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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C8).D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C4⋊C4⋊7D5 — (C2×C8).D10
 Lower central C5 — C10 — C2×C20 — (C2×C8).D10
 Upper central C1 — C22 — C2×C4 — Q8⋊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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, Dic5, C20, C20, D10, C2×C10, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2.D8, C42⋊C2, C22⋊Q8, C52C8, C40, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×D5, C23.20D4, C2×C52C8, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, 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, C22, D4, C23, D5, C2×D4, C4○D4, D10, 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 70 36 88 105 159 46 122)(2 99 47 61 106 133 37 150)(3 72 38 90 107 141 48 124)(4 81 49 63 108 135 39 152)(5 74 40 92 109 143 50 126)(6 83 51 65 110 137 21 154)(7 76 22 94 111 145 52 128)(8 85 53 67 112 139 23 156)(9 78 24 96 113 147 54 130)(10 87 55 69 114 121 25 158)(11 80 26 98 115 149 56 132)(12 89 57 71 116 123 27 160)(13 62 28 100 117 151 58 134)(14 91 59 73 118 125 29 142)(15 64 30 82 119 153 60 136)(16 93 41 75 120 127 31 144)(17 66 32 84 101 155 42 138)(18 95 43 77 102 129 33 146)(19 68 34 86 103 157 44 140)(20 97 45 79 104 131 35 148)
(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 10 115 104)(2 103 116 9)(3 8 117 102)(4 101 118 7)(5 6 119 120)(11 20 105 114)(12 113 106 19)(13 18 107 112)(14 111 108 17)(15 16 109 110)(21 60 41 40)(22 39 42 59)(23 58 43 38)(24 37 44 57)(25 56 45 36)(26 35 46 55)(27 54 47 34)(28 33 48 53)(29 52 49 32)(30 31 50 51)(61 68 160 147)(62 146 141 67)(63 66 142 145)(64 144 143 65)(69 80 148 159)(70 158 149 79)(71 78 150 157)(72 156 151 77)(73 76 152 155)(74 154 153 75)(81 84 125 128)(82 127 126 83)(85 100 129 124)(86 123 130 99)(87 98 131 122)(88 121 132 97)(89 96 133 140)(90 139 134 95)(91 94 135 138)(92 137 136 93)

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - - + - + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 C8.C22 D4⋊2D5 D4×D5 SD16⋊D5 Q8.D10 kernel (C2×C8).D10 C20.Q8 C40⋊5C4 D10⋊1C8 Q8⋊Dic5 C5×Q8⋊C4 C4⋊C4⋊7D5 D10⋊3Q8 C2×Dic5 C22×D5 Q8⋊C4 C20 C4⋊C4 C2×C8 C2×Q8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 1 1 2 4 2 2 2 4 8 1 2 2 4 4

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

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 2 28 0 0 13 39 0 0 0 0 14 0 0 0 0 3
,
 32 28 0 0 13 28 0 0 0 0 0 38 0 0 14 0
,
 28 32 0 0 28 13 0 0 0 0 0 3 0 0 14 0
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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