Copied to
clipboard

## G = C2×D10.Q8order 320 = 26·5

### Direct product of C2 and D10.Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C2×D10.Q8
 Chief series C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — C2×C4.F5 — C2×D10.Q8
 Lower central C5 — C10 — C20 — C2×D10.Q8
 Upper central C1 — C22 — C2×C4 — C2×C8

Generators and relations for C2×D10.Q8
G = < a,b,c,d,e | a2=b10=c2=1, d4=b5, e2=b-1cd2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b3, cd=dc, ece-1=b7c, ede-1=b5d3 >

Subgroups: 346 in 106 conjugacy classes, 52 normal (28 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, D5 [×2], C10, C10 [×2], C2×C8, C2×C8 [×7], M4(2) [×6], C22×C4, Dic5 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C8.C4 [×4], C22×C8, C2×M4(2) [×2], C52C8 [×2], C40 [×2], C5⋊C8 [×4], C4×D5 [×4], C2×Dic5, C2×C20, C22×D5, C2×C8.C4, C8×D5 [×4], C2×C52C8, C2×C40, C4.F5 [×4], C4.F5 [×2], C2×C5⋊C8 [×2], C2×C4×D5, D10.Q8 [×4], D5×C2×C8, C2×C4.F5 [×2], C2×D10.Q8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, F5, C8.C4 [×2], C2×C4⋊C4, C2×F5 [×3], C2×C8.C4, C4⋊F5 [×2], C22×F5, D10.Q8 [×2], C2×C4⋊F5, C2×D10.Q8

Smallest permutation representation of C2×D10.Q8
On 160 points
Generators in S160
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 22)(12 23)(13 24)(14 25)(15 26)(16 27)(17 28)(18 29)(19 30)(20 21)(41 77)(42 78)(43 79)(44 80)(45 71)(46 72)(47 73)(48 74)(49 75)(50 76)(51 62)(52 63)(53 64)(54 65)(55 66)(56 67)(57 68)(58 69)(59 70)(60 61)(81 119)(82 120)(83 111)(84 112)(85 113)(86 114)(87 115)(88 116)(89 117)(90 118)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(121 159)(122 160)(123 151)(124 152)(125 153)(126 154)(127 155)(128 156)(129 157)(130 158)(131 144)(132 145)(133 146)(134 147)(135 148)(136 149)(137 150)(138 141)(139 142)(140 143)
(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 32)(2 31)(3 40)(4 39)(5 38)(6 37)(7 36)(8 35)(9 34)(10 33)(11 23)(12 22)(13 21)(14 30)(15 29)(16 28)(17 27)(18 26)(19 25)(20 24)(41 78)(42 77)(43 76)(44 75)(45 74)(46 73)(47 72)(48 71)(49 80)(50 79)(51 63)(52 62)(53 61)(54 70)(55 69)(56 68)(57 67)(58 66)(59 65)(60 64)(81 113)(82 112)(83 111)(84 120)(85 119)(86 118)(87 117)(88 116)(89 115)(90 114)(91 108)(92 107)(93 106)(94 105)(95 104)(96 103)(97 102)(98 101)(99 110)(100 109)(121 153)(122 152)(123 151)(124 160)(125 159)(126 158)(127 157)(128 156)(129 155)(130 154)(131 148)(132 147)(133 146)(134 145)(135 144)(136 143)(137 142)(138 141)(139 150)(140 149)
(1 57 17 42 6 52 12 47)(2 58 18 43 7 53 13 48)(3 59 19 44 8 54 14 49)(4 60 20 45 9 55 15 50)(5 51 11 46 10 56 16 41)(21 71 31 66 26 76 36 61)(22 72 32 67 27 77 37 62)(23 73 33 68 28 78 38 63)(24 74 34 69 29 79 39 64)(25 75 35 70 30 80 40 65)(81 121 96 136 86 126 91 131)(82 122 97 137 87 127 92 132)(83 123 98 138 88 128 93 133)(84 124 99 139 89 129 94 134)(85 125 100 140 90 130 95 135)(101 141 116 156 106 146 111 151)(102 142 117 157 107 147 112 152)(103 143 118 158 108 148 113 153)(104 144 119 159 109 149 114 154)(105 145 120 160 110 150 115 155)
(1 104 28 86 6 109 23 81)(2 101 27 89 7 106 22 84)(3 108 26 82 8 103 21 87)(4 105 25 85 9 110 30 90)(5 102 24 88 10 107 29 83)(11 112 34 98 16 117 39 93)(12 119 33 91 17 114 38 96)(13 116 32 94 18 111 37 99)(14 113 31 97 19 118 36 92)(15 120 40 100 20 115 35 95)(41 142 64 128 46 147 69 123)(42 149 63 121 47 144 68 126)(43 146 62 124 48 141 67 129)(44 143 61 127 49 148 66 122)(45 150 70 130 50 145 65 125)(51 152 74 138 56 157 79 133)(52 159 73 131 57 154 78 136)(53 156 72 134 58 151 77 139)(54 153 71 137 59 158 76 132)(55 160 80 140 60 155 75 135)

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

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

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

44 conjugacy classes

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

44 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + + - + - + + + image C1 C2 C2 C2 C4 C4 C4 D4 Q8 D4 Q8 C8.C4 F5 C2×F5 C2×F5 C4⋊F5 C4⋊F5 D10.Q8 kernel C2×D10.Q8 D10.Q8 D5×C2×C8 C2×C4.F5 C8×D5 C2×C5⋊2C8 C2×C40 C4×D5 C4×D5 C2×Dic5 C22×D5 C10 C2×C8 C8 C2×C4 C4 C22 C2 # reps 1 4 1 2 4 2 2 1 1 1 1 8 1 2 1 2 2 8

Matrix representation of C2×D10.Q8 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 1 1 1 1 0 0 40 0 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40 40 40 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 31 0 4 4 0 0 37 27 37 0 0 0 0 37 27 37 0 0 4 4 0 31
,
 15 26 0 0 0 0 26 26 0 0 0 0 0 0 23 24 32 40 0 0 8 16 17 40 0 0 1 24 25 33 0 0 1 9 17 18

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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,40,0,0,0,0,1,0,0,0,40,0,1,0,0,0,0,40,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,40,0,1,0,0,0,40,1,0],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,31,37,0,4,0,0,0,27,37,4,0,0,4,37,27,0,0,0,4,0,37,31],[15,26,0,0,0,0,26,26,0,0,0,0,0,0,23,8,1,1,0,0,24,16,24,9,0,0,32,17,25,17,0,0,40,40,33,18] >;

C2×D10.Q8 in GAP, Magma, Sage, TeX

C_2\times D_{10}.Q_8
% in TeX

G:=Group("C2xD10.Q8");
// GroupNames label

G:=SmallGroup(320,1061);
// by ID

G=gap.SmallGroup(320,1061);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,268,136,1684,102,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=c^2=1,d^4=b^5,e^2=b^-1*c*d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,e*b*e^-1=b^3,c*d=d*c,e*c*e^-1=b^7*c,e*d*e^-1=b^5*d^3>;
// generators/relations

׿
×
𝔽