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G = C2×C20.17D4order 320 = 26·5

Direct product of C2 and C20.17D4

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

Aliases: C2×C20.17D4, C24.37D10, C20.251(C2×D4), (C2×C20).209D4, (C2×D4).229D10, C103(C4.4D4), (C22×D4).11D5, (C2×C20).540C23, (C2×C10).292C24, (C4×Dic5)⋊67C22, C10.140(C22×D4), (C22×C4).378D10, C23.D558C22, (C2×Dic10)⋊67C22, (C22×Dic10)⋊20C2, (D4×C10).269C22, (C23×C10).74C22, C23.134(C22×D5), C22.306(C23×D5), C22.78(D42D5), (C22×C10).228C23, (C22×C20).273C22, (C2×Dic5).292C23, (C22×Dic5).254C22, (D4×C2×C10).8C2, C54(C2×C4.4D4), (C2×C4×Dic5)⋊11C2, C4.23(C2×C5⋊D4), C10.104(C2×C4○D4), (C2×C10).579(C2×D4), C2.68(C2×D42D5), (C2×C23.D5)⋊25C2, C2.13(C22×C5⋊D4), (C2×C4).153(C5⋊D4), (C2×C4).623(C22×D5), C22.109(C2×C5⋊D4), (C2×C10).176(C4○D4), SmallGroup(320,1469)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20.17D4
C1C5C10C2×C10C2×Dic5C22×Dic5C2×C4×Dic5 — C2×C20.17D4
C5C2×C10 — C2×C20.17D4
C1C23C22×D4

Generators and relations for C2×C20.17D4
 G = < a,b,c,d | a2=b20=c4=1, d2=b10, ab=ba, ac=ca, ad=da, cbc-1=b9, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 958 in 330 conjugacy classes, 127 normal (15 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×4], C4 [×8], C22, C22 [×6], C22 [×20], C5, C2×C4 [×6], C2×C4 [×16], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C10, C10 [×6], C10 [×4], C42 [×4], C22⋊C4 [×16], C22×C4, C22×C4 [×4], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×8], C24 [×2], Dic5 [×8], C20 [×4], C2×C10, C2×C10 [×6], C2×C10 [×20], C2×C42, C2×C22⋊C4 [×4], C4.4D4 [×8], C22×D4, C22×Q8, Dic10 [×8], C2×Dic5 [×8], C2×Dic5 [×8], C2×C20 [×6], C5×D4 [×8], C22×C10, C22×C10 [×4], C22×C10 [×12], C2×C4.4D4, C4×Dic5 [×4], C23.D5 [×16], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×C20, D4×C10 [×4], D4×C10 [×4], C23×C10 [×2], C2×C4×Dic5, C20.17D4 [×8], C2×C23.D5 [×4], C22×Dic10, D4×C2×C10, C2×C20.17D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C2×C4.4D4, D42D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20.17D4 [×4], C2×D42D5 [×2], C22×C5⋊D4, C2×C20.17D4

Smallest permutation representation of C2×C20.17D4
On 160 points
Generators in S160
(1 88)(2 89)(3 90)(4 91)(5 92)(6 93)(7 94)(8 95)(9 96)(10 97)(11 98)(12 99)(13 100)(14 81)(15 82)(16 83)(17 84)(18 85)(19 86)(20 87)(21 136)(22 137)(23 138)(24 139)(25 140)(26 121)(27 122)(28 123)(29 124)(30 125)(31 126)(32 127)(33 128)(34 129)(35 130)(36 131)(37 132)(38 133)(39 134)(40 135)(41 147)(42 148)(43 149)(44 150)(45 151)(46 152)(47 153)(48 154)(49 155)(50 156)(51 157)(52 158)(53 159)(54 160)(55 141)(56 142)(57 143)(58 144)(59 145)(60 146)(61 108)(62 109)(63 110)(64 111)(65 112)(66 113)(67 114)(68 115)(69 116)(70 117)(71 118)(72 119)(73 120)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)
(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)
(1 26 103 51)(2 35 104 60)(3 24 105 49)(4 33 106 58)(5 22 107 47)(6 31 108 56)(7 40 109 45)(8 29 110 54)(9 38 111 43)(10 27 112 52)(11 36 113 41)(12 25 114 50)(13 34 115 59)(14 23 116 48)(15 32 117 57)(16 21 118 46)(17 30 119 55)(18 39 120 44)(19 28 101 53)(20 37 102 42)(61 142 93 126)(62 151 94 135)(63 160 95 124)(64 149 96 133)(65 158 97 122)(66 147 98 131)(67 156 99 140)(68 145 100 129)(69 154 81 138)(70 143 82 127)(71 152 83 136)(72 141 84 125)(73 150 85 134)(74 159 86 123)(75 148 87 132)(76 157 88 121)(77 146 89 130)(78 155 90 139)(79 144 91 128)(80 153 92 137)
(1 121 11 131)(2 140 12 130)(3 139 13 129)(4 138 14 128)(5 137 15 127)(6 136 16 126)(7 135 17 125)(8 134 18 124)(9 133 19 123)(10 132 20 122)(21 83 31 93)(22 82 32 92)(23 81 33 91)(24 100 34 90)(25 99 35 89)(26 98 36 88)(27 97 37 87)(28 96 38 86)(29 95 39 85)(30 94 40 84)(41 76 51 66)(42 75 52 65)(43 74 53 64)(44 73 54 63)(45 72 55 62)(46 71 56 61)(47 70 57 80)(48 69 58 79)(49 68 59 78)(50 67 60 77)(101 159 111 149)(102 158 112 148)(103 157 113 147)(104 156 114 146)(105 155 115 145)(106 154 116 144)(107 153 117 143)(108 152 118 142)(109 151 119 141)(110 150 120 160)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P5A5B10A···10N10O···10AD20A···20H
order12···2222244444···444445510···1010···1020···20
size11···14444222210···1020202020222···24···44···4

68 irreducible representations

dim11111122222224
type+++++++++++-
imageC1C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D4D42D5
kernelC2×C20.17D4C2×C4×Dic5C20.17D4C2×C23.D5C22×Dic10D4×C2×C10C2×C20C22×D4C2×C10C22×C4C2×D4C24C2×C4C22
# reps118411428284168

Matrix representation of C2×C20.17D4 in GL5(𝔽41)

400000
01000
00100
000400
000040
,
400000
00100
040000
000230
0001825
,
10000
09000
00900
0002339
0001918
,
10000
09000
003200
0002339
0001818

G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,23,18,0,0,0,0,25],[1,0,0,0,0,0,9,0,0,0,0,0,9,0,0,0,0,0,23,19,0,0,0,39,18],[1,0,0,0,0,0,9,0,0,0,0,0,32,0,0,0,0,0,23,18,0,0,0,39,18] >;

C2×C20.17D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{17}D_4
% in TeX

G:=Group("C2xC20.17D4");
// GroupNames label

G:=SmallGroup(320,1469);
// by ID

G=gap.SmallGroup(320,1469);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,100,1571,185,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^9,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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×
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