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G = D7×C22order 308 = 22·7·11

Direct product of C22 and D7

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

Aliases: D7×C22, C14⋊C22, C1543C2, C774C22, C7⋊(C2×C22), SmallGroup(308,7)

Series: Derived Chief Lower central Upper central

C1C7 — D7×C22
C1C7C77C11×D7 — D7×C22
C7 — D7×C22
C1C22

Generators and relations for D7×C22
 G = < a,b,c | a22=b7=c2=1, ab=ba, ac=ca, cbc=b-1 >

7C2
7C2
7C22
7C22
7C22
7C2×C22

Smallest permutation representation of D7×C22
On 154 points
Generators in S154
(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)(111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132)(133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154)
(1 154 50 103 75 130 39)(2 133 51 104 76 131 40)(3 134 52 105 77 132 41)(4 135 53 106 78 111 42)(5 136 54 107 79 112 43)(6 137 55 108 80 113 44)(7 138 56 109 81 114 23)(8 139 57 110 82 115 24)(9 140 58 89 83 116 25)(10 141 59 90 84 117 26)(11 142 60 91 85 118 27)(12 143 61 92 86 119 28)(13 144 62 93 87 120 29)(14 145 63 94 88 121 30)(15 146 64 95 67 122 31)(16 147 65 96 68 123 32)(17 148 66 97 69 124 33)(18 149 45 98 70 125 34)(19 150 46 99 71 126 35)(20 151 47 100 72 127 36)(21 152 48 101 73 128 37)(22 153 49 102 74 129 38)
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 37)(11 38)(12 39)(13 40)(14 41)(15 42)(16 43)(17 44)(18 23)(19 24)(20 25)(21 26)(22 27)(45 81)(46 82)(47 83)(48 84)(49 85)(50 86)(51 87)(52 88)(53 67)(54 68)(55 69)(56 70)(57 71)(58 72)(59 73)(60 74)(61 75)(62 76)(63 77)(64 78)(65 79)(66 80)(89 100)(90 101)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(111 146)(112 147)(113 148)(114 149)(115 150)(116 151)(117 152)(118 153)(119 154)(120 133)(121 134)(122 135)(123 136)(124 137)(125 138)(126 139)(127 140)(128 141)(129 142)(130 143)(131 144)(132 145)

G:=sub<Sym(154)| (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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154), (1,154,50,103,75,130,39)(2,133,51,104,76,131,40)(3,134,52,105,77,132,41)(4,135,53,106,78,111,42)(5,136,54,107,79,112,43)(6,137,55,108,80,113,44)(7,138,56,109,81,114,23)(8,139,57,110,82,115,24)(9,140,58,89,83,116,25)(10,141,59,90,84,117,26)(11,142,60,91,85,118,27)(12,143,61,92,86,119,28)(13,144,62,93,87,120,29)(14,145,63,94,88,121,30)(15,146,64,95,67,122,31)(16,147,65,96,68,123,32)(17,148,66,97,69,124,33)(18,149,45,98,70,125,34)(19,150,46,99,71,126,35)(20,151,47,100,72,127,36)(21,152,48,101,73,128,37)(22,153,49,102,74,129,38), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,23)(19,24)(20,25)(21,26)(22,27)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143)(131,144)(132,145)>;

