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G = C2×D5⋊C16order 320 = 26·5

Direct product of C2 and D5⋊C16

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

Aliases: C2×D5⋊C16, D103C16, D5⋊(C2×C16), C101(C2×C16), C5⋊C164C22, C51(C22×C16), (C4×D5).7C8, C8.41(C2×F5), (C2×C8).24F5, C20.23(C2×C8), C40.38(C2×C4), (C2×C40).22C4, (C8×D5).10C4, C4.18(D5⋊C8), D10.16(C2×C8), C10.8(C22×C8), (C22×D5).7C8, C4.45(C22×F5), C20.85(C22×C4), C52C8.35C23, (C2×Dic5).12C8, Dic5.16(C2×C8), (C8×D5).61C22, C22.12(D5⋊C8), (C2×C5⋊C16)⋊9C2, C2.2(C2×D5⋊C8), (D5×C2×C8).29C2, (C2×C4×D5).45C4, (C2×C10).10(C2×C8), C52C8.51(C2×C4), (C4×D5).91(C2×C4), (C2×C4).163(C2×F5), (C2×C20).172(C2×C4), (C2×C52C8).348C22, SmallGroup(320,1051)

Series: Derived Chief Lower central Upper central

C1C5 — C2×D5⋊C16
C1C5C10C20C52C8C5⋊C16C2×C5⋊C16 — C2×D5⋊C16
C5 — C2×D5⋊C16
C1C2×C8

Generators and relations for C2×D5⋊C16
 G = < a,b,c,d | a2=b5=c2=d16=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b3, dcd-1=b2c >

Subgroups: 250 in 98 conjugacy classes, 60 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C22, C22 [×6], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×5], C23, D5 [×4], C10, C10 [×2], C16 [×4], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], C20 [×2], D10 [×6], C2×C10, C2×C16 [×6], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C22×C16, C5⋊C16 [×4], C8×D5 [×4], C2×C52C8, C2×C40, C2×C4×D5, D5⋊C16 [×4], C2×C5⋊C16 [×2], D5×C2×C8, C2×D5⋊C16
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], C23, C16 [×4], C2×C8 [×6], C22×C4, F5, C2×C16 [×6], C22×C8, C2×F5 [×3], C22×C16, D5⋊C8 [×2], C22×F5, D5⋊C16 [×2], C2×D5⋊C8, C2×D5⋊C16

Smallest permutation representation of C2×D5⋊C16
On 160 points
Generators in S160
(1 48)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 103)(18 104)(19 105)(20 106)(21 107)(22 108)(23 109)(24 110)(25 111)(26 112)(27 97)(28 98)(29 99)(30 100)(31 101)(32 102)(49 116)(50 117)(51 118)(52 119)(53 120)(54 121)(55 122)(56 123)(57 124)(58 125)(59 126)(60 127)(61 128)(62 113)(63 114)(64 115)(65 145)(66 146)(67 147)(68 148)(69 149)(70 150)(71 151)(72 152)(73 153)(74 154)(75 155)(76 156)(77 157)(78 158)(79 159)(80 160)(81 144)(82 129)(83 130)(84 131)(85 132)(86 133)(87 134)(88 135)(89 136)(90 137)(91 138)(92 139)(93 140)(94 141)(95 142)(96 143)
(1 145 126 140 19)(2 141 146 20 127)(3 21 142 128 147)(4 113 22 148 143)(5 149 114 144 23)(6 129 150 24 115)(7 25 130 116 151)(8 117 26 152 131)(9 153 118 132 27)(10 133 154 28 119)(11 29 134 120 155)(12 121 30 156 135)(13 157 122 136 31)(14 137 158 32 123)(15 17 138 124 159)(16 125 18 160 139)(33 94 66 106 60)(34 107 95 61 67)(35 62 108 68 96)(36 69 63 81 109)(37 82 70 110 64)(38 111 83 49 71)(39 50 112 72 84)(40 73 51 85 97)(41 86 74 98 52)(42 99 87 53 75)(43 54 100 76 88)(44 77 55 89 101)(45 90 78 102 56)(46 103 91 57 79)(47 58 104 80 92)(48 65 59 93 105)
(1 19)(2 127)(3 147)(4 143)(5 23)(6 115)(7 151)(8 131)(9 27)(10 119)(11 155)(12 135)(13 31)(14 123)(15 159)(16 139)(17 124)(20 141)(21 128)(24 129)(25 116)(28 133)(29 120)(32 137)(33 60)(34 67)(35 96)(36 109)(37 64)(38 71)(39 84)(40 97)(41 52)(42 75)(43 88)(44 101)(45 56)(46 79)(47 92)(48 105)(49 111)(50 72)(53 99)(54 76)(57 103)(58 80)(61 107)(62 68)(65 93)(69 81)(73 85)(77 89)(82 110)(86 98)(90 102)(94 106)(113 148)(117 152)(121 156)(125 160)(132 153)(136 157)(140 145)(144 149)
(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)(145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)

