Copied to
clipboard

?

G = C2×Dic5.14D4  order 320 = 26·5

Direct product of C2 and Dic5.14D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Dic5.14D4, C23⋊3Dic10, C24.51D10, (C22×C10)⋊4Q8, C10⋊1(C22⋊Q8), C10.5(C22×Q8), (C2×C10).25C24, C4⋊Dic5⋊49C22, C22⋊C4.84D10, Dic5.81(C2×D4), C22⋊2(C2×Dic10), C22.122(D4×D5), C10.32(C22×D4), (C2×C20).125C23, (C2×Dic5).245D4, (C22×Dic10)⋊5C2, (C22×C4).167D10, (C23×Dic5).8C2, C2.7(C22×Dic10), C22.67(C23×D5), (C2×Dic10)⋊47C22, C10.D4⋊46C22, (C23×C10).51C22, (C22×C20).70C22, C23.318(C22×D5), C23.D5.83C22, C22.64(D4⋊2D5), (C22×C10).117C23, (C2×Dic5).185C23, (C22×Dic5).225C22, C2.7(C2×D4×D5), C5⋊1(C2×C22⋊Q8), (C2×C10)⋊4(C2×Q8), (C2×C4⋊Dic5)⋊17C2, C10.66(C2×C4○D4), C2.7(C2×D4⋊2D5), (C2×C10).378(C2×D4), (C2×C22⋊C4).16D5, (C2×C10.D4)⋊21C2, (C10×C22⋊C4).17C2, (C2×C4).132(C22×D5), (C2×C23.D5).20C2, (C2×C10).166(C4○D4), (C5×C22⋊C4).96C22, SmallGroup(320,1153)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C2×Dic5.14D4
Chief series
C1 — C5 — C10 — C2×C10 — C2×Dic5 — C22×Dic5 — C23×Dic5 — C2×Dic5.14D4
Lower central
C5 — C2×C10 — C2×Dic5.14D4

Subgroups: 974 in 322 conjugacy classes, 135 normal (31 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×14], C22, C22 [×10], C22 [×12], C5, C2×C4 [×4], C2×C4 [×30], Q8 [×8], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C22⋊C4 [×4], C4⋊C4 [×12], C22×C4 [×2], C22×C4 [×12], C2×Q8 [×8], C24, Dic5 [×4], Dic5 [×6], C20 [×4], C2×C10, C2×C10 [×10], C2×C10 [×12], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×3], C22⋊Q8 [×8], C23×C4, C22×Q8, Dic10 [×8], C2×Dic5 [×12], C2×Dic5 [×14], C2×C20 [×4], C2×C20 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C2×C22⋊Q8, C10.D4 [×8], C4⋊Dic5 [×4], C23.D5 [×4], C5×C22⋊C4 [×4], C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×4], C22×Dic5 [×4], C22×Dic5 [×4], C22×C20 [×2], C23×C10, Dic5.14D4 [×8], C2×C10.D4 [×2], C2×C4⋊Dic5, C2×C23.D5, C10×C22⋊C4, C22×Dic10, C23×Dic5, C2×Dic5.14D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], D5, C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, D10 [×7], C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, Dic10 [×4], C22×D5 [×7], C2×C22⋊Q8, C2×Dic10 [×6], D4×D5 [×2], D4⋊2D5 [×2], C23×D5, Dic5.14D4 [×4], C22×Dic10, C2×D4×D5, C2×D4⋊2D5, C2×Dic5.14D4

Generators and relations
 G = < a,b,c,d,e | a2=b10=d4=e2=1, c2=b5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, dcd-1=b5c, ce=ec, ede=b5d-1 >

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

(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)

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
34400000
100000
001100
005600
0000400
0000040
,
2350000
21390000
0020300
0032100
0000320
000009
,
4000000
0400000
0039900
004200
0000040
0000400
,
100000
010000
001000
000100
000010
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[34,1,0,0,0,0,40,0,0,0,0,0,0,0,1,5,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[2,21,0,0,0,0,35,39,0,0,0,0,0,0,20,3,0,0,0,0,3,21,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,39,4,0,0,0,0,9,2,0,0,0,0,0,0,0,40,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;
 

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P5A5B10A···10N10O···10V20A···20P
order12···2222244444···444445510···1010···1020···20
size11···12222444410···1020202020222···24···44···4

68 irreducible representations

dim111111112222222244
type+++++++++-++++-+-
imageC1C2C2C2C2C2C2C2D4Q8D5C4○D4D10D10D10Dic10D4×D5D4⋊2D5
kernelC2×Dic5.14D4Dic5.14D4C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C22×Dic10C23×Dic5C2×Dic5C22×C10C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C23C22C22
# reps1821111144248421644

In GAP, Magma, Sage, TeX 

C_2\times Dic_5._{14}D_4
 
% in TeX
 
G:=Group("C2xDic5.14D4");
 
// GroupNames label
 
G:=SmallGroup(320,1153);
 
// by ID
 
G=gap.SmallGroup(320,1153);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,675,297,80,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^10=d^4=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,d*c*d^-1=b^5*c,c*e=e*c,e*d*e=b^5*d^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