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G = D4.2Dic10order 320 = 26·5

2nd non-split extension by D4 of Dic10 acting via Dic10/Dic5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4.2Dic10, C4⋊C4.7D10, C405C47C2, (C2×C8).7D10, C52(D4.Q8), (C5×D4).2Q8, C20.4(C2×Q8), D4⋊C4.3D5, (C2×C40).7C22, C20.Q83C2, C4.Dic103C2, C20.8Q85C2, (D4×Dic5).6C2, C4.4(C2×Dic10), (C2×D4).129D10, C10.21(C4○D8), C2.6(D83D5), C2.9(D40⋊C2), D4⋊Dic5.5C2, C22.168(D4×D5), C20.148(C4○D4), C4.77(D42D5), C10.54(C8⋊C22), (C2×C20).206C23, (C2×Dic5).194D4, (D4×C10).27C22, C10.10(C22⋊Q8), C4⋊Dic5.65C22, (C4×Dic5).16C22, C2.15(Dic5.14D4), (C5×D4⋊C4).3C2, (C2×C10).219(C2×D4), (C5×C4⋊C4).11C22, (C2×C52C8).12C22, (C2×C4).313(C22×D5), SmallGroup(320,393)

Series: Derived Chief Lower central Upper central

C1C2×C20 — D4.2Dic10
C1C5C10C2×C10C2×C20C4×Dic5D4×Dic5 — D4.2Dic10
C5C10C2×C20 — D4.2Dic10
C1C22C2×C4D4⋊C4

Generators and relations for D4.2Dic10
 G = < a,b,c,d | a4=b2=c20=1, d2=a2c10, bab=cac-1=a-1, ad=da, cbc-1=a-1b, bd=db, dcd-1=c-1 >

Subgroups: 374 in 102 conjugacy classes, 41 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×7], D4 [×2], D4, C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8, C2×C8, C22×C4, C2×D4, Dic5 [×4], C20 [×2], C20, C2×C10, C2×C10 [×4], D4⋊C4, D4⋊C4, C4⋊C8, C4.Q8, C2.D8, C4×D4, C42.C2, C52C8, C40, C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C22×C10, D4.Q8, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], C4⋊Dic5, C23.D5, C5×C4⋊C4, C2×C40, C22×Dic5, D4×C10, C20.Q8, C20.8Q8, C405C4, D4⋊Dic5, C5×D4⋊C4, C4.Dic10, D4×Dic5, D4.2Dic10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], Q8 [×2], C23, D5, C2×D4, C2×Q8, C4○D4, D10 [×3], C22⋊Q8, C4○D8, C8⋊C22, Dic10 [×2], C22×D5, D4.Q8, C2×Dic10, D4×D5, D42D5, Dic5.14D4, D83D5, D40⋊C2, D4.2Dic10

Smallest permutation representation of D4.2Dic10
On 160 points
Generators in S160
(1 32 143 83)(2 84 144 33)(3 34 145 85)(4 86 146 35)(5 36 147 87)(6 88 148 37)(7 38 149 89)(8 90 150 39)(9 40 151 91)(10 92 152 21)(11 22 153 93)(12 94 154 23)(13 24 155 95)(14 96 156 25)(15 26 157 97)(16 98 158 27)(17 28 159 99)(18 100 160 29)(19 30 141 81)(20 82 142 31)(41 104 63 132)(42 133 64 105)(43 106 65 134)(44 135 66 107)(45 108 67 136)(46 137 68 109)(47 110 69 138)(48 139 70 111)(49 112 71 140)(50 121 72 113)(51 114 73 122)(52 123 74 115)(53 116 75 124)(54 125 76 117)(55 118 77 126)(56 127 78 119)(57 120 79 128)(58 129 80 101)(59 102 61 130)(60 131 62 103)
(1 22)(2 154)(3 24)(4 156)(5 26)(6 158)(7 28)(8 160)(9 30)(10 142)(11 32)(12 144)(13 34)(14 146)(15 36)(16 148)(17 38)(18 150)(19 40)(20 152)(21 31)(23 33)(25 35)(27 37)(29 39)(41 122)(42 52)(43 124)(44 54)(45 126)(46 56)(47 128)(48 58)(49 130)(50 60)(51 132)(53 134)(55 136)(57 138)(59 140)(61 112)(62 72)(63 114)(64 74)(65 116)(66 76)(67 118)(68 78)(69 120)(70 80)(71 102)(73 104)(75 106)(77 108)(79 110)(81 151)(82 92)(83 153)(84 94)(85 155)(86 96)(87 157)(88 98)(89 159)(90 100)(91 141)(93 143)(95 145)(97 147)(99 149)(101 139)(103 121)(105 123)(107 125)(109 127)(111 129)(113 131)(115 133)(117 135)(119 137)
(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)
(1 116 153 134)(2 115 154 133)(3 114 155 132)(4 113 156 131)(5 112 157 130)(6 111 158 129)(7 110 159 128)(8 109 160 127)(9 108 141 126)(10 107 142 125)(11 106 143 124)(12 105 144 123)(13 104 145 122)(14 103 146 121)(15 102 147 140)(16 101 148 139)(17 120 149 138)(18 119 150 137)(19 118 151 136)(20 117 152 135)(21 66 82 54)(22 65 83 53)(23 64 84 52)(24 63 85 51)(25 62 86 50)(26 61 87 49)(27 80 88 48)(28 79 89 47)(29 78 90 46)(30 77 91 45)(31 76 92 44)(32 75 93 43)(33 74 94 42)(34 73 95 41)(35 72 96 60)(36 71 97 59)(37 70 98 58)(38 69 99 57)(39 68 100 56)(40 67 81 55)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222244444444455888810···1010101010202020202020202040···40
size111144228101020202040224420202···28888444488884···4

47 irreducible representations

dim1111111122222222244444
type+++++++++-++++-+-+-+
imageC1C2C2C2C2C2C2C2D4Q8D5C4○D4D10D10D10C4○D8Dic10C8⋊C22D42D5D4×D5D83D5D40⋊C2
kernelD4.2Dic10C20.Q8C20.8Q8C405C4D4⋊Dic5C5×D4⋊C4C4.Dic10D4×Dic5C2×Dic5C5×D4D4⋊C4C20C4⋊C4C2×C8C2×D4C10D4C10C4C22C2C2
# reps1111111122222224812244

Matrix representation of D4.2Dic10 in GL6(𝔽41)

4000000
0400000
000100
0040000
000010
000001
,
100000
0400000
0004000
0040000
000010
000001
,
0370000
3100000
00121200
00122900
0000140
0000366
,
3200000
090000
0032000
0003200
00003540
0000356

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,31,0,0,0,0,37,0,0,0,0,0,0,0,12,12,0,0,0,0,12,29,0,0,0,0,0,0,1,36,0,0,0,0,40,6],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,35,35,0,0,0,0,40,6] >;

D4.2Dic10 in GAP, Magma, Sage, TeX

D_4._2{\rm Dic}_{10}
% in TeX

G:=Group("D4.2Dic10");
// GroupNames label

G:=SmallGroup(320,393);
// by ID

G=gap.SmallGroup(320,393);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,926,219,226,851,438,102,12550]);
// Polycyclic

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

׿
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