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G = (C2×C8).D10order 320 = 26·5

19th non-split extension by C2×C8 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Q8⋊C47D5, C405C411C2, (C2×C8).19D10, C4⋊C4.153D10, Q8⋊Dic59C2, (C2×Q8).20D10, D101C8.4C2, C4.57(C4○D20), C10.71(C4○D8), (C2×C40).19C22, D103Q8.5C2, C20.Q812C2, (C22×D5).28D4, C22.204(D4×D5), C20.163(C4○D4), C4.88(D42D5), (C2×C20).254C23, (C2×Dic5).212D4, C52(C23.20D4), C4⋊Dic5.98C22, (Q8×C10).37C22, C2.10(Q8.D10), C2.18(SD16⋊D5), C10.36(C8.C22), C2.18(D10.12D4), C10.26(C22.D4), C4⋊C47D5.2C2, (C5×Q8⋊C4)⋊7C2, (C2×C4×D5).30C22, (C2×C10).267(C2×D4), (C5×C4⋊C4).55C22, (C2×C52C8).44C22, (C2×C4).361(C22×D5), SmallGroup(320,441)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C8).D10
C1C5C10C2×C10C2×C20C2×C4×D5C4⋊C47D5 — (C2×C8).D10
C5C10C2×C20 — (C2×C8).D10
C1C22C2×C4Q8⋊C4

Generators and relations for (C2×C8).D10
 G = < a,b,c,d | a2=b8=1, c10=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=ab3, dbd-1=b-1, dcd-1=b4c9 >

Subgroups: 366 in 96 conjugacy classes, 37 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×7], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C22×C4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, C4×D5 [×2], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4, C2×C40, C2×C4×D5, Q8×C10, C20.Q8, C405C4, D101C8, Q8⋊Dic5, C5×Q8⋊C4, C4⋊C47D5, D103Q8, (C2×C8).D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C4○D8, C8.C22, C22×D5, C23.20D4, C4○D20, D4×D5, D42D5, D10.12D4, SD16⋊D5, Q8.D10, (C2×C8).D10

Smallest permutation representation of (C2×C8).D10
On 160 points
Generators in S160
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(81 91)(82 92)(83 93)(84 94)(85 95)(86 96)(87 97)(88 98)(89 99)(90 100)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(1 140 92 65 103 39 44 145)(2 76 45 131 104 156 93 30)(3 122 94 67 105 21 46 147)(4 78 47 133 106 158 95 32)(5 124 96 69 107 23 48 149)(6 80 49 135 108 160 97 34)(7 126 98 71 109 25 50 151)(8 62 51 137 110 142 99 36)(9 128 100 73 111 27 52 153)(10 64 53 139 112 144 81 38)(11 130 82 75 113 29 54 155)(12 66 55 121 114 146 83 40)(13 132 84 77 115 31 56 157)(14 68 57 123 116 148 85 22)(15 134 86 79 117 33 58 159)(16 70 59 125 118 150 87 24)(17 136 88 61 119 35 60 141)(18 72 41 127 120 152 89 26)(19 138 90 63 101 37 42 143)(20 74 43 129 102 154 91 28)
(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 10 113 102)(2 101 114 9)(3 8 115 120)(4 119 116 7)(5 6 117 118)(11 20 103 112)(12 111 104 19)(13 18 105 110)(14 109 106 17)(15 16 107 108)(21 137 132 26)(22 25 133 136)(23 135 134 24)(27 131 138 40)(28 39 139 130)(29 129 140 38)(30 37 121 128)(31 127 122 36)(32 35 123 126)(33 125 124 34)(41 94 99 56)(42 55 100 93)(43 92 81 54)(44 53 82 91)(45 90 83 52)(46 51 84 89)(47 88 85 50)(48 49 86 87)(57 98 95 60)(58 59 96 97)(61 148 151 78)(62 77 152 147)(63 146 153 76)(64 75 154 145)(65 144 155 74)(66 73 156 143)(67 142 157 72)(68 71 158 141)(69 160 159 70)(79 150 149 80)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444455888810···102020202020···2040···40
size111120224481010202040224420202···244448···84···4

47 irreducible representations

dim1111111122222222244444
type++++++++++++++--+-+
imageC1C2C2C2C2C2C2C2D4D4D5C4○D4D10D10D10C4○D8C4○D20C8.C22D42D5D4×D5SD16⋊D5Q8.D10
kernel(C2×C8).D10C20.Q8C405C4D101C8Q8⋊Dic5C5×Q8⋊C4C4⋊C47D5D103Q8C2×Dic5C22×D5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10C4C10C4C22C2C2
# reps1111111111242224812244

Matrix representation of (C2×C8).D10 in GL4(𝔽41) generated by

40000
04000
00400
00040
,
22800
133900
00140
0003
,
322800
132800
00038
00140
,
283200
281300
0003
00140
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[2,13,0,0,28,39,0,0,0,0,14,0,0,0,0,3],[32,13,0,0,28,28,0,0,0,0,0,14,0,0,38,0],[28,28,0,0,32,13,0,0,0,0,0,14,0,0,3,0] >;

(C2×C8).D10 in GAP, Magma, Sage, TeX

(C_2\times C_8).D_{10}
% in TeX

G:=Group("(C2xC8).D10");
// GroupNames label

G:=SmallGroup(320,441);
// by ID

G=gap.SmallGroup(320,441);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,64,254,219,184,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^10=a,d^2=a*b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=a*b^3,d*b*d^-1=b^-1,d*c*d^-1=b^4*c^9>;
// generators/relations

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