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G = C2×Dic5.14D4order 320 = 26·5

Direct product of C2 and Dic5.14D4

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

Aliases: C2×Dic5.14D4, C233Dic10, C24.51D10, (C22×C10)⋊4Q8, C101(C22⋊Q8), C10.5(C22×Q8), (C2×C10).25C24, C4⋊Dic549C22, C22⋊C4.84D10, Dic5.81(C2×D4), C222(C2×Dic10), C10.32(C22×D4), C22.122(D4×D5), (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.D446C22, (C22×C20).70C22, (C23×C10).51C22, C23.318(C22×D5), C23.D5.83C22, C22.64(D42D5), (C22×C10).117C23, (C2×Dic5).185C23, (C22×Dic5).225C22, C2.7(C2×D4×D5), C51(C2×C22⋊Q8), (C2×C10)⋊4(C2×Q8), (C2×C4⋊Dic5)⋊17C2, C10.66(C2×C4○D4), C2.7(C2×D42D5), (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

C1C2×C10 — C2×Dic5.14D4
C1C5C10C2×C10C2×Dic5C22×Dic5C23×Dic5 — C2×Dic5.14D4
C5C2×C10 — C2×Dic5.14D4
C1C23C2×C22⋊C4

Generators and relations for C2×Dic5.14D4
 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 >

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], D42D5 [×2], C23×D5, Dic5.14D4 [×4], C22×Dic10, C2×D4×D5, C2×D42D5, C2×Dic5.14D4

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

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

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

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

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×D5D42D5
kernelC2×Dic5.14D4Dic5.14D4C2×C10.D4C2×C4⋊Dic5C2×C23.D5C10×C22⋊C4C22×Dic10C23×Dic5C2×Dic5C22×C10C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C23C22C22
# reps1821111144248421644

Matrix representation of C2×Dic5.14D4 in 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] >;

C2×Dic5.14D4 in GAP, Magma, Sage, TeX

C_2\times {\rm 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

׿
×
𝔽