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## G = D4×D29order 464 = 24·29

### Direct product of D4 and D29

Aliases: D4×D29, C41D58, C116⋊C22, D1163C2, C221D58, D582C22, C58.5C23, Dic291C22, C292(C2×D4), (C2×C58)⋊C22, (C4×D29)⋊1C2, (D4×C29)⋊2C2, C29⋊D41C2, (C22×D29)⋊2C2, C2.6(C22×D29), SmallGroup(464,39)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C58 — D4×D29
 Chief series C1 — C29 — C58 — D58 — C22×D29 — D4×D29
 Lower central C29 — C58 — D4×D29
 Upper central C1 — C2 — D4

Generators and relations for D4×D29
G = < a,b,c,d | a4=b2=c29=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 770 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C2×D4, C29, D29, D29, C58, C58, Dic29, C116, D58, D58, D58, C2×C58, C4×D29, D116, C29⋊D4, D4×C29, C22×D29, D4×D29
Quotients: C1, C2, C22, D4, C23, C2×D4, D29, D58, C22×D29, D4×D29

Smallest permutation representation of D4×D29
On 116 points
Generators in S116
(1 108 53 61)(2 109 54 62)(3 110 55 63)(4 111 56 64)(5 112 57 65)(6 113 58 66)(7 114 30 67)(8 115 31 68)(9 116 32 69)(10 88 33 70)(11 89 34 71)(12 90 35 72)(13 91 36 73)(14 92 37 74)(15 93 38 75)(16 94 39 76)(17 95 40 77)(18 96 41 78)(19 97 42 79)(20 98 43 80)(21 99 44 81)(22 100 45 82)(23 101 46 83)(24 102 47 84)(25 103 48 85)(26 104 49 86)(27 105 50 87)(28 106 51 59)(29 107 52 60)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 81)(22 82)(23 83)(24 84)(25 85)(26 86)(27 87)(28 59)(29 60)(30 114)(31 115)(32 116)(33 88)(34 89)(35 90)(36 91)(37 92)(38 93)(39 94)(40 95)(41 96)(42 97)(43 98)(44 99)(45 100)(46 101)(47 102)(48 103)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)
(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 29)(2 28)(3 27)(4 26)(5 25)(6 24)(7 23)(8 22)(9 21)(10 20)(11 19)(12 18)(13 17)(14 16)(30 46)(31 45)(32 44)(33 43)(34 42)(35 41)(36 40)(37 39)(47 58)(48 57)(49 56)(50 55)(51 54)(52 53)(59 62)(60 61)(63 87)(64 86)(65 85)(66 84)(67 83)(68 82)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)(88 98)(89 97)(90 96)(91 95)(92 94)(99 116)(100 115)(101 114)(102 113)(103 112)(104 111)(105 110)(106 109)(107 108)

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

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

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

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 29A ··· 29N 58A ··· 58N 58O ··· 58AP 116A ··· 116N order 1 2 2 2 2 2 2 2 4 4 29 ··· 29 58 ··· 58 58 ··· 58 116 ··· 116 size 1 1 2 2 29 29 58 58 2 58 2 ··· 2 2 ··· 2 4 ··· 4 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D4 D29 D58 D58 D4×D29 kernel D4×D29 C4×D29 D116 C29⋊D4 D4×C29 C22×D29 D29 D4 C4 C22 C1 # reps 1 1 1 2 1 2 2 14 14 28 14

Matrix representation of D4×D29 in GL4(𝔽233) generated by

 1 0 0 0 0 1 0 0 0 0 1 36 0 0 220 232
,
 1 0 0 0 0 1 0 0 0 0 1 36 0 0 0 232
,
 0 1 0 0 232 122 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(233))| [1,0,0,0,0,1,0,0,0,0,1,220,0,0,36,232],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,36,232],[0,232,0,0,1,122,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

D4×D29 in GAP, Magma, Sage, TeX

D_4\times D_{29}
% in TeX

G:=Group("D4xD29");
// GroupNames label

G:=SmallGroup(464,39);
// by ID

G=gap.SmallGroup(464,39);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-29,97,11204]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^29=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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