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## G = C2×C8.D10order 320 = 26·5

### Direct product of C2 and C8.D10

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×C8.D10
 Chief series C1 — C5 — C10 — C20 — D20 — C2×D20 — C2×C4○D20 — C2×C8.D10
 Lower central C5 — C10 — C20 — C2×C8.D10
 Upper central C1 — C22 — C22×C4 — C2×M4(2)

Generators and relations for C2×C8.D10
G = < a,b,c,d | a2=b8=1, c10=d2=b4, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd-1=b-1, dcd-1=c9 >

Subgroups: 958 in 258 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, D5, C10, C10, C10, C2×C8, M4(2), SD16, Q16, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C2×C10, C2×M4(2), C2×SD16, C2×Q16, C8.C22, C22×Q8, C2×C4○D4, C40, Dic10, Dic10, C4×D5, D20, D20, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C22×D5, C22×C10, C2×C8.C22, C40⋊C2, Dic20, C2×C40, C5×M4(2), C2×Dic10, C2×Dic10, C2×Dic10, C2×C4×D5, C2×D20, C4○D20, C4○D20, C22×Dic5, C2×C5⋊D4, C22×C20, C2×C40⋊C2, C2×Dic20, C8.D10, C10×M4(2), C22×Dic10, C2×C4○D20, C2×C8.D10
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C24, D10, C8.C22, C22×D4, D20, C22×D5, C2×C8.C22, C2×D20, C23×D5, C8.D10, C22×D20, C2×C8.D10

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

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4J 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I 20J 20K 20L 40A ··· 40P order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 40 ··· 40 size 1 1 1 1 2 2 20 20 2 2 2 2 20 ··· 20 2 2 4 4 4 4 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

62 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + + + + - - image C1 C2 C2 C2 C2 C2 C2 D4 D4 D5 D10 D10 D10 D20 D20 C8.C22 C8.D10 kernel C2×C8.D10 C2×C40⋊C2 C2×Dic20 C8.D10 C10×M4(2) C22×Dic10 C2×C4○D20 C2×C20 C22×C10 C2×M4(2) C2×C8 M4(2) C22×C4 C2×C4 C23 C10 C2 # reps 1 2 2 8 1 1 1 3 1 2 4 8 2 12 4 2 8

Matrix representation of C2×C8.D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 2 28 0 0 0 0 13 39 0 0
,
 0 34 0 0 0 0 6 6 0 0 0 0 0 0 6 35 34 8 0 0 6 1 33 27 0 0 33 7 35 6 0 0 34 32 35 40
,
 1 0 0 0 0 0 5 40 0 0 0 0 0 0 0 1 35 32 0 0 1 0 27 6 0 0 35 32 28 2 0 0 27 6 39 13

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,2,13,0,0,0,0,28,39,0,0,1,0,0,0,0,0,0,1,0,0],[0,6,0,0,0,0,34,6,0,0,0,0,0,0,6,6,33,34,0,0,35,1,7,32,0,0,34,33,35,35,0,0,8,27,6,40],[1,5,0,0,0,0,0,40,0,0,0,0,0,0,0,1,35,27,0,0,1,0,32,6,0,0,35,27,28,39,0,0,32,6,2,13] >;

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(320,1419);
// by ID

G=gap.SmallGroup(320,1419);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,297,80,1684,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=1,c^10=d^2=b^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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