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G = (C2×C8).207D10order 320 = 26·5

16th non-split extension by C2×C8 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4⋊C4.22D10, Q8⋊C41D5, (C8×Dic5)⋊22C2, (C2×C8).207D10, Q8⋊Dic52C2, (C2×Q8).14D10, C4.Dic105C2, D206C4.3C2, D205C4.8C2, C20.18(C4○D4), C10.68(C4○D8), C4.31(C4○D20), C22.194(D4×D5), C2.7(Q8.D10), C4.57(D42D5), (C2×C20).240C23, (C2×C40).195C22, (C2×Dic5).137D4, C20.23D4.6C2, (C2×D20).63C22, C4⋊Dic5.88C22, (Q8×C10).23C22, C10.29(C4.4D4), C2.16(SD163D5), C53(C42.78C22), (C4×Dic5).258C22, C2.19(Dic5.5D4), (C5×Q8⋊C4)⋊17C2, (C2×C10).253(C2×D4), (C5×C4⋊C4).41C22, (C2×C4).347(C22×D5), (C2×C52C8).225C22, SmallGroup(320,427)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C8).207D10
C1C5C10C20C2×C20C4×Dic5C4.Dic10 — (C2×C8).207D10
C5C10C2×C20 — (C2×C8).207D10
C1C22C2×C4Q8⋊C4

Generators and relations for (C2×C8).207D10
 G = < a,b,c,d | a2=b8=1, c10=a, d2=ab4, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=ab3, dcd-1=ab4c9 >

Subgroups: 414 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 [×5], D4 [×2], Q8 [×2], C23, D5, C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×3], C2×C8, C2×C8, C2×D4, C2×Q8, Dic5 [×3], C20 [×2], C20 [×2], D10 [×3], C2×C10, C4×C8, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4, C4.4D4, C42.C2, C52C8, C40, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C42.78C22, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, C4⋊Dic5, D10⋊C4 [×2], C5×C4⋊C4, C2×C40, C2×D20, Q8×C10, D206C4, C8×Dic5, D205C4, Q8⋊Dic5, C5×Q8⋊C4, C4.Dic10, C20.23D4, (C2×C8).207D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C4○D8 [×2], C22×D5, C42.78C22, C4○D20, D4×D5, D42D5, Dic5.5D4, SD163D5, Q8.D10, (C2×C8).207D10

Smallest permutation representation of (C2×C8).207D10
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)
(1 124 101 30 96 70 142 58)(2 21 143 135 97 49 102 61)(3 126 103 32 98 72 144 60)(4 23 145 137 99 51 104 63)(5 128 105 34 100 74 146 42)(6 25 147 139 81 53 106 65)(7 130 107 36 82 76 148 44)(8 27 149 121 83 55 108 67)(9 132 109 38 84 78 150 46)(10 29 151 123 85 57 110 69)(11 134 111 40 86 80 152 48)(12 31 153 125 87 59 112 71)(13 136 113 22 88 62 154 50)(14 33 155 127 89 41 114 73)(15 138 115 24 90 64 156 52)(16 35 157 129 91 43 116 75)(17 140 117 26 92 66 158 54)(18 37 159 131 93 45 118 77)(19 122 119 28 94 68 160 56)(20 39 141 133 95 47 120 79)
(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 52 86 34)(2 23 87 41)(3 50 88 32)(4 21 89 59)(5 48 90 30)(6 39 91 57)(7 46 92 28)(8 37 93 55)(9 44 94 26)(10 35 95 53)(11 42 96 24)(12 33 97 51)(13 60 98 22)(14 31 99 49)(15 58 100 40)(16 29 81 47)(17 56 82 38)(18 27 83 45)(19 54 84 36)(20 25 85 43)(61 114 125 145)(62 154 126 103)(63 112 127 143)(64 152 128 101)(65 110 129 141)(66 150 130 119)(67 108 131 159)(68 148 132 117)(69 106 133 157)(70 146 134 115)(71 104 135 155)(72 144 136 113)(73 102 137 153)(74 142 138 111)(75 120 139 151)(76 160 140 109)(77 118 121 149)(78 158 122 107)(79 116 123 147)(80 156 124 105)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222444444444558888888810···102020202020···2040···40
size11114022881010101040222222101010102···244448···84···4

50 irreducible representations

dim11111111222222224444
type+++++++++++++-++
imageC1C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C4○D8C4○D20D42D5D4×D5SD163D5Q8.D10
kernel(C2×C8).207D10D206C4C8×Dic5D205C4Q8⋊Dic5C5×Q8⋊C4C4.Dic10C20.23D4C2×Dic5Q8⋊C4C20C4⋊C4C2×C8C2×Q8C10C4C4C22C2C2
# reps11111111224222882244

Matrix representation of (C2×C8).207D10 in GL4(𝔽41) generated by

40000
04000
0010
0001
,
9000
0900
002615
002626
,
251600
253900
001212
001229
,
282800
321300
001526
002626
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,26,26,0,0,15,26],[25,25,0,0,16,39,0,0,0,0,12,12,0,0,12,29],[28,32,0,0,28,13,0,0,0,0,15,26,0,0,26,26] >;

(C2×C8).207D10 in GAP, Magma, Sage, TeX

(C_2\times C_8)._{207}D_{10}
% in TeX

G:=Group("(C2xC8).207D10");
// GroupNames label

G:=SmallGroup(320,427);
// by ID

G=gap.SmallGroup(320,427);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,701,120,1094,135,184,570,297,136,12550]);
// 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=d*b*d^-1=a*b^3,d*c*d^-1=a*b^4*c^9>;
// generators/relations

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