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G = D7×D17order 476 = 22·7·17

Direct product of D7 and D17

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: D7×D17, C71D34, D119⋊C2, C171D14, C119⋊C22, (C7×D17)⋊C2, (D7×C17)⋊C2, SmallGroup(476,7)

Series: Derived Chief Lower central Upper central

C1C119 — D7×D17
C1C17C119C7×D17 — D7×D17
C119 — D7×D17
C1

Generators and relations for D7×D17
 G = < a,b,c,d | a7=b2=c17=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

7C2
17C2
119C2
119C22
17C14
17D7
7C34
7D17
17D14
7D34

Smallest permutation representation of D7×D17
On 119 points
Generators in S119
(1 113 96 74 58 42 28)(2 114 97 75 59 43 29)(3 115 98 76 60 44 30)(4 116 99 77 61 45 31)(5 117 100 78 62 46 32)(6 118 101 79 63 47 33)(7 119 102 80 64 48 34)(8 103 86 81 65 49 18)(9 104 87 82 66 50 19)(10 105 88 83 67 51 20)(11 106 89 84 68 35 21)(12 107 90 85 52 36 22)(13 108 91 69 53 37 23)(14 109 92 70 54 38 24)(15 110 93 71 55 39 25)(16 111 94 72 56 40 26)(17 112 95 73 57 41 27)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 18)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 25)(16 26)(17 27)(35 106)(36 107)(37 108)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 103)(50 104)(51 105)(52 90)(53 91)(54 92)(55 93)(56 94)(57 95)(58 96)(59 97)(60 98)(61 99)(62 100)(63 101)(64 102)(65 86)(66 87)(67 88)(68 89)
(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)
(1 17)(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(18 20)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(41 42)(49 51)(52 63)(53 62)(54 61)(55 60)(56 59)(57 58)(64 68)(65 67)(69 78)(70 77)(71 76)(72 75)(73 74)(79 85)(80 84)(81 83)(86 88)(89 102)(90 101)(91 100)(92 99)(93 98)(94 97)(95 96)(103 105)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)

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

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

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

50 conjugacy classes

class 1 2A2B2C7A7B7C14A14B14C17A···17H34A···34H119A···119X
order122277714141417···1734···34119···119
size17171192223434342···214···144···4

50 irreducible representations

dim111122224
type+++++++++
imageC1C2C2C2D7D14D17D34D7×D17
kernelD7×D17D7×C17C7×D17D119D17C17D7C7C1
# reps1111338824

Matrix representation of D7×D17 in GL4(𝔽239) generated by

1000
0100
0001
0023834
,
1000
0100
0001
0010
,
0100
23810000
0010
0001
,
0100
1000
0010
0001
G:=sub<GL(4,GF(239))| [1,0,0,0,0,1,0,0,0,0,0,238,0,0,1,34],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[0,238,0,0,1,100,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1] >;

D7×D17 in GAP, Magma, Sage, TeX

D_7\times D_{17}
% in TeX

G:=Group("D7xD17");
// GroupNames label

G:=SmallGroup(476,7);
// by ID

G=gap.SmallGroup(476,7);
# by ID

G:=PCGroup([4,-2,-2,-7,-17,150,7171]);
// Polycyclic

G:=Group<a,b,c,d|a^7=b^2=c^17=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

Export

Subgroup lattice of D7×D17 in TeX

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