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G = (C2×C20).289D4order 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(C41D4), C53(C23.4Q8), C2.8(D103Q8), C10.50(C22⋊Q8), C2.22(D10⋊Q8), (C22×C20).31C22, (C23×D5).21C22, C23.381(C22×D5), C10.10C4243C2, C22.109(C4○D20), (C22×C10).356C23, C22.52(Q82D5), 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

C1C22×C10 — (C2×C20).289D4
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — (C2×C20).289D4
C5C22×C10 — (C2×C20).289D4
C1C23C2×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 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C24, Dic5 [×5], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.4Q8, C10.D4 [×4], D10⋊C4 [×6], C5×C4⋊C4 [×2], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42, C2×C10.D4 [×2], C2×D10⋊C4, C2×D10⋊C4 [×2], C10×C4⋊C4, (C2×C20).289D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22⋊Q8 [×3], C22.D4 [×3], C41D4, C5⋊D4 [×2], C22×D5, C23.4Q8, C4○D20 [×2], D4×D5 [×2], Q8×D5, Q82D5, C2×C5⋊D4, D10.13D4 [×2], D10⋊Q8 [×2], C23.23D10, C20⋊D4, D103Q8, (C2×C20).289D4

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

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

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

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

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×D5Q82D5
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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