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## G = C2×C20.C8order 320 = 26·5

### Direct product of C2 and C20.C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C2×C20.C8
 Chief series C1 — C5 — C10 — C20 — C5⋊2C8 — C5⋊C16 — C2×C5⋊C16 — C2×C20.C8
 Lower central C5 — C10 — C2×C20.C8
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C20.C8
G = < a,b,c | a2=b20=1, c8=b10, ab=ba, ac=ca, cbc-1=b3 >

Subgroups: 186 in 90 conjugacy classes, 60 normal (24 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C16, C2×C8, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C2×C16, M5(2), C22×C8, C52C8, C52C8, C2×C20, C2×C20, C22×C10, C2×M5(2), C5⋊C16, C2×C52C8, C2×C52C8, C22×C20, C2×C5⋊C16, C20.C8, C22×C52C8, C2×C20.C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C23, C2×C8, C22×C4, F5, M5(2), C22×C8, C5⋊C8, C2×F5, C2×M5(2), C2×C5⋊C8, C22×F5, C20.C8, C22×C5⋊C8, C2×C20.C8

Smallest permutation representation of C2×C20.C8
On 160 points
Generators in S160
(1 65)(2 66)(3 67)(4 68)(5 69)(6 70)(7 71)(8 72)(9 73)(10 74)(11 75)(12 76)(13 77)(14 78)(15 79)(16 80)(17 61)(18 62)(19 63)(20 64)(21 50)(22 51)(23 52)(24 53)(25 54)(26 55)(27 56)(28 57)(29 58)(30 59)(31 60)(32 41)(33 42)(34 43)(35 44)(36 45)(37 46)(38 47)(39 48)(40 49)(81 106)(82 107)(83 108)(84 109)(85 110)(86 111)(87 112)(88 113)(89 114)(90 115)(91 116)(92 117)(93 118)(94 119)(95 120)(96 101)(97 102)(98 103)(99 104)(100 105)(121 143)(122 144)(123 145)(124 146)(125 147)(126 148)(127 149)(128 150)(129 151)(130 152)(131 153)(132 154)(133 155)(134 156)(135 157)(136 158)(137 159)(138 160)(139 141)(140 142)
(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)
(1 133 110 57 70 150 90 23 11 123 120 47 80 160 100 33)(2 140 119 60 71 157 99 26 12 130 109 50 61 147 89 36)(3 127 108 43 72 144 88 29 13 137 118 53 62 154 98 39)(4 134 117 46 73 151 97 32 14 124 107 56 63 141 87 22)(5 121 106 49 74 158 86 35 15 131 116 59 64 148 96 25)(6 128 115 52 75 145 95 38 16 138 105 42 65 155 85 28)(7 135 104 55 76 152 84 21 17 125 114 45 66 142 94 31)(8 122 113 58 77 159 93 24 18 132 103 48 67 149 83 34)(9 129 102 41 78 146 82 27 19 139 112 51 68 156 92 37)(10 136 111 44 79 153 91 30 20 126 101 54 69 143 81 40)

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

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

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

56 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5 8A ··· 8H 8I 8J 8K 8L 10A ··· 10G 16A ··· 16P 20A ··· 20H order 1 2 2 2 2 2 4 4 4 4 4 4 5 8 ··· 8 8 8 8 8 10 ··· 10 16 ··· 16 20 ··· 20 size 1 1 1 1 2 2 1 1 1 1 2 2 4 5 ··· 5 10 10 10 10 4 ··· 4 10 ··· 10 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 4 4 4 4 4 type + + + + + - + - image C1 C2 C2 C2 C4 C4 C8 C8 M5(2) F5 C5⋊C8 C2×F5 C5⋊C8 C20.C8 kernel C2×C20.C8 C2×C5⋊C16 C20.C8 C22×C5⋊2C8 C2×C5⋊2C8 C22×C20 C2×C20 C22×C10 C10 C22×C4 C2×C4 C2×C4 C23 C2 # reps 1 2 4 1 6 2 12 4 8 1 3 3 1 8

Matrix representation of C2×C20.C8 in GL6(𝔽241)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240 0 0 0 0 0 0 240
,
 177 0 0 0 0 0 236 64 0 0 0 0 0 0 131 64 0 0 0 0 177 0 0 0 0 0 116 117 0 110 0 0 58 7 195 195
,
 177 96 0 0 0 0 43 64 0 0 0 0 0 0 0 0 190 240 0 0 64 64 2 1 0 0 83 185 177 0 0 0 207 215 131 0

G:=sub<GL(6,GF(241))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240,0,0,0,0,0,0,240],[177,236,0,0,0,0,0,64,0,0,0,0,0,0,131,177,116,58,0,0,64,0,117,7,0,0,0,0,0,195,0,0,0,0,110,195],[177,43,0,0,0,0,96,64,0,0,0,0,0,0,0,64,83,207,0,0,0,64,185,215,0,0,190,2,177,131,0,0,240,1,0,0] >;

C2×C20.C8 in GAP, Magma, Sage, TeX

C_2\times C_{20}.C_8
% in TeX

G:=Group("C2xC20.C8");
// GroupNames label

G:=SmallGroup(320,1081);
// by ID

G=gap.SmallGroup(320,1081);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,758,80,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^2=b^20=1,c^8=b^10,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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