G:=sub<Sym(160)| (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,113)(63,114)(64,115)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,145,126,140,19)(2,141,146,20,127)(3,21,142,128,147)(4,113,22,148,143)(5,149,114,144,23)(6,129,150,24,115)(7,25,130,116,151)(8,117,26,152,131)(9,153,118,132,27)(10,133,154,28,119)(11,29,134,120,155)(12,121,30,156,135)(13,157,122,136,31)(14,137,158,32,123)(15,17,138,124,159)(16,125,18,160,139)(33,94,66,106,60)(34,107,95,61,67)(35,62,108,68,96)(36,69,63,81,109)(37,82,70,110,64)(38,111,83,49,71)(39,50,112,72,84)(40,73,51,85,97)(41,86,74,98,52)(42,99,87,53,75)(43,54,100,76,88)(44,77,55,89,101)(45,90,78,102,56)(46,103,91,57,79)(47,58,104,80,92)(48,65,59,93,105), (1,19)(2,127)(3,147)(4,143)(5,23)(6,115)(7,151)(8,131)(9,27)(10,119)(11,155)(12,135)(13,31)(14,123)(15,159)(16,139)(17,124)(20,141)(21,128)(24,129)(25,116)(28,133)(29,120)(32,137)(33,60)(34,67)(35,96)(36,109)(37,64)(38,71)(39,84)(40,97)(41,52)(42,75)(43,88)(44,101)(45,56)(46,79)(47,92)(48,105)(49,111)(50,72)(53,99)(54,76)(57,103)(58,80)(61,107)(62,68)(65,93)(69,81)(73,85)(77,89)(82,110)(86,98)(90,102)(94,106)(113,148)(117,152)(121,156)(125,160)(132,153)(136,157)(140,145)(144,149), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)>;

G:=Group( (1,48)(2,33)(3,34)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,103)(18,104)(19,105)(20,106)(21,107)(22,108)(23,109)(24,110)(25,111)(26,112)(27,97)(28,98)(29,99)(30,100)(31,101)(32,102)(49,116)(50,117)(51,118)(52,119)(53,120)(54,121)(55,122)(56,123)(57,124)(58,125)(59,126)(60,127)(61,128)(62,113)(63,114)(64,115)(65,145)(66,146)(67,147)(68,148)(69,149)(70,150)(71,151)(72,152)(73,153)(74,154)(75,155)(76,156)(77,157)(78,158)(79,159)(80,160)(81,144)(82,129)(83,130)(84,131)(85,132)(86,133)(87,134)(88,135)(89,136)(90,137)(91,138)(92,139)(93,140)(94,141)(95,142)(96,143), (1,145,126,140,19)(2,141,146,20,127)(3,21,142,128,147)(4,113,22,148,143)(5,149,114,144,23)(6,129,150,24,115)(7,25,130,116,151)(8,117,26,152,131)(9,153,118,132,27)(10,133,154,28,119)(11,29,134,120,155)(12,121,30,156,135)(13,157,122,136,31)(14,137,158,32,123)(15,17,138,124,159)(16,125,18,160,139)(33,94,66,106,60)(34,107,95,61,67)(35,62,108,68,96)(36,69,63,81,109)(37,82,70,110,64)(38,111,83,49,71)(39,50,112,72,84)(40,73,51,85,97)(41,86,74,98,52)(42,99,87,53,75)(43,54,100,76,88)(44,77,55,89,101)(45,90,78,102,56)(46,103,91,57,79)(47,58,104,80,92)(48,65,59,93,105), (1,19)(2,127)(3,147)(4,143)(5,23)(6,115)(7,151)(8,131)(9,27)(10,119)(11,155)(12,135)(13,31)(14,123)(15,159)(16,139)(17,124)(20,141)(21,128)(24,129)(25,116)(28,133)(29,120)(32,137)(33,60)(34,67)(35,96)(36,109)(37,64)(38,71)(39,84)(40,97)(41,52)(42,75)(43,88)(44,101)(45,56)(46,79)(47,92)(48,105)(49,111)(50,72)(53,99)(54,76)(57,103)(58,80)(61,107)(62,68)(65,93)(69,81)(73,85)(77,89)(82,110)(86,98)(90,102)(94,106)(113,148)(117,152)(121,156)(125,160)(132,153)(136,157)(140,145)(144,149), (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)(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160) );

