direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C2×C20.17D4, C24.37D10, C20.251(C2×D4), (C2×C20).209D4, (C2×D4).229D10, C10⋊3(C4.4D4), (C22×D4).11D5, (C2×C20).540C23, (C2×C10).292C24, (C4×Dic5)⋊67C22, C10.140(C22×D4), (C22×C4).378D10, C23.D5⋊58C22, (C2×Dic10)⋊67C22, (C22×Dic10)⋊20C2, (D4×C10).269C22, (C23×C10).74C22, C23.134(C22×D5), C22.306(C23×D5), C22.78(D4⋊2D5), (C22×C10).228C23, (C22×C20).273C22, (C2×Dic5).292C23, (C22×Dic5).254C22, (D4×C2×C10).8C2, C5⋊4(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×D4⋊2D5), (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
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, D4⋊2D5 [×4], C2×C5⋊D4 [×6], C23×D5, C20.17D4 [×4], C2×D4⋊2D5 [×2], C22×C5⋊D4, C2×C20.17D4
(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 | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | 4N | 4O | 4P | 5A | 5B | 10A | ··· | 10N | 10O | ··· | 10AD | 20A | ··· | 20H |
order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 10 | ··· | 10 | 20 | 20 | 20 | 20 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
68 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
type | + | + | + | + | + | + | + | + | + | + | + | - | ||
image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D5 | C4○D4 | D10 | D10 | D10 | C5⋊D4 | D4⋊2D5 |
kernel | C2×C20.17D4 | C2×C4×Dic5 | C20.17D4 | C2×C23.D5 | C22×Dic10 | D4×C2×C10 | C2×C20 | C22×D4 | C2×C10 | C22×C4 | C2×D4 | C24 | C2×C4 | C22 |
# reps | 1 | 1 | 8 | 4 | 1 | 1 | 4 | 2 | 8 | 2 | 8 | 4 | 16 | 8 |
Matrix representation of C2×C20.17D4 ►in GL5(𝔽41)
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 | 1 | 0 | 0 |
0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 23 | 0 |
0 | 0 | 0 | 18 | 25 |
1 | 0 | 0 | 0 | 0 |
0 | 9 | 0 | 0 | 0 |
0 | 0 | 9 | 0 | 0 |
0 | 0 | 0 | 23 | 39 |
0 | 0 | 0 | 19 | 18 |
1 | 0 | 0 | 0 | 0 |
0 | 9 | 0 | 0 | 0 |
0 | 0 | 32 | 0 | 0 |
0 | 0 | 0 | 23 | 39 |
0 | 0 | 0 | 18 | 18 |
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