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G = C88D20order 320 = 26·5

2nd semidirect product of C8 and D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C88D20, C4013D4, D103SD16, C4.Q88D5, C53(C88D4), C4⋊C4.38D10, C4.50(C2×D20), D102Q86C2, (C2×C8).260D10, C4⋊D20.6C2, C20.130(C2×D4), D206C415C2, C2.24(D5×SD16), C10.55(C4○D8), C20.29(C4○D4), C10.Q1616C2, C4.3(Q82D5), C10.40(C2×SD16), (C22×D5).84D4, C22.216(D4×D5), C10.43(C4⋊D4), C2.16(C4⋊D20), (C2×C40).161C22, (C2×C20).280C23, (C2×Dic5).144D4, (C2×D20).78C22, C2.22(SD163D5), (C2×Dic10).87C22, (D5×C2×C8)⋊7C2, (C5×C4.Q8)⋊9C2, (C2×C40⋊C2)⋊28C2, (C2×C10).285(C2×D4), (C5×C4⋊C4).73C22, (C2×C4×D5).303C22, (C2×C4).383(C22×D5), (C2×C52C8).236C22, SmallGroup(320,491)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C88D20
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C88D20
C5C10C2×C20 — C88D20
C1C22C2×C4C4.Q8

Generators and relations for C88D20
 G = < a,b,c | a8=b20=c2=1, bab-1=cac=a3, cbc=b-1 >

Subgroups: 598 in 124 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, Dic5, C20, C20, D10, D10, C2×C10, D4⋊C4, Q8⋊C4, C4.Q8, C4⋊D4, C22⋊Q8, C22×C8, C2×SD16, C52C8, C40, Dic10, C4×D5, D20, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C88D4, C8×D5, C40⋊C2, C2×C52C8, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C40, C2×Dic10, C2×C4×D5, C2×D20, C2×D20, D206C4, C10.Q16, C5×C4.Q8, C4⋊D20, D102Q8, D5×C2×C8, C2×C40⋊C2, C88D20
Quotients: C1, C2, C22, D4, C23, D5, SD16, C2×D4, C4○D4, D10, C4⋊D4, C2×SD16, C4○D8, D20, C22×D5, C88D4, C2×D20, D4×D5, Q82D5, C4⋊D20, D5×SD16, SD163D5, C88D20

Smallest permutation representation of C88D20
On 160 points
Generators in S160
(1 104 49 149 62 140 39 91)(2 150 40 105 63 92 50 121)(3 106 51 151 64 122 21 93)(4 152 22 107 65 94 52 123)(5 108 53 153 66 124 23 95)(6 154 24 109 67 96 54 125)(7 110 55 155 68 126 25 97)(8 156 26 111 69 98 56 127)(9 112 57 157 70 128 27 99)(10 158 28 113 71 100 58 129)(11 114 59 159 72 130 29 81)(12 160 30 115 73 82 60 131)(13 116 41 141 74 132 31 83)(14 142 32 117 75 84 42 133)(15 118 43 143 76 134 33 85)(16 144 34 119 77 86 44 135)(17 120 45 145 78 136 35 87)(18 146 36 101 79 88 46 137)(19 102 47 147 80 138 37 89)(20 148 38 103 61 90 48 139)
(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 15)(2 14)(3 13)(4 12)(5 11)(6 10)(7 9)(16 20)(17 19)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 77)(62 76)(63 75)(64 74)(65 73)(66 72)(67 71)(68 70)(78 80)(81 124)(82 123)(83 122)(84 121)(85 140)(86 139)(87 138)(88 137)(89 136)(90 135)(91 134)(92 133)(93 132)(94 131)(95 130)(96 129)(97 128)(98 127)(99 126)(100 125)(101 146)(102 145)(103 144)(104 143)(105 142)(106 141)(107 160)(108 159)(109 158)(110 157)(111 156)(112 155)(113 154)(114 153)(115 152)(116 151)(117 150)(118 149)(119 148)(120 147)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222224444444558888888810···102020202020···2040···40
size11111010402288101040222222101010102···244448···84···4

50 irreducible representations

dim1111111122222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2D4D4D4D5C4○D4SD16D10D10C4○D8D20Q82D5D4×D5D5×SD16SD163D5
kernelC88D20D206C4C10.Q16C5×C4.Q8C4⋊D20D102Q8D5×C2×C8C2×C40⋊C2C40C2×Dic5C22×D5C4.Q8C20D10C4⋊C4C2×C8C10C8C4C22C2C2
# reps1111111121122442482244

Matrix representation of C88D20 in GL6(𝔽41)

100000
010000
001000
000100
0000030
00002630
,
010000
40340000
001500
00164000
000010
0000140
,
010000
100000
0040000
0025100
000010
0000140

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,26,0,0,0,0,30,30],[0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,16,0,0,0,0,5,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,40,25,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

C88D20 in GAP, Magma, Sage, TeX

C_8\rtimes_8D_{20}
% in TeX

G:=Group("C8:8D20");
// GroupNames label

G:=SmallGroup(320,491);
// by ID

G=gap.SmallGroup(320,491);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,254,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^8=b^20=c^2=1,b*a*b^-1=c*a*c=a^3,c*b*c=b^-1>;
// generators/relations

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