G:=Group( (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)(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132)(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154), (1,154,50,103,75,130,39)(2,133,51,104,76,131,40)(3,134,52,105,77,132,41)(4,135,53,106,78,111,42)(5,136,54,107,79,112,43)(6,137,55,108,80,113,44)(7,138,56,109,81,114,23)(8,139,57,110,82,115,24)(9,140,58,89,83,116,25)(10,141,59,90,84,117,26)(11,142,60,91,85,118,27)(12,143,61,92,86,119,28)(13,144,62,93,87,120,29)(14,145,63,94,88,121,30)(15,146,64,95,67,122,31)(16,147,65,96,68,123,32)(17,148,66,97,69,124,33)(18,149,45,98,70,125,34)(19,150,46,99,71,126,35)(20,151,47,100,72,127,36)(21,152,48,101,73,128,37)(22,153,49,102,74,129,38), (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,37)(11,38)(12,39)(13,40)(14,41)(15,42)(16,43)(17,44)(18,23)(19,24)(20,25)(21,26)(22,27)(45,81)(46,82)(47,83)(48,84)(49,85)(50,86)(51,87)(52,88)(53,67)(54,68)(55,69)(56,70)(57,71)(58,72)(59,73)(60,74)(61,75)(62,76)(63,77)(64,78)(65,79)(66,80)(89,100)(90,101)(91,102)(92,103)(93,104)(94,105)(95,106)(96,107)(97,108)(98,109)(99,110)(111,146)(112,147)(113,148)(114,149)(115,150)(116,151)(117,152)(118,153)(119,154)(120,133)(121,134)(122,135)(123,136)(124,137)(125,138)(126,139)(127,140)(128,141)(129,142)(130,143)(131,144)(132,145) );

G=PermutationGroup([(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),(111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132),(133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154)], [(1,154,50,103,75,130,39),(2,133,51,104,76,131,40),(3,134,52,105,77,132,41),(4,135,53,106,78,111,42),(5,136,54,107,79,112,43),(6,137,55,108,80,113,44),(7,138,56,109,81,114,23),(8,139,57,110,82,115,24),(9,140,58,89,83,116,25),(10,141,59,90,84,117,26),(11,142,60,91,85,118,27),(12,143,61,92,86,119,28),(13,144,62,93,87,120,29),(14,145,63,94,88,121,30),(15,146,64,95,67,122,31),(16,147,65,96,68,123,32),(17,148,66,97,69,124,33),(18,149,45,98,70,125,34),(19,150,46,99,71,126,35),(20,151,47,100,72,127,36),(21,152,48,101,73,128,37),(22,153,49,102,74,129,38)], [(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,37),(11,38),(12,39),(13,40),(14,41),(15,42),(16,43),(17,44),(18,23),(19,24),(20,25),(21,26),(22,27),(45,81),(46,82),(47,83),(48,84),(49,85),(50,86),(51,87),(52,88),(53,67),(54,68),(55,69),(56,70),(57,71),(58,72),(59,73),(60,74),(61,75),(62,76),(63,77),(64,78),(65,79),(66,80),(89,100),(90,101),(91,102),(92,103),(93,104),(94,105),(95,106),(96,107),(97,108),(98,109),(99,110),(111,146),(112,147),(113,148),(114,149),(115,150),(116,151),(117,152),(118,153),(119,154),(120,133),(121,134),(122,135),(123,136),(124,137),(125,138),(126,139),(127,140),(128,141),(129,142),(130,143),(131,144),(132,145)])

110 conjugacy classes

class 1 2A2B2C7A7B7C11A···11J14A14B14C22A···22J22K···22AD77A···77AD154A···154AD
order122277711···1114141422···2222···2277···77154···154
size11772221···12221···17···72···22···2

110 irreducible representations

dim1111112222
type+++++
imageC1C2C2C11C22C22D7D14C11×D7D7×C22
kernelD7×C22C11×D7C154D14D7C14C22C11C2C1
# reps121102010333030

Matrix representation of D7×C22 in GL3(𝔽463) generated by

46200
02470
00247
,
100
001
046275
,
100
001
010
G:=sub<GL(3,GF(463))| [462,0,0,0,247,0,0,0,247],[1,0,0,0,0,462,0,1,75],[1,0,0,0,0,1,0,1,0] >;

D7×C22 in GAP, Magma, Sage, TeX

D_7\times C_{22}
% in TeX

G:=Group("D7xC22");
// GroupNames label

G:=SmallGroup(308,7);
// by ID

G=gap.SmallGroup(308,7);
# by ID

G:=PCGroup([4,-2,-2,-11,-7,4227]);
// Polycyclic

G:=Group<a,b,c|a^22=b^7=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of D7×C22 in TeX

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