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 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×8], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4 [×4], C2×C8 [×2], C22×C4, C22×C4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C2.D8 [×2], C2×C4⋊C4, C22⋊Q8, C52C8 [×2], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×C10, C23.48D4, C2×C52C8 [×2], C4⋊Dic5 [×2], C4⋊Dic5, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, Q8×C10, C10.D8 [×2], C20.55D4, Q8⋊Dic5 [×2], C2×C4⋊Dic5, C5×C22⋊Q8, (C2×C10).Q16
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, Q16 [×2], C2×D4, C4○D4 [×2], D10 [×3], C22.D4, C2×Q16, C8⋊C22, C5⋊D4 [×2], C22×D5, C23.48D4, C5⋊Q16 [×2], D42D5 [×2], 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)(11 160)(12 151)(13 152)(14 153)(15 154)(16 155)(17 156)(18 157)(19 158)(20 159)(31 45)(32 46)(33 47)(34 48)(35 49)(36 50)(37 41)(38 42)(39 43)(40 44)(51 70)(52 61)(53 62)(54 63)(55 64)(56 65)(57 66)(58 67)(59 68)(60 69)(71 81)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(91 102)(92 103)(93 104)(94 105)(95 106)(96 107)(97 108)(98 109)(99 110)(100 101)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(131 147)(132 148)(133 149)(134 150)(135 141)(136 142)(137 143)(138 144)(139 145)(140 146)
(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)
(1 154 36 145 30 20 50 134)(2 153 37 144 21 19 41 133)(3 152 38 143 22 18 42 132)(4 151 39 142 23 17 43 131)(5 160 40 141 24 16 44 140)(6 159 31 150 25 15 45 139)(7 158 32 149 26 14 46 138)(8 157 33 148 27 13 47 137)(9 156 34 147 28 12 48 136)(10 155 35 146 29 11 49 135)(51 114 86 94 65 125 71 105)(52 113 87 93 66 124 72 104)(53 112 88 92 67 123 73 103)(54 111 89 91 68 122 74 102)(55 120 90 100 69 121 75 101)(56 119 81 99 70 130 76 110)(57 118 82 98 61 129 77 109)(58 117 83 97 62 128 78 108)(59 116 84 96 63 127 79 107)(60 115 85 95 64 126 80 106)
(1 65 25 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 70 30 51)(7 61 21 52)(8 62 22 53)(9 63 23 54)(10 64 24 55)(11 106 160 100)(12 107 151 91)(13 108 152 92)(14 109 153 93)(15 110 154 94)(16 101 155 95)(17 102 156 96)(18 103 157 97)(19 104 158 98)(20 105 159 99)(31 81 50 71)(32 82 41 72)(33 83 42 73)(34 84 43 74)(35 85 44 75)(36 86 45 76)(37 87 46 77)(38 88 47 78)(39 89 48 79)(40 90 49 80)(111 136 127 142)(112 137 128 143)(113 138 129 144)(114 139 130 145)(115 140 121 146)(116 131 122 147)(117 132 123 148)(118 133 124 149)(119 134 125 150)(120 135 126 141)

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

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

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

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