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G = D20.44D4order 320 = 26·5

14th non-split extension by D20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D20.44D4, C40.7C23, Q16.6D10, C20.26C24, SD16.1D10, Dic10.44D4, D20.19C23, M4(2).18D10, Dic20.1C22, Dic10.19C23, C54(Q8○D8), (D5×Q16)⋊2C2, C5⋊D4.7D4, D4⋊D5.C22, C4.118(D4×D5), C40⋊C2.C22, C8⋊D5.C22, C8.C225D5, C8.7(C22×D5), Q16⋊D54C2, C4○D4.15D10, D10.58(C2×D4), SD16⋊D54C2, C20.247(C2×D4), C8.D104C2, (C2×Q8).92D10, (C8×D5).2C22, C4.26(C23×D5), C22.17(D4×D5), Q8⋊D5.2C22, (C5×SD16).C22, SD163D54C2, D4.8D106C2, D20.2C44C2, (Q8×D5).3C22, C52C8.28C23, Dic5.64(C2×D4), (C4×D5).17C23, D4.19(C22×D5), (C5×D4).19C23, D4.D5.2C22, D4.10D108C2, (C5×Q8).19C23, (C5×Q16).1C22, Q8.19(C22×D5), C5⋊Q16.3C22, (C2×C20).117C23, Q8.10D106C2, C4○D20.32C22, D42D5.3C22, C10.127(C22×D4), Q82D5.3C22, (Q8×C10).153C22, (C5×M4(2)).1C22, (C2×Dic10).207C22, C2.100(C2×D4×D5), (C2×C5⋊Q16)⋊29C2, (C2×C10).72(C2×D4), (C5×C8.C22)⋊4C2, (C5×C4○D4).28C22, (C2×C4).101(C22×D5), (C2×C52C8).182C22, SmallGroup(320,1451)

Series: Derived Chief Lower central Upper central

C1C20 — D20.44D4
C1C5C10C20C4×D5C4○D20Q8.10D10 — D20.44D4
C5C10C20 — D20.44D4
C1C2C2×C4C8.C22

Generators and relations for D20.44D4
 G = < a,b,c,d | a20=b2=1, c4=d2=a10, bab=a-1, cac-1=dad-1=a11, cbc-1=dbd-1=a10b, dcd-1=a10c3 >

Subgroups: 870 in 248 conjugacy classes, 99 normal (45 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×14], D4, D4 [×10], Q8, Q8 [×2], Q8 [×10], D5 [×3], C10, C10 [×2], C2×C8 [×3], M4(2), M4(2) [×2], D8, SD16 [×2], SD16 [×4], Q16 [×2], Q16 [×7], C2×Q8, C2×Q8 [×7], C4○D4, C4○D4 [×12], Dic5 [×2], Dic5 [×3], C20 [×2], C20 [×3], D10 [×2], D10, C2×C10, C2×C10, C8○D4, C2×Q16 [×3], C4○D8 [×3], C8.C22, C8.C22 [×5], 2- 1+4 [×2], C52C8 [×2], C40 [×2], Dic10 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×2], C4×D5 [×7], D20 [×2], D20 [×2], C2×Dic5 [×3], C5⋊D4 [×2], C5⋊D4 [×3], C2×C20, C2×C20 [×2], C5×D4, C5×D4, C5×Q8, C5×Q8 [×2], C5×Q8, Q8○D8, C8×D5 [×2], C8⋊D5 [×2], C40⋊C2 [×2], Dic20 [×2], C2×C52C8, D4⋊D5, D4.D5, Q8⋊D5, C5⋊Q16, C5⋊Q16 [×4], C5×M4(2), C5×SD16 [×2], C5×Q16 [×2], C2×Dic10, C2×Dic10, C4○D20 [×2], C4○D20 [×3], D42D5 [×2], D42D5 [×2], Q8×D5 [×4], Q8×D5, Q82D5 [×2], Q82D5, Q8×C10, C5×C4○D4, D20.2C4, C8.D10, SD16⋊D5 [×2], SD163D5 [×2], D5×Q16 [×2], Q16⋊D5 [×2], C2×C5⋊Q16, D4.8D10, C5×C8.C22, Q8.10D10, D4.10D10, D20.44D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C22×D5 [×7], Q8○D8, D4×D5 [×2], C23×D5, C2×D4×D5, D20.44D4

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

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E10A10B10C10D10E10F20A20B20C20D20E···20J40A40B40C40D
order1222222444444444455888881010101010102020202020···2040404040
size1124101020224441010202020224410102022448844448···88888

44 irreducible representations

dim1111111111112222222224448
type+++++++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4D5D10D10D10D10D10Q8○D8D4×D5D4×D5D20.44D4
kernelD20.44D4D20.2C4C8.D10SD16⋊D5SD163D5D5×Q16Q16⋊D5C2×C5⋊Q16D4.8D10C5×C8.C22Q8.10D10D4.10D10Dic10D20C5⋊D4C8.C22M4(2)SD16Q16C2×Q8C4○D4C5C4C22C1
# reps1112222111111122244222222

Matrix representation of D20.44D4 in GL8(𝔽41)

61000000
400000000
00610000
004000000
0000002912
0000001212
0000122900
0000292900
,
0403570000
400260000
003560000
00160000
0000013232
0000400329
00009901
0000932400
,
3540600000
10060000
3540610000
104000000
00009901
000032910
0000400329
0000013232
,
6135390000
400260000
6135400000
400100000
0000400329
0000013232
00003232040
0000932400

G:=sub<GL(8,GF(41))| [6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,12,29,0,0,0,0,0,0,29,29,0,0,0,0,29,12,0,0,0,0,0,0,12,12,0,0],[0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,35,2,35,1,0,0,0,0,7,6,6,6,0,0,0,0,0,0,0,0,0,40,9,9,0,0,0,0,1,0,9,32,0,0,0,0,32,32,0,40,0,0,0,0,32,9,1,0],[35,1,35,1,0,0,0,0,40,0,40,0,0,0,0,0,6,0,6,40,0,0,0,0,0,6,1,0,0,0,0,0,0,0,0,0,9,32,40,0,0,0,0,0,9,9,0,1,0,0,0,0,0,1,32,32,0,0,0,0,1,0,9,32],[6,40,6,40,0,0,0,0,1,0,1,0,0,0,0,0,35,2,35,1,0,0,0,0,39,6,40,0,0,0,0,0,0,0,0,0,40,0,32,9,0,0,0,0,0,1,32,32,0,0,0,0,32,32,0,40,0,0,0,0,9,32,40,0] >;

D20.44D4 in GAP, Magma, Sage, TeX

D_{20}._{44}D_4
% in TeX

G:=Group("D20.44D4");
// GroupNames label

G:=SmallGroup(320,1451);
// by ID

G=gap.SmallGroup(320,1451);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,184,570,185,136,438,235,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=b^2=1,c^4=d^2=a^10,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^11,c*b*c^-1=d*b*d^-1=a^10*b,d*c*d^-1=a^10*c^3>;
// generators/relations

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