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G = C2×C5⋊D16order 320 = 26·5

Direct product of C2 and C5⋊D16

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

Aliases: C2×C5⋊D16, D87D10, C102D16, C20.20D8, C40.20D4, D4013C22, C40.23C23, C53(C2×D16), (C2×D8)⋊1D5, (C10×D8)⋊4C2, (C2×D40)⋊17C2, C4.8(D4⋊D5), (C2×C10).41D8, C10.62(C2×D8), (C5×D8)⋊7C22, C52C167C22, (C2×C20).179D4, C20.159(C2×D4), (C2×C8).233D10, C8.13(C5⋊D4), C8.29(C22×D5), (C2×C40).85C22, C22.21(D4⋊D5), (C2×C52C16)⋊6C2, C4.1(C2×C5⋊D4), C2.17(C2×D4⋊D5), (C2×C4).142(C5⋊D4), SmallGroup(320,773)

Series: Derived Chief Lower central Upper central

C1C40 — C2×C5⋊D16
C1C5C10C20C40D40C2×D40 — C2×C5⋊D16
C5C10C20C40 — C2×C5⋊D16
C1C22C2×C4C2×C8C2×D8

Generators and relations for C2×C5⋊D16
 G = < a,b,c,d | a2=b5=c16=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 542 in 98 conjugacy classes, 39 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C22, C22 [×8], C5, C8 [×2], C2×C4, D4 [×6], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×2], C16 [×2], C2×C8, D8 [×2], D8 [×4], C2×D4 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×4], C2×C16, D16 [×4], C2×D8, C2×D8, C40 [×2], D20 [×3], C2×C20, C5×D4 [×3], C22×D5, C22×C10, C2×D16, C52C16 [×2], D40 [×2], D40, C2×C40, C5×D8 [×2], C5×D8, C2×D20, D4×C10, C2×C52C16, C5⋊D16 [×4], C2×D40, C10×D8, C2×C5⋊D16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, D8 [×2], C2×D4, D10 [×3], D16 [×2], C2×D8, C5⋊D4 [×2], C22×D5, C2×D16, D4⋊D5 [×2], C2×C5⋊D4, C5⋊D16 [×2], C2×D4⋊D5, C2×C5⋊D16

Smallest permutation representation of C2×C5⋊D16
On 160 points
Generators in S160
(1 72)(2 73)(3 74)(4 75)(5 76)(6 77)(7 78)(8 79)(9 80)(10 65)(11 66)(12 67)(13 68)(14 69)(15 70)(16 71)(17 157)(18 158)(19 159)(20 160)(21 145)(22 146)(23 147)(24 148)(25 149)(26 150)(27 151)(28 152)(29 153)(30 154)(31 155)(32 156)(33 131)(34 132)(35 133)(36 134)(37 135)(38 136)(39 137)(40 138)(41 139)(42 140)(43 141)(44 142)(45 143)(46 144)(47 129)(48 130)(49 91)(50 92)(51 93)(52 94)(53 95)(54 96)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 89)(64 90)(97 117)(98 118)(99 119)(100 120)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 113)(110 114)(111 115)(112 116)
(1 123 17 144 52)(2 53 129 18 124)(3 125 19 130 54)(4 55 131 20 126)(5 127 21 132 56)(6 57 133 22 128)(7 113 23 134 58)(8 59 135 24 114)(9 115 25 136 60)(10 61 137 26 116)(11 117 27 138 62)(12 63 139 28 118)(13 119 29 140 64)(14 49 141 30 120)(15 121 31 142 50)(16 51 143 32 122)(33 160 106 75 81)(34 82 76 107 145)(35 146 108 77 83)(36 84 78 109 147)(37 148 110 79 85)(38 86 80 111 149)(39 150 112 65 87)(40 88 66 97 151)(41 152 98 67 89)(42 90 68 99 153)(43 154 100 69 91)(44 92 70 101 155)(45 156 102 71 93)(46 94 72 103 157)(47 158 104 73 95)(48 96 74 105 159)
(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)
(2 16)(3 15)(4 14)(5 13)(6 12)(7 11)(8 10)(17 144)(18 143)(19 142)(20 141)(21 140)(22 139)(23 138)(24 137)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 154)(34 153)(35 152)(36 151)(37 150)(38 149)(39 148)(40 147)(41 146)(42 145)(43 160)(44 159)(45 158)(46 157)(47 156)(48 155)(49 126)(50 125)(51 124)(52 123)(53 122)(54 121)(55 120)(56 119)(57 118)(58 117)(59 116)(60 115)(61 114)(62 113)(63 128)(64 127)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)(81 100)(82 99)(83 98)(84 97)(85 112)(86 111)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B5A5B8A8B8C8D10A···10F10G···10N16A···16H20A20B20C20D40A···40H
order122222224455888810···1010···1016···162020202040···40
size1111884040222222222···28···810···1044444···4

50 irreducible representations

dim111112222222222444
type++++++++++++++++
imageC1C2C2C2C2D4D4D5D8D8D10D10D16C5⋊D4C5⋊D4D4⋊D5D4⋊D5C5⋊D16
kernelC2×C5⋊D16C2×C52C16C5⋊D16C2×D40C10×D8C40C2×C20C2×D8C20C2×C10C2×C8D8C10C8C2×C4C4C22C2
# reps114111122224844228

Matrix representation of C2×C5⋊D16 in GL5(𝔽241)

2400000
01000
00100
00010
00001
,
10000
018924000
01000
00010
00001
,
2400000
021413800
0962700
0002785
00015627
,
2400000
01000
018924000
00010
0000240

G:=sub<GL(5,GF(241))| [240,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,189,1,0,0,0,240,0,0,0,0,0,0,1,0,0,0,0,0,1],[240,0,0,0,0,0,214,96,0,0,0,138,27,0,0,0,0,0,27,156,0,0,0,85,27],[240,0,0,0,0,0,1,189,0,0,0,0,240,0,0,0,0,0,1,0,0,0,0,0,240] >;

C2×C5⋊D16 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{16}
% in TeX

G:=Group("C2xC5:D16");
// GroupNames label

G:=SmallGroup(320,773);
// by ID

G=gap.SmallGroup(320,773);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,254,675,185,192,1684,438,102,12550]);
// Polycyclic

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

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