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G = (C2×C20).289D4  order 320 = 26·5

263rd non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).289D4, (C22×D5).4Q8, C22.50(Q8×D5), C2.7(C20⋊D4), (C2×Dic5).69D4, C22.248(D4×D5), (C22×C4).44D10, C10.16(C4⋊1D4), C5⋊3(C23.4Q8), C2.8(D10⋊3Q8), C10.50(C22⋊Q8), C2.22(D10⋊Q8), (C22×C20).31C22, (C23×D5).21C22, C23.381(C22×D5), C10.10C42⋊43C2, C22.109(C4○D20), (C22×C10).356C23, C22.52(Q8⋊2D5), C2.21(D10.13D4), C10.54(C22.D4), (C22×Dic5).61C22, C2.14(C23.23D10), (C2×C4⋊C4)⋊10D5, (C10×C4⋊C4)⋊26C2, (C2×C10).85(C2×Q8), (C2×C10).453(C2×D4), (C2×C4).42(C5⋊D4), (C2×C10.D4)⋊14C2, C22.141(C2×C5⋊D4), (C2×D10⋊C4).16C2, (C2×C10).191(C4○D4), SmallGroup(320,619)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — (C2×C20).289D4
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C2×C20).289D4
Lower central
C5 — C22×C10 — (C2×C20).289D4
Upper central
C1 — C23 — C2×C4⋊C4

Generators and relations for (C2×C20).289D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=ab-1, dbd=ab9, dcd=ab10c-1 >

Subgroups: 774 in 186 conjugacy classes, 61 normal (25 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23.4Q8, C10.D4, D10⋊C4, C5×C4⋊C4, C22×Dic5, C22×Dic5, C22×C20, C22×C20, C23×D5, C10.10C42, C2×C10.D4, C2×D10⋊C4, C2×D10⋊C4, C10×C4⋊C4, (C2×C20).289D4
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C22⋊Q8, C22.D4, C4⋊1D4, C5⋊D4, C22×D5, C23.4Q8, C4○D20, D4×D5, Q8×D5, Q8⋊2D5, C2×C5⋊D4, D10.13D4, D10⋊Q8, C23.23D10, C20⋊D4, D10⋊3Q8, (C2×C20).289D4

Smallest permutation representation of (C2×C20).289D4
►On 160 points
Generators in S160
(1 45)(2 46)(3 47)(4 48)(5 49)(6 50)(7 51)(8 52)(9 53)(10 54)(11 55)(12 56)(13 57)(14 58)(15 59)(16 60)(17 41)(18 42)(19 43)(20 44)(21 158)(22 159)(23 160)(24 141)(25 142)(26 143)(27 144)(28 145)(29 146)(30 147)(31 148)(32 149)(33 150)(34 151)(35 152)(36 153)(37 154)(38 155)(39 156)(40 157)(61 134)(62 135)(63 136)(64 137)(65 138)(66 139)(67 140)(68 121)(69 122)(70 123)(71 124)(72 125)(73 126)(74 127)(75 128)(76 129)(77 130)(78 131)(79 132)(80 133)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 101)(97 102)(98 103)(99 104)(100 105)

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim1111122222222444
type+++++++-+++-+
imageC1C2C2C2C2D4D4Q8D5C4○D4D10C5⋊D4C4○D20D4×D5Q8×D5Q8⋊2D5
kernel(C2×C20).289D4C10.10C42C2×C10.D4C2×D10⋊C4C10×C4⋊C4C2×Dic5C2×C20C22×D5C2×C4⋊C4C2×C10C22×C4C2×C4C22C22C22C22
# reps11231422266816422

Matrix representation of (C2×C20).289D4 ►in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
21110000
12200000
0004000
001600
000098
00003132
,
40160000
510000
00182000
00352300
0000110
0000040
,
100000
010000
001600
0004000
000010
0000840

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[21,12,0,0,0,0,11,20,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,9,31,0,0,0,0,8,32],[40,5,0,0,0,0,16,1,0,0,0,0,0,0,18,35,0,0,0,0,20,23,0,0,0,0,0,0,1,0,0,0,0,0,10,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,8,0,0,0,0,0,40] >;
 

(C2×C20).289D4 in GAP, Magma, Sage, TeX 

(C_2\times C_{20})._{289}D_4
 
% in TeX
 
G:=Group("(C2xC20).289D4");
 
// GroupNames label
 
G:=SmallGroup(320,619);
 
// by ID
 
G=gap.SmallGroup(320,619);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,254,387,268,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^-1,d*b*d=a*b^9,d*c*d=a*b^10*c^-1>;
 
// generators/relations
 

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