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G = (C2×C20).56D4order 320 = 26·5

30th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).56D4, (C2×C4).45D20, (C22×D5).5Q8, C22.51(Q8×D5), (C2×Dic5).70D4, C22.250(D4×D5), C10.63(C4⋊D4), C2.26(C4⋊D20), C2.11(C207D4), C53(C23.Q8), C2.9(D103Q8), C22.130(C2×D20), (C22×C4).106D10, C10.51(C22⋊Q8), C2.23(D10⋊Q8), C2.17(D102Q8), (C22×C20).69C22, (C23×D5).23C22, C23.383(C22×D5), C10.10C4221C2, C10.30(C422C2), C2.13(Dic5⋊D4), C22.111(C4○D20), (C22×C10).358C23, C22.54(Q82D5), C22.105(D42D5), (C22×Dic5).63C22, (C2×C4⋊C4)⋊12D5, (C10×C4⋊C4)⋊23C2, (C2×C4⋊Dic5)⋊14C2, (C2×C10).86(C2×Q8), (C2×C10).338(C2×D4), (C2×C4).44(C5⋊D4), C2.15(C4⋊C4⋊D5), (C2×C10).87(C4○D4), (C2×C10.D4)⋊42C2, C22.143(C2×C5⋊D4), (C2×D10⋊C4).26C2, SmallGroup(320,621)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).56D4
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — (C2×C20).56D4
C5C22×C10 — (C2×C20).56D4
C1C23C2×C4⋊C4

Generators and relations for (C2×C20).56D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b-1, dbd=ab-1, dcd=ac-1 >

Subgroups: 774 in 186 conjugacy classes, 63 normal (51 characteristic)
C1, C2 [×7], C2 [×2], C4 [×9], C22 [×7], C22 [×10], C5, C2×C4 [×4], C2×C4 [×17], C23, C23 [×8], D5 [×2], C10 [×7], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×3], C22×C4 [×3], C24, Dic5 [×4], C20 [×5], D10 [×10], C2×C10 [×7], C2.C42, C2×C22⋊C4 [×3], C2×C4⋊C4, C2×C4⋊C4 [×2], C2×Dic5 [×2], C2×Dic5 [×8], C2×C20 [×4], C2×C20 [×7], C22×D5 [×2], C22×D5 [×6], C22×C10, C23.Q8, C10.D4 [×2], C4⋊Dic5 [×2], D10⋊C4 [×6], C5×C4⋊C4 [×2], C22×Dic5 [×3], C22×C20 [×3], C23×D5, C10.10C42, C2×C10.D4, C2×C4⋊Dic5, C2×D10⋊C4 [×3], C10×C4⋊C4, (C2×C20).56D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C4⋊D4 [×3], C22⋊Q8 [×3], C422C2, D20 [×2], C5⋊D4 [×2], C22×D5, C23.Q8, C2×D20, C4○D20, D4×D5, D42D5, Q8×D5, Q82D5, C2×C5⋊D4, C4⋊D20, D10⋊Q8, D102Q8, C4⋊C4⋊D5, C207D4, Dic5⋊D4, D103Q8, (C2×C20).56D4

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

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

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

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

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

dim1111112222222224444
type++++++++-++++--+
imageC1C2C2C2C2C2D4D4Q8D5C4○D4D10D20C5⋊D4C4○D20D4×D5D42D5Q8×D5Q82D5
kernel(C2×C20).56D4C10.10C42C2×C10.D4C2×C4⋊Dic5C2×D10⋊C4C10×C4⋊C4C2×Dic5C2×C20C22×D5C2×C4⋊C4C2×C10C22×C4C2×C4C2×C4C22C22C22C22C22
# reps1111312422668882222

Matrix representation of (C2×C20).56D4 in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
35390000
3960000
0014200
00253000
000090
0000032
,
010000
4000000
0031800
0043800
0000040
0000400
,
100000
010000
0034100
0034700
000010
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,39,0,0,0,0,39,6,0,0,0,0,0,0,14,25,0,0,0,0,2,30,0,0,0,0,0,0,9,0,0,0,0,0,0,32],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,18,38,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C20).56D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{56}D_4
% in TeX

G:=Group("(C2xC20).56D4");
// GroupNames label

G:=SmallGroup(320,621);
// by ID

G=gap.SmallGroup(320,621);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,387,184,12550]);
// 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=b^-1,d*b*d=a*b^-1,d*c*d=a*c^-1>;
// generators/relations

׿
×
𝔽