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G = Dic10.16D4order 320 = 26·5

16th non-split extension by Dic10 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.16D4, C4.65(D4×D5), (C5×D4).10D4, D103Q85C2, (C2×SD16)⋊14D5, (C2×C8).149D10, C20.177(C2×D4), D4.9(C5⋊D4), C55(D4.7D4), (C2×Q8).55D10, D101C835C2, C10.60C22≀C2, (C10×SD16)⋊23C2, (C2×D4).147D10, C10.65(C4○D8), D4⋊Dic535C2, (C22×D5).46D4, C22.270(D4×D5), C20.44D436C2, (C2×C40).296C22, (C2×C20).450C23, (C2×Dic5).241D4, (D4×C10).99C22, (Q8×C10).79C22, C2.28(C23⋊D10), C2.30(SD16⋊D5), C10.50(C8.C22), C4⋊Dic5.177C22, C2.31(SD163D5), (C2×Dic10).132C22, C4.45(C2×C5⋊D4), (C2×C5⋊Q16)⋊19C2, (C2×C4×D5).54C22, (C2×D42D5).6C2, (C2×C10).362(C2×D4), (C2×C4).539(C22×D5), (C2×C52C8).159C22, SmallGroup(320,800)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.16D4
C1C5C10C2×C10C2×C20C2×C4×D5C2×D42D5 — Dic10.16D4
C5C10C2×C20 — Dic10.16D4
C1C22C2×C4C2×SD16

Generators and relations for Dic10.16D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, dad=a9, cbc-1=a5b, dbd=a10b, dcd=a10c-1 >

Subgroups: 606 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 [×2], D4 [×5], Q8 [×5], C23 [×2], D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4 [×2], C2×C8, C2×C8, SD16 [×2], Q16 [×2], C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4 [×4], Dic5 [×4], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×4], C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10 [×2], Dic10, C4×D5 [×2], C2×Dic5, C2×Dic5 [×6], C5⋊D4 [×4], C2×C20, C2×C20, C5×D4 [×2], C5×D4, C5×Q8 [×2], C22×D5, C22×C10, D4.7D4, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5⋊Q16 [×2], C2×C40, C5×SD16 [×2], C2×Dic10, C2×C4×D5, D42D5 [×4], C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D101C8, D4⋊Dic5, C2×C5⋊Q16, D103Q8, C10×SD16, C2×D42D5, Dic10.16D4
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, SD16⋊D5, SD163D5, C23⋊D10, Dic10.16D4

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

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H5A5B8A8B8C8D10A···10F10G10H10I10J20A20B20C20D20E20F20G20H40A···40H
order12222224444444455888810···1010101010202020202020202040···40
size111144202281010202040224420202···28888444488884···4

47 irreducible representations

dim11111111222222222244444
type++++++++++++++++-++-
imageC1C2C2C2C2C2C2C2D4D4D4D4D5D10D10D10C4○D8C5⋊D4C8.C22D4×D5D4×D5SD16⋊D5SD163D5
kernelDic10.16D4C20.44D4D101C8D4⋊Dic5C2×C5⋊Q16D103Q8C10×SD16C2×D42D5Dic10C2×Dic5C5×D4C22×D5C2×SD16C2×C8C2×D4C2×Q8C10D4C10C4C22C2C2
# reps11111111212122224812244

Matrix representation of Dic10.16D4 in GL6(𝔽41)

4010000
3370000
0040000
0004000
0000320
000009
,
3410000
3470000
001000
00404000
0000027
000030
,
3410000
3470000
0025900
00171600
000009
0000320
,
3410000
3470000
0040000
001100
000010
0000040

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,0,0,3,0,0,0,0,27,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,25,17,0,0,0,0,9,16,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

Dic10.16D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{16}D_4
% in TeX

G:=Group("Dic10.16D4");
// GroupNames label

G:=SmallGroup(320,800);
// by ID

G=gap.SmallGroup(320,800);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,184,851,438,102,12550]);
// 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,d*a*d=a^9,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=a^10*c^-1>;
// generators/relations

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