G=PermutationGroup([(1,48),(2,33),(3,34),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,103),(18,104),(19,105),(20,106),(21,107),(22,108),(23,109),(24,110),(25,111),(26,112),(27,97),(28,98),(29,99),(30,100),(31,101),(32,102),(49,116),(50,117),(51,118),(52,119),(53,120),(54,121),(55,122),(56,123),(57,124),(58,125),(59,126),(60,127),(61,128),(62,113),(63,114),(64,115),(65,145),(66,146),(67,147),(68,148),(69,149),(70,150),(71,151),(72,152),(73,153),(74,154),(75,155),(76,156),(77,157),(78,158),(79,159),(80,160),(81,144),(82,129),(83,130),(84,131),(85,132),(86,133),(87,134),(88,135),(89,136),(90,137),(91,138),(92,139),(93,140),(94,141),(95,142),(96,143)], [(1,145,126,140,19),(2,141,146,20,127),(3,21,142,128,147),(4,113,22,148,143),(5,149,114,144,23),(6,129,150,24,115),(7,25,130,116,151),(8,117,26,152,131),(9,153,118,132,27),(10,133,154,28,119),(11,29,134,120,155),(12,121,30,156,135),(13,157,122,136,31),(14,137,158,32,123),(15,17,138,124,159),(16,125,18,160,139),(33,94,66,106,60),(34,107,95,61,67),(35,62,108,68,96),(36,69,63,81,109),(37,82,70,110,64),(38,111,83,49,71),(39,50,112,72,84),(40,73,51,85,97),(41,86,74,98,52),(42,99,87,53,75),(43,54,100,76,88),(44,77,55,89,101),(45,90,78,102,56),(46,103,91,57,79),(47,58,104,80,92),(48,65,59,93,105)], [(1,19),(2,127),(3,147),(4,143),(5,23),(6,115),(7,151),(8,131),(9,27),(10,119),(11,155),(12,135),(13,31),(14,123),(15,159),(16,139),(17,124),(20,141),(21,128),(24,129),(25,116),(28,133),(29,120),(32,137),(33,60),(34,67),(35,96),(36,109),(37,64),(38,71),(39,84),(40,97),(41,52),(42,75),(43,88),(44,101),(45,56),(46,79),(47,92),(48,105),(49,111),(50,72),(53,99),(54,76),(57,103),(58,80),(61,107),(62,68),(65,93),(69,81),(73,85),(77,89),(82,110),(86,98),(90,102),(94,106),(113,148),(117,152),(121,156),(125,160),(132,153),(136,157),(140,145),(144,149)], [(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),(145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)])

80 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H 5 8A···8H8I···8P10A10B10C16A···16AF20A20B20C20D40A···40H
order122222224444444458···88···810101016···162020202040···40
size111155551111555541···15···54445···544444···4

80 irreducible representations

dim11111111111444444
type+++++++
imageC1C2C2C2C4C4C4C8C8C8C16F5C2×F5C2×F5D5⋊C8D5⋊C8D5⋊C16
kernelC2×D5⋊C16D5⋊C16C2×C5⋊C16D5×C2×C8C8×D5C2×C40C2×C4×D5C4×D5C2×Dic5C22×D5D10C2×C8C8C2×C4C4C22C2
# reps142142284432121228

Matrix representation of C2×D5⋊C16 in GL5(𝔽241)

2400000
0240000
0024000
0002400
0000240
,
10000
0240240240240
01000
00100
00010
,
2400000
0240240240240
00001
00010
00100
,
2400000
015407676
076760154
0165781650
08716316387

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240,0,0,0,0,0,240],[1,0,0,0,0,0,240,1,0,0,0,240,0,1,0,0,240,0,0,1,0,240,0,0,0],[240,0,0,0,0,0,240,0,0,0,0,240,0,0,1,0,240,0,1,0,0,240,1,0,0],[240,0,0,0,0,0,154,76,165,87,0,0,76,78,163,0,76,0,165,163,0,76,154,0,87] >;

C2×D5⋊C16 in GAP, Magma, Sage, TeX

C_2\times D_5\rtimes C_{16}
% in TeX

G:=Group("C2xD5:C16");
// GroupNames label

G:=SmallGroup(320,1051);
// by ID

G=gap.SmallGroup(320,1051);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,100,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^2=d^16=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^3,d*c*d^-1=b^2*c>;
// generators/relations

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