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## G = D5×D23order 460 = 22·5·23

### Direct product of D5 and D23

Aliases: D5×D23, C51D46, D115⋊C2, C231D10, C115⋊C22, (C5×D23)⋊C2, (D5×C23)⋊C2, SmallGroup(460,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C115 — D5×D23
 Chief series C1 — C23 — C115 — C5×D23 — D5×D23
 Lower central C115 — D5×D23
 Upper central C1

Generators and relations for D5×D23
G = < a,b,c,d | a5=b2=c23=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

5C2
23C2
115C2
115C22
23C10
23D5
5C46
5D23
23D10
5D46

Smallest permutation representation of D5×D23
On 115 points
Generators in S115
(1 109 82 52 40)(2 110 83 53 41)(3 111 84 54 42)(4 112 85 55 43)(5 113 86 56 44)(6 114 87 57 45)(7 115 88 58 46)(8 93 89 59 24)(9 94 90 60 25)(10 95 91 61 26)(11 96 92 62 27)(12 97 70 63 28)(13 98 71 64 29)(14 99 72 65 30)(15 100 73 66 31)(16 101 74 67 32)(17 102 75 68 33)(18 103 76 69 34)(19 104 77 47 35)(20 105 78 48 36)(21 106 79 49 37)(22 107 80 50 38)(23 108 81 51 39)
(1 40)(2 41)(3 42)(4 43)(5 44)(6 45)(7 46)(8 24)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 37)(22 38)(23 39)(47 104)(48 105)(49 106)(50 107)(51 108)(52 109)(53 110)(54 111)(55 112)(56 113)(57 114)(58 115)(59 93)(60 94)(61 95)(62 96)(63 97)(64 98)(65 99)(66 100)(67 101)(68 102)(69 103)
(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)
(1 23)(2 22)(3 21)(4 20)(5 19)(6 18)(7 17)(8 16)(9 15)(10 14)(11 13)(24 32)(25 31)(26 30)(27 29)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)(47 56)(48 55)(49 54)(50 53)(51 52)(57 69)(58 68)(59 67)(60 66)(61 65)(62 64)(71 92)(72 91)(73 90)(74 89)(75 88)(76 87)(77 86)(78 85)(79 84)(80 83)(81 82)(93 101)(94 100)(95 99)(96 98)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 5A 5B 10A 10B 23A ··· 23K 46A ··· 46K 115A ··· 115V order 1 2 2 2 5 5 10 10 23 ··· 23 46 ··· 46 115 ··· 115 size 1 5 23 115 2 2 46 46 2 ··· 2 10 ··· 10 4 ··· 4

52 irreducible representations

 dim 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 D5 D10 D23 D46 D5×D23 kernel D5×D23 D5×C23 C5×D23 D115 D23 C23 D5 C5 C1 # reps 1 1 1 1 2 2 11 11 22

Matrix representation of D5×D23 in GL4(𝔽461) generated by

 1 0 0 0 0 1 0 0 0 0 1 460 0 0 24 438
,
 1 0 0 0 0 1 0 0 0 0 1 460 0 0 0 460
,
 304 1 0 0 234 160 0 0 0 0 1 0 0 0 0 1
,
 20 406 0 0 309 441 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(461))| [1,0,0,0,0,1,0,0,0,0,1,24,0,0,460,438],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,460,460],[304,234,0,0,1,160,0,0,0,0,1,0,0,0,0,1],[20,309,0,0,406,441,0,0,0,0,1,0,0,0,0,1] >;

D5×D23 in GAP, Magma, Sage, TeX

D_5\times D_{23}
% in TeX

G:=Group("D5xD23");
// GroupNames label

G:=SmallGroup(460,7);
// by ID

G=gap.SmallGroup(460,7);
# by ID

G:=PCGroup([4,-2,-2,-5,-23,102,7043]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^2=c^23=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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