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G = Dic1014D4order 320 = 26·5

2nd semidirect product of Dic10 and D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic1014D4, C23.40D20, C22⋊C810D5, C51(Q8⋊D4), (C2×C10)⋊2SD16, C4.123(D4×D5), (C2×C4).35D20, (C2×C20).46D4, C20.335(C2×D4), C207D4.4C2, (C2×C8).111D10, C10.9(C2×SD16), C10.11C22≀C2, C222(C40⋊C2), (C22×C4).88D10, (C22×C10).58D4, C20.44D411C2, (C2×C40).122C22, (C2×C20).748C23, (C22×Dic10)⋊2C2, (C2×D20).15C22, C22.111(C2×D20), C4⋊Dic5.15C22, C2.14(C22⋊D20), C2.14(C8.D10), C10.11(C8.C22), (C22×C20).54C22, (C2×Dic10).218C22, (C2×C40⋊C2)⋊12C2, (C5×C22⋊C8)⋊12C2, C2.12(C2×C40⋊C2), (C2×C10).131(C2×D4), (C2×C4).693(C22×D5), SmallGroup(320,365)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic1014D4
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — Dic1014D4
C5C10C2×C20 — Dic1014D4
C1C22C22×C4C22⋊C8

Generators and relations for Dic1014D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >

Subgroups: 734 in 158 conjugacy classes, 47 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, SD16, C22×C4, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C22×Q8, C40, Dic10, Dic10, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, Q8⋊D4, C40⋊C2, C4⋊Dic5, D10⋊C4, C2×C40, C2×Dic10, C2×Dic10, C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4, C5×C22⋊C8, C2×C40⋊C2, C207D4, C22×Dic10, Dic1014D4
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, D10, C22≀C2, C2×SD16, C8.C22, D20, C22×D5, Q8⋊D4, C40⋊C2, C2×D20, D4×D5, C22⋊D20, C2×C40⋊C2, C8.D10, Dic1014D4

Smallest permutation representation of Dic1014D4
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)
(1 142 11 152)(2 141 12 151)(3 160 13 150)(4 159 14 149)(5 158 15 148)(6 157 16 147)(7 156 17 146)(8 155 18 145)(9 154 19 144)(10 153 20 143)(21 88 31 98)(22 87 32 97)(23 86 33 96)(24 85 34 95)(25 84 35 94)(26 83 36 93)(27 82 37 92)(28 81 38 91)(29 100 39 90)(30 99 40 89)(41 105 51 115)(42 104 52 114)(43 103 53 113)(44 102 54 112)(45 101 55 111)(46 120 56 110)(47 119 57 109)(48 118 58 108)(49 117 59 107)(50 116 60 106)(61 125 71 135)(62 124 72 134)(63 123 73 133)(64 122 74 132)(65 121 75 131)(66 140 76 130)(67 139 77 129)(68 138 78 128)(69 137 79 127)(70 136 80 126)
(1 79 85 118)(2 78 86 117)(3 77 87 116)(4 76 88 115)(5 75 89 114)(6 74 90 113)(7 73 91 112)(8 72 92 111)(9 71 93 110)(10 70 94 109)(11 69 95 108)(12 68 96 107)(13 67 97 106)(14 66 98 105)(15 65 99 104)(16 64 100 103)(17 63 81 102)(18 62 82 101)(19 61 83 120)(20 80 84 119)(21 46 149 135)(22 45 150 134)(23 44 151 133)(24 43 152 132)(25 42 153 131)(26 41 154 130)(27 60 155 129)(28 59 156 128)(29 58 157 127)(30 57 158 126)(31 56 159 125)(32 55 160 124)(33 54 141 123)(34 53 142 122)(35 52 143 121)(36 51 144 140)(37 50 145 139)(38 49 146 138)(39 48 147 137)(40 47 148 136)
(1 79)(2 80)(3 61)(4 62)(5 63)(6 64)(7 65)(8 66)(9 67)(10 68)(11 69)(12 70)(13 71)(14 72)(15 73)(16 74)(17 75)(18 76)(19 77)(20 78)(21 45)(22 46)(23 47)(24 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 41)(38 42)(39 43)(40 44)(81 114)(82 115)(83 116)(84 117)(85 118)(86 119)(87 120)(88 101)(89 102)(90 103)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 111)(99 112)(100 113)(121 156)(122 157)(123 158)(124 159)(125 160)(126 141)(127 142)(128 143)(129 144)(130 145)(131 146)(132 147)(133 148)(134 149)(135 150)(136 151)(137 152)(138 153)(139 154)(140 155)

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

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

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

59 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222224444444455888810···101010101020···202020202040···40
size1111224022420202020402244442···244442···244444···4

59 irreducible representations

dim1111112222222222444
type++++++++++++++-+-
imageC1C2C2C2C2C2D4D4D4D5SD16D10D10D20D20C40⋊C2C8.C22D4×D5C8.D10
kernelDic1014D4C20.44D4C5×C22⋊C8C2×C40⋊C2C207D4C22×Dic10Dic10C2×C20C22×C10C22⋊C8C2×C10C2×C8C22×C4C2×C4C23C22C10C4C2
# reps12121141124424416144

Matrix representation of Dic1014D4 in GL4(𝔽41) generated by

13200
392500
00400
00040
,
153100
392600
00040
00400
,
392800
16200
0001
00400
,
1000
0100
00040
00400
G:=sub<GL(4,GF(41))| [13,39,0,0,2,25,0,0,0,0,40,0,0,0,0,40],[15,39,0,0,31,26,0,0,0,0,0,40,0,0,40,0],[39,16,0,0,28,2,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;

Dic1014D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{14}D_4
% in TeX

G:=Group("Dic10:14D4");
// GroupNames label

G:=SmallGroup(320,365);
// by ID

G=gap.SmallGroup(320,365);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,58,1123,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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