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G = D20.37D4order 320 = 26·5

7th non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.37D4, Dic10.36D4, C22⋊Q82D5, C4⋊C4.66D10, C4.102(D4×D5), (C2×C20).266D4, C20.153(C2×D4), C54(D4.7D4), (C2×Q8).28D10, D206C438C2, C10.48C22≀C2, C10.Q1637C2, (C22×C10).93D4, C10.100(C4○D8), C20.55D414C2, (C2×C20).366C23, (C22×C4).128D10, C23.27(C5⋊D4), (Q8×C10).46C22, C2.16(C23⋊D10), (C2×D20).251C22, C10.90(C8.C22), C2.19(D4.8D10), C2.11(C20.C23), (C22×C20).170C22, (C2×Dic10).278C22, (C2×Q8⋊D5)⋊9C2, (C2×C5⋊Q16)⋊8C2, (C5×C22⋊Q8)⋊2C2, (C2×C4○D20).10C2, (C2×C10).497(C2×D4), (C2×C4).173(C5⋊D4), (C5×C4⋊C4).113C22, (C2×C4).466(C22×D5), C22.172(C2×C5⋊D4), (C2×C52C8).115C22, SmallGroup(320,674)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D20.37D4
C1C5C10C20C2×C20C2×D20C2×C4○D20 — D20.37D4
C5C10C2×C20 — D20.37D4
C1C22C22×C4C22⋊Q8

Generators and relations for D20.37D4
 G = < a,b,c,d | a20=b2=c4=1, d2=a10, bab=a-1, cac-1=a11, ad=da, cbc-1=a15b, bd=db, dcd-1=c-1 >

Subgroups: 622 in 152 conjugacy classes, 43 normal (39 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×5], C22, C22 [×7], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×7], Q8 [×5], C23, C23, D5 [×2], C10 [×3], C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4, C22×C4, C2×D4 [×2], C2×Q8, C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×3], D10 [×4], C2×C10, C2×C10 [×3], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×D5, C22×C10, D4.7D4, C2×C52C8 [×2], Q8⋊D5 [×2], C5⋊Q16 [×2], C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C2×C5⋊D4, C22×C20, Q8×C10, D206C4, C10.Q16, C20.55D4, C2×Q8⋊D5, C2×C5⋊Q16, C5×C22⋊Q8, C2×C4○D20, D20.37D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, C2×D4 [×3], D10 [×3], C22≀C2, C4○D8, C8.C22, C5⋊D4 [×2], C22×D5, D4.7D4, D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C20.C23, D4.8D10, D20.37D4

Smallest permutation representation of D20.37D4
On 160 points
Generators in S160
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)(35 40)(36 39)(37 38)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 62)(63 80)(64 79)(65 78)(66 77)(67 76)(68 75)(69 74)(70 73)(71 72)(82 100)(83 99)(84 98)(85 97)(86 96)(87 95)(88 94)(89 93)(90 92)(101 113)(102 112)(103 111)(104 110)(105 109)(106 108)(114 120)(115 119)(116 118)(121 126)(122 125)(123 124)(127 140)(128 139)(129 138)(130 137)(131 136)(132 135)(133 134)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)
(1 99 72 145)(2 90 73 156)(3 81 74 147)(4 92 75 158)(5 83 76 149)(6 94 77 160)(7 85 78 151)(8 96 79 142)(9 87 80 153)(10 98 61 144)(11 89 62 155)(12 100 63 146)(13 91 64 157)(14 82 65 148)(15 93 66 159)(16 84 67 150)(17 95 68 141)(18 86 69 152)(19 97 70 143)(20 88 71 154)(21 118 127 54)(22 109 128 45)(23 120 129 56)(24 111 130 47)(25 102 131 58)(26 113 132 49)(27 104 133 60)(28 115 134 51)(29 106 135 42)(30 117 136 53)(31 108 137 44)(32 119 138 55)(33 110 139 46)(34 101 140 57)(35 112 121 48)(36 103 122 59)(37 114 123 50)(38 105 124 41)(39 116 125 52)(40 107 126 43)
(1 28 11 38)(2 29 12 39)(3 30 13 40)(4 31 14 21)(5 32 15 22)(6 33 16 23)(7 34 17 24)(8 35 18 25)(9 36 19 26)(10 37 20 27)(41 99 51 89)(42 100 52 90)(43 81 53 91)(44 82 54 92)(45 83 55 93)(46 84 56 94)(47 85 57 95)(48 86 58 96)(49 87 59 97)(50 88 60 98)(61 123 71 133)(62 124 72 134)(63 125 73 135)(64 126 74 136)(65 127 75 137)(66 128 76 138)(67 129 77 139)(68 130 78 140)(69 131 79 121)(70 132 80 122)(101 141 111 151)(102 142 112 152)(103 143 113 153)(104 144 114 154)(105 145 115 155)(106 146 116 156)(107 147 117 157)(108 148 118 158)(109 149 119 159)(110 150 120 160)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20P
order12222224444444455888810···101010101020···2020···20
size111142020222288202022202020202···244444···48···8

47 irreducible representations

dim11111111222222222224444
type++++++++++++++++-+
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8C5⋊D4C5⋊D4C8.C22D4×D5C20.C23D4.8D10
kernelD20.37D4D206C4C10.Q16C20.55D4C2×Q8⋊D5C2×C5⋊Q16C5×C22⋊Q8C2×C4○D20Dic10D20C2×C20C22×C10C22⋊Q8C4⋊C4C22×C4C2×Q8C10C2×C4C23C10C4C2C2
# reps11111111221122224441444

Matrix representation of D20.37D4 in GL6(𝔽41)

010000
4000000
0040100
0053500
0000400
0000040
,
010000
100000
0040000
005100
0000400
0000361
,
29290000
29120000
001000
000100
00003336
0000138
,
3200000
0320000
001000
000100
000010
0000540

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,1,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,5,0,0,0,0,0,1,0,0,0,0,0,0,40,36,0,0,0,0,0,1],[29,29,0,0,0,0,29,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,33,13,0,0,0,0,36,8],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,5,0,0,0,0,0,40] >;

D20.37D4 in GAP, Magma, Sage, TeX

D_{20}._{37}D_4
% in TeX

G:=Group("D20.37D4");
// GroupNames label

G:=SmallGroup(320,674);
// by ID

G=gap.SmallGroup(320,674);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,254,219,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=c^4=1,d^2=a^10,b*a*b=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^15*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
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