Copied to
clipboard

## G = (C2×C10).Q16order 320 = 26·5

### 5th non-split extension by C2×C10 of Q16 acting via Q16/C4=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C10).Q16
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4⋊Dic5 — C2×C4⋊Dic5 — (C2×C10).Q16
 Lower central C5 — C10 — C2×C20 — (C2×C10).Q16
 Upper central C1 — C22 — C22×C4 — C22⋊Q8

Generators and relations for (C2×C10).Q16
G = < a,b,c,d | a2=b10=c8=1, d2=b5c4, ab=ba, cac-1=dad-1=ab5, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 334 in 104 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×Q8, Dic5, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊C8, Q8⋊C4, C2.D8, C2×C4⋊C4, C22⋊Q8, C52C8, C2×Dic5, C2×C20, C2×C20, C5×Q8, C22×C10, C23.48D4, C2×C52C8, C4⋊Dic5, C4⋊Dic5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, Q8×C10, C10.D8, C20.55D4, Q8⋊Dic5, C2×C4⋊Dic5, C5×C22⋊Q8, (C2×C10).Q16
Quotients: C1, C2, C22, D4, C23, D5, Q16, C2×D4, C4○D4, D10, C22.D4, C2×Q16, C8⋊C22, C5⋊D4, C22×D5, C23.48D4, C5⋊Q16, D42D5, C2×C5⋊D4, C23.18D10, C2×C5⋊Q16, D4⋊D10, (C2×C10).Q16

Smallest permutation representation of (C2×C10).Q16
On 160 points
Generators in S160
(1 30)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 158)(12 159)(13 160)(14 151)(15 152)(16 153)(17 154)(18 155)(19 156)(20 157)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 70)(58 61)(59 62)(60 63)(71 85)(72 86)(73 87)(74 88)(75 89)(76 90)(77 81)(78 82)(79 83)(80 84)(91 104)(92 105)(93 106)(94 107)(95 108)(96 109)(97 110)(98 101)(99 102)(100 103)(111 124)(112 125)(113 126)(114 127)(115 128)(116 129)(117 130)(118 121)(119 122)(120 123)(131 149)(132 150)(133 141)(134 142)(135 143)(136 144)(137 145)(138 146)(139 147)(140 148)
(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)
(1 152 32 145 30 20 50 132)(2 151 33 144 21 19 41 131)(3 160 34 143 22 18 42 140)(4 159 35 142 23 17 43 139)(5 158 36 141 24 16 44 138)(6 157 37 150 25 15 45 137)(7 156 38 149 26 14 46 136)(8 155 39 148 27 13 47 135)(9 154 40 147 28 12 48 134)(10 153 31 146 29 11 49 133)(51 118 90 98 69 121 71 101)(52 117 81 97 70 130 72 110)(53 116 82 96 61 129 73 109)(54 115 83 95 62 128 74 108)(55 114 84 94 63 127 75 107)(56 113 85 93 64 126 76 106)(57 112 86 92 65 125 77 105)(58 111 87 91 66 124 78 104)(59 120 88 100 67 123 79 103)(60 119 89 99 68 122 80 102)
(1 65 25 52)(2 66 26 53)(3 67 27 54)(4 68 28 55)(5 69 29 56)(6 70 30 57)(7 61 21 58)(8 62 22 59)(9 63 23 60)(10 64 24 51)(11 106 158 98)(12 107 159 99)(13 108 160 100)(14 109 151 91)(15 110 152 92)(16 101 153 93)(17 102 154 94)(18 103 155 95)(19 104 156 96)(20 105 157 97)(31 85 44 71)(32 86 45 72)(33 87 46 73)(34 88 47 74)(35 89 48 75)(36 90 49 76)(37 81 50 77)(38 82 41 78)(39 83 42 79)(40 84 43 80)(111 136 129 144)(112 137 130 145)(113 138 121 146)(114 139 122 147)(115 140 123 148)(116 131 124 149)(117 132 125 150)(118 133 126 141)(119 134 127 142)(120 135 128 143)

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

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

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

47 conjugacy classes

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

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - + + + + - - + image C1 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 Q16 D10 D10 D10 C5⋊D4 C5⋊D4 C8⋊C22 D4⋊2D5 C5⋊Q16 D4⋊D10 kernel (C2×C10).Q16 C10.D8 C20.55D4 Q8⋊Dic5 C2×C4⋊Dic5 C5×C22⋊Q8 C2×C20 C22×C10 C22⋊Q8 C20 C2×C10 C4⋊C4 C22×C4 C2×Q8 C2×C4 C23 C10 C4 C22 C2 # reps 1 2 1 2 1 1 1 1 2 4 4 2 2 2 4 4 1 4 4 4

Matrix representation of (C2×C10).Q16 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 14 40
,
 7 7 0 0 0 0 34 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 7 1 0 0 0 0 0 0 17 17 0 0 0 0 12 0 0 0 0 0 0 0 34 1 0 0 0 0 34 7
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 4 12 0 0 0 0 2 37 0 0 0 0 0 0 19 9 0 0 0 0 19 22

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,14,0,0,0,0,0,40],[7,34,0,0,0,0,7,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,7,0,0,0,0,0,1,0,0,0,0,0,0,17,12,0,0,0,0,17,0,0,0,0,0,0,0,34,34,0,0,0,0,1,7],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,4,2,0,0,0,0,12,37,0,0,0,0,0,0,19,19,0,0,0,0,9,22] >;

(C2×C10).Q16 in GAP, Magma, Sage, TeX

(C_2\times C_{10}).Q_{16}
% in TeX

G:=Group("(C2xC10).Q16");
// GroupNames label

G:=SmallGroup(320,671);
// by ID

G=gap.SmallGroup(320,671);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,336,254,219,268,1123,297,136,12550]);
// Polycyclic

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

׿
×
𝔽