)(19 124 33 86)(20 94 34 87)(21 95 35 88)(22 96 36 89)(23 97 37 90)(24 98 38 91)(25 99 39 92)(26 100 40 93)(27 101 41 63)(28 102 42 64)(29 103 43 65)(30 104 44 66)(31 105 45 67)
(1 68)(2 69)(3 70)(4 71)(5 72)(6 73)(7 74)(8 75)(9 76)(10 77)(11 78)(12 79)(13 80)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 88)(22 89)(23 90)(24 91)(25 92)(26 93)(27 63)(28 64)(29 65)(30 66)(31 67)(32 123)(33 124)(34 94)(35 95)(36 96)(37 97)(38 98)(39 99)(40 100)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(61 121)(62 122)
(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 31)(2 30)(3 29)(4 28)(5 27)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(14 18)(15 17)(32 59)(33 58)(34 57)(35 56)(36 55)(37 54)(38 53)(39 52)(40 51)(41 50)(42 49)(43 48)(44 47)(45 46)(60 62)(63 72)(64 71)(65 70)(66 69)(67 68)(73 93)(74 92)(75 91)(76 90)(77 89)(78 88)(79 87)(80 86)(81 85)(82 84)(94 117)(95 116)(96 115)(97 114)(98 113)(99 112)(100 111)(101 110)(102 109)(103 108)(104 107)(105 106)(118 124)(119 123)(120 122)

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

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

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

85 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 31A ··· 31O 62A ··· 62O 62P ··· 62AS 124A ··· 124O order 1 2 2 2 2 2 2 2 4 4 31 ··· 31 62 ··· 62 62 ··· 62 124 ··· 124 size 1 1 2 2 31 31 62 62 2 62 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

85 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D31 D62 D62 D4×D31 kernel D4×D31 C4×D31 D124 C31⋊D4 D4×C31 C22×D31 D31 D4 C4 C22 C1 # reps 1 1 1 2 1 2 2 15 15 30 15

Matrix representation of D4×D31 in GL4(𝔽373) generated by

 372 0 0 0 0 372 0 0 0 0 32 356 0 0 170 341
,
 372 0 0 0 0 372 0 0 0 0 32 356 0 0 126 341
,
 350 1 0 0 39 128 0 0 0 0 1 0 0 0 0 1
,
 346 369 0 0 182 27 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(373))| [372,0,0,0,0,372,0,0,0,0,32,170,0,0,356,341],[372,0,0,0,0,372,0,0,0,0,32,126,0,0,356,341],[350,39,0,0,1,128,0,0,0,0,1,0,0,0,0,1],[346,182,0,0,369,27,0,0,0,0,1,0,0,0,0,1] >;

D4×D31 in GAP, Magma, Sage, TeX

D_4\times D_{31}
% in TeX

G:=Group("D4xD31");
// GroupNames label

G:=SmallGroup(496,31);
// by ID

G=gap.SmallGroup(496,31);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-31,97,12004]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^31=d^2=1,b*a*b=a^-1,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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