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G = C2×C4⋊Dic5order 160 = 25·5

Direct product of C2 and C4⋊Dic5

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

Aliases: C2×C4⋊Dic5, C22.15D20, C23.30D10, C22.5Dic10, C103(C4⋊C4), (C2×C20)⋊10C4, C2010(C2×C4), (C2×C4)⋊3Dic5, C42(C2×Dic5), C2.2(C2×D20), C10.9(C2×Q8), (C2×C10).6Q8, C10.15(C2×D4), (C2×C10).20D4, (C2×C4).84D10, (C22×C4).6D5, (C22×C20).7C2, C2.3(C2×Dic10), (C2×C20).92C22, C10.36(C22×C4), (C2×C10).43C23, C2.4(C22×Dic5), (C22×Dic5).5C2, C22.14(C2×Dic5), C22.21(C22×D5), (C22×C10).35C22, (C2×Dic5).37C22, C54(C2×C4⋊C4), (C2×C10).54(C2×C4), SmallGroup(160,146)

Series: Derived Chief Lower central Upper central

C1C10 — C2×C4⋊Dic5
C1C5C10C2×C10C2×Dic5C22×Dic5 — C2×C4⋊Dic5
C5C10 — C2×C4⋊Dic5
C1C23C22×C4

Generators and relations for C2×C4⋊Dic5
 G = < a,b,c,d | a2=b4=c10=1, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 216 in 92 conjugacy classes, 65 normal (15 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×4], C22, C22 [×6], C5, C2×C4 [×6], C2×C4 [×8], C23, C10 [×3], C10 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×2], Dic5 [×4], C20 [×4], C2×C10, C2×C10 [×6], C2×C4⋊C4, C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C4⋊Dic5 [×4], C22×Dic5 [×2], C22×C20, C2×C4⋊Dic5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, Dic5 [×4], D10 [×3], C2×C4⋊C4, Dic10 [×2], D20 [×2], C2×Dic5 [×6], C22×D5, C4⋊Dic5 [×4], C2×Dic10, C2×D20, C22×Dic5, C2×C4⋊Dic5

Smallest permutation representation of C2×C4⋊Dic5
Regular action on 160 points
Generators in S160
(1 62)(2 63)(3 64)(4 65)(5 66)(6 67)(7 68)(8 69)(9 70)(10 61)(11 115)(12 116)(13 117)(14 118)(15 119)(16 120)(17 111)(18 112)(19 113)(20 114)(21 53)(22 54)(23 55)(24 56)(25 57)(26 58)(27 59)(28 60)(29 51)(30 52)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 81)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 71)(50 72)(91 144)(92 145)(93 146)(94 147)(95 148)(96 149)(97 150)(98 141)(99 142)(100 143)(101 138)(102 139)(103 140)(104 131)(105 132)(106 133)(107 134)(108 135)(109 136)(110 137)(121 158)(122 159)(123 160)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)
(1 47 27 34)(2 48 28 35)(3 49 29 36)(4 50 30 37)(5 41 21 38)(6 42 22 39)(7 43 23 40)(8 44 24 31)(9 45 25 32)(10 46 26 33)(11 140 160 148)(12 131 151 149)(13 132 152 150)(14 133 153 141)(15 134 154 142)(16 135 155 143)(17 136 156 144)(18 137 157 145)(19 138 158 146)(20 139 159 147)(51 87 64 71)(52 88 65 72)(53 89 66 73)(54 90 67 74)(55 81 68 75)(56 82 69 76)(57 83 70 77)(58 84 61 78)(59 85 62 79)(60 86 63 80)(91 111 109 129)(92 112 110 130)(93 113 101 121)(94 114 102 122)(95 115 103 123)(96 116 104 124)(97 117 105 125)(98 118 106 126)(99 119 107 127)(100 120 108 128)
(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 95 6 100)(2 94 7 99)(3 93 8 98)(4 92 9 97)(5 91 10 96)(11 90 16 85)(12 89 17 84)(13 88 18 83)(14 87 19 82)(15 86 20 81)(21 109 26 104)(22 108 27 103)(23 107 28 102)(24 106 29 101)(25 105 30 110)(31 118 36 113)(32 117 37 112)(33 116 38 111)(34 115 39 120)(35 114 40 119)(41 129 46 124)(42 128 47 123)(43 127 48 122)(44 126 49 121)(45 125 50 130)(51 138 56 133)(52 137 57 132)(53 136 58 131)(54 135 59 140)(55 134 60 139)(61 149 66 144)(62 148 67 143)(63 147 68 142)(64 146 69 141)(65 145 70 150)(71 158 76 153)(72 157 77 152)(73 156 78 151)(74 155 79 160)(75 154 80 159)

