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G = Q8.D20order 320 = 26·5

2nd non-split extension by Q8 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8.7D20, D20.10D4, C4.94(D4×D5), C4.7(C2×D20), (C5×Q8).2D4, C4⋊C4.26D10, Q8⋊C412D5, D102Q85C2, (C2×C8).124D10, C20.123(C2×D4), C53(D4.7D4), D101C814C2, D206C411C2, C10.25C22≀C2, C10.48(C4○D8), (C2×Q8).109D10, (C22×D5).25D4, C22.200(D4×D5), (C2×C40).135C22, (C2×C20).250C23, (C2×Dic5).210D4, (C2×D20).68C22, (Q8×C10).33C22, C2.28(C22⋊D20), C2.15(Q16⋊D5), C10.61(C8.C22), C2.17(SD163D5), (C2×Dic10).76C22, (C2×C5⋊Q16)⋊4C2, (C2×C40⋊C2)⋊18C2, (C2×C4×D5).27C22, (C5×Q8⋊C4)⋊12C2, (C2×C10).263(C2×D4), (C5×C4⋊C4).51C22, (C2×Q82D5).4C2, (C2×C52C8).41C22, (C2×C4).357(C22×D5), SmallGroup(320,437)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Q8.D20
C1C5C10C20C2×C20C2×C4×D5D102Q8 — Q8.D20
C5C10C2×C20 — Q8.D20
C1C22C2×C4Q8⋊C4

Generators and relations for Q8.D20
 G = < a,b,c,d | a4=c20=1, b2=d2=a2, bab-1=cac-1=dad-1=a-1, cbc-1=a-1b, dbd-1=ab, dcd-1=a2c-1 >

Subgroups: 702 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×10], D4 [×7], Q8 [×2], Q8 [×3], C23 [×2], D5 [×3], C10 [×3], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8, C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×3], D10 [×7], C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10 [×2], C4×D5 [×6], D20 [×2], D20 [×5], C2×Dic5, C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, C22×D5, D4.7D4, C40⋊C2 [×2], C2×C52C8, C4⋊Dic5, D10⋊C4, C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q82D5 [×4], Q8×C10, D206C4, D101C8, C5×Q8⋊C4, D102Q8, C2×C40⋊C2, C2×C5⋊Q16, C2×Q82D5, Q8.D20
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8.C22, D20 [×2], C22×D5, D4.7D4, C2×D20, D4×D5 [×2], C22⋊D20, SD163D5, Q16⋊D5, Q8.D20

Smallest permutation representation of Q8.D20
On 160 points
Generators in S160
(1 98 119 51)(2 52 120 99)(3 100 101 53)(4 54 102 81)(5 82 103 55)(6 56 104 83)(7 84 105 57)(8 58 106 85)(9 86 107 59)(10 60 108 87)(11 88 109 41)(12 42 110 89)(13 90 111 43)(14 44 112 91)(15 92 113 45)(16 46 114 93)(17 94 115 47)(18 48 116 95)(19 96 117 49)(20 50 118 97)(21 127 146 67)(22 68 147 128)(23 129 148 69)(24 70 149 130)(25 131 150 71)(26 72 151 132)(27 133 152 73)(28 74 153 134)(29 135 154 75)(30 76 155 136)(31 137 156 77)(32 78 157 138)(33 139 158 79)(34 80 159 140)(35 121 160 61)(36 62 141 122)(37 123 142 63)(38 64 143 124)(39 125 144 65)(40 66 145 126)
(1 144 119 39)(2 66 120 126)(3 146 101 21)(4 68 102 128)(5 148 103 23)(6 70 104 130)(7 150 105 25)(8 72 106 132)(9 152 107 27)(10 74 108 134)(11 154 109 29)(12 76 110 136)(13 156 111 31)(14 78 112 138)(15 158 113 33)(16 80 114 140)(17 160 115 35)(18 62 116 122)(19 142 117 37)(20 64 118 124)(22 81 147 54)(24 83 149 56)(26 85 151 58)(28 87 153 60)(30 89 155 42)(32 91 157 44)(34 93 159 46)(36 95 141 48)(38 97 143 50)(40 99 145 52)(41 75 88 135)(43 77 90 137)(45 79 92 139)(47 61 94 121)(49 63 96 123)(51 65 98 125)(53 67 100 127)(55 69 82 129)(57 71 84 131)(59 73 86 133)
(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 28 119 153)(2 152 120 27)(3 26 101 151)(4 150 102 25)(5 24 103 149)(6 148 104 23)(7 22 105 147)(8 146 106 21)(9 40 107 145)(10 144 108 39)(11 38 109 143)(12 142 110 37)(13 36 111 141)(14 160 112 35)(15 34 113 159)(16 158 114 33)(17 32 115 157)(18 156 116 31)(19 30 117 155)(20 154 118 29)(41 64 88 124)(42 123 89 63)(43 62 90 122)(44 121 91 61)(45 80 92 140)(46 139 93 79)(47 78 94 138)(48 137 95 77)(49 76 96 136)(50 135 97 75)(51 74 98 134)(52 133 99 73)(53 72 100 132)(54 131 81 71)(55 70 82 130)(56 129 83 69)(57 68 84 128)(58 127 85 67)(59 66 86 126)(60 125 87 65)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444455888810···102020202020···2040···40
size111120202022448101040224420202···244448···84···4

47 irreducible representations

dim11111111222222222244444
type+++++++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8D20C8.C22D4×D5D4×D5SD163D5Q16⋊D5
kernelQ8.D20D206C4D101C8C5×Q8⋊C4D102Q8C2×C40⋊C2C2×C5⋊Q16C2×Q82D5D20C2×Dic5C5×Q8C22×D5Q8⋊C4C4⋊C4C2×C8C2×Q8C10Q8C10C4C22C2C2
# reps11111111212122224812244

Matrix representation of Q8.D20 in GL4(𝔽41) generated by

1000
0100
0019
001840
,
1000
0100
00940
00032
,
143900
163000
001715
003024
,
30200
221100
00012
00170
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,18,0,0,9,40],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,40,32],[14,16,0,0,39,30,0,0,0,0,17,30,0,0,15,24],[30,22,0,0,2,11,0,0,0,0,0,17,0,0,12,0] >;

Q8.D20 in GAP, Magma, Sage, TeX

Q_8.D_{20}
% in TeX

G:=Group("Q8.D20");
// GroupNames label

G:=SmallGroup(320,437);
// by ID

G=gap.SmallGroup(320,437);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,758,135,184,570,297,136,12550]);
// Polycyclic

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

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