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G = C7×D17order 238 = 2·7·17

Direct product of C7 and D17

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

Aliases: C7×D17, C17⋊C14, C1192C2, SmallGroup(238,2)

Series: Derived Chief Lower central Upper central

C1C17 — C7×D17
C1C17C119 — C7×D17
C17 — C7×D17
C1C7

Generators and relations for C7×D17
 G = < a,b,c | a7=b17=c2=1, ab=ba, ac=ca, cbc=b-1 >

17C2
17C14

Smallest permutation representation of C7×D17
On 119 points
Generators in S119
(1 119 92 82 54 35 28)(2 103 93 83 55 36 29)(3 104 94 84 56 37 30)(4 105 95 85 57 38 31)(5 106 96 69 58 39 32)(6 107 97 70 59 40 33)(7 108 98 71 60 41 34)(8 109 99 72 61 42 18)(9 110 100 73 62 43 19)(10 111 101 74 63 44 20)(11 112 102 75 64 45 21)(12 113 86 76 65 46 22)(13 114 87 77 66 47 23)(14 115 88 78 67 48 24)(15 116 89 79 68 49 25)(16 117 90 80 52 50 26)(17 118 91 81 53 51 27)
(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 51)(36 50)(37 49)(38 48)(39 47)(40 46)(41 45)(42 44)(52 55)(53 54)(56 68)(57 67)(58 66)(59 65)(60 64)(61 63)(69 77)(70 76)(71 75)(72 74)(78 85)(79 84)(80 83)(81 82)(86 97)(87 96)(88 95)(89 94)(90 93)(91 92)(98 102)(99 101)(103 117)(104 116)(105 115)(106 114)(107 113)(108 112)(109 111)(118 119)

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

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

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

C7×D17 is a maximal subgroup of   C17⋊Dic7

70 conjugacy classes

class 1  2 7A···7F14A···14F17A···17H119A···119AV
order127···714···1417···17119···119
size1171···117···172···22···2

70 irreducible representations

dim111122
type+++
imageC1C2C7C14D17C7×D17
kernelC7×D17C119D17C17C7C1
# reps1166848

Matrix representation of C7×D17 in GL2(𝔽239) generated by

440
044
,
1291
90236
,
49138
232190
G:=sub<GL(2,GF(239))| [44,0,0,44],[129,90,1,236],[49,232,138,190] >;

C7×D17 in GAP, Magma, Sage, TeX

C_7\times D_{17}
% in TeX

G:=Group("C7xD17");
// GroupNames label

G:=SmallGroup(238,2);
// by ID

G=gap.SmallGroup(238,2);
# by ID

G:=PCGroup([3,-2,-7,-17,2018]);
// Polycyclic

G:=Group<a,b,c|a^7=b^17=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C7×D17 in TeX

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