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

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

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

C2×C4⋊Dic5 is a maximal subgroup of
C20.31C42  C20.39C42  M4(2)⋊Dic5  (C2×C20)⋊1C8  C10.49(C4×D4)  C2.(C4×D20)  C4⋊Dic515C4  C10.52(C4×D4)  (C2×Dic5)⋊Q8  C2.(C20⋊Q8)  (C2×C20).28D4  (C2×C4).Dic10  C10.(C4⋊Q8)  D103(C4⋊C4)  C10.55(C4×D4)  (C2×C4).21D20  (C2×C20).33D4  C23.34D20  C23.35D20  C23.38D20  C22.D40  C207(C4⋊C4)  (C2×C20)⋊10Q8  C428Dic5  C429Dic5  (C2×C4)⋊6D20  C24.6D10  C24.7D10  C24.47D10  C24.8D10  C23.14D20  C24.16D10  C204(C4⋊C4)  C4⋊C4×Dic5  C205(C4⋊C4)  C20.48(C4⋊C4)  C4⋊C45Dic5  (C2×C20).53D4  (C2×C20).54D4  C206(C4⋊C4)  (C2×C20).55D4  D104(C4⋊C4)  (C2×C20).56D4  C20.64(C4⋊C4)  (C2×C10).D8  C4⋊D4.D5  C22⋊Q8.D5  (C2×C10).Q16  C23.47D20  C23.49D20  C24.64D10  C24.19D10  (Q8×C10)⋊17C4  C4○D4⋊Dic5  C2×C4×Dic10  C2×C4×D20  C2×D5×C4⋊C4  C42.90D10  C42.91D10  C42.105D10  D46Dic10  D46D20  C42.119D10  C10.732- 1+4  C10.1152+ 1+4  C10.1182+ 1+4  C10.772- 1+4  C10.852- 1+4  C2×D4×Dic5  C2×Q8×Dic5  C10.1062- 1+4  C10.1472+ 1+4
C2×C4⋊Dic5 is a maximal quotient of
C2013M4(2)  C428Dic5  C429Dic5  C24.47D10  C206(C4⋊C4)  C42.43D10  C23.22D20  C23.47D20  M4(2).Dic5  C24.64D10

52 conjugacy classes

class 1 2A···2G4A4B4C4D4E···4L5A5B10A···10N20A···20P
order12···244444···45510···1020···20
size11···1222210···10222···22···2

52 irreducible representations

dim1111122222222
type+++++-+-++-+
imageC1C2C2C2C4D4Q8D5Dic5D10D10Dic10D20
kernelC2×C4⋊Dic5C4⋊Dic5C22×Dic5C22×C20C2×C20C2×C10C2×C10C22×C4C2×C4C2×C4C23C22C22
# reps1421822284288

Matrix representation of C2×C4⋊Dic5 in GL6(𝔽41)

100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
001000
000100
0000320
000019
,
3510000
4000000
0035100
0040000
0000160
00003218
,
1420000
4270000
00273900
00371400
00002540
0000916

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,1,0,0,0,0,0,9],[35,40,0,0,0,0,1,0,0,0,0,0,0,0,35,40,0,0,0,0,1,0,0,0,0,0,0,0,16,32,0,0,0,0,0,18],[14,4,0,0,0,0,2,27,0,0,0,0,0,0,27,37,0,0,0,0,39,14,0,0,0,0,0,0,25,9,0,0,0,0,40,16] >;

C2×C4⋊Dic5 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes {\rm Dic}_5
% in TeX

G:=Group("C2xC4:Dic5");
// GroupNames label

G:=SmallGroup(160,146);
// by ID

G=gap.SmallGroup(160,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,48,362,86,4613]);
// Polycyclic

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

׿
×
𝔽