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G = C5×C4.D8order 320 = 26·5

Direct product of C5 and C4.D8

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Aliases: C5×C4.D8, C20.60D8, C20.51SD16, C4⋊C82C10, C4.9(C5×D8), (C2×D4).2C20, (D4×C10).18C4, (C2×C20).503D4, C41D4.1C10, C42.3(C2×C10), C4.11(C5×SD16), (C4×C20).243C22, C10.49(D4⋊C4), C10.20(C4.D4), (C5×C4⋊C8)⋊4C2, (C2×C4).11(C2×C20), (C5×C41D4).8C2, (C2×C4).109(C5×D4), C2.4(C5×D4⋊C4), C2.4(C5×C4.D4), (C2×C20).351(C2×C4), C22.39(C5×C22⋊C4), (C2×C10).190(C22⋊C4), SmallGroup(320,136)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C4.D8
C1C2C22C2×C4C42C4×C20C5×C4⋊C8 — C5×C4.D8
C1C22C2×C4 — C5×C4.D8
C1C2×C10C4×C20 — C5×C4.D8

Generators and relations for C5×C4.D8
 G = < a,b,c,d | a5=b4=c8=1, d2=b, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=bc-1 >

Subgroups: 210 in 84 conjugacy classes, 38 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4, C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], D4 [×6], C23 [×2], C10, C10 [×2], C10 [×2], C42, C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C20 [×4], C20, C2×C10, C2×C10 [×6], C4⋊C8 [×2], C41D4, C40 [×2], C2×C20, C2×C20 [×2], C5×D4 [×6], C22×C10 [×2], C4.D8, C4×C20, C2×C40 [×2], D4×C10 [×2], D4×C10 [×2], C5×C4⋊C8 [×2], C5×C41D4, C5×C4.D8
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, D8 [×2], SD16 [×2], C20 [×2], C2×C10, C4.D4, D4⋊C4 [×2], C2×C20, C5×D4 [×2], C4.D8, C5×C22⋊C4, C5×D8 [×2], C5×SD16 [×2], C5×C4.D4, C5×D4⋊C4 [×2], C5×C4.D8

Smallest permutation representation of C5×C4.D8
On 160 points
Generators in S160
(1 53 41 37 25)(2 54 42 38 26)(3 55 43 39 27)(4 56 44 40 28)(5 49 45 33 29)(6 50 46 34 30)(7 51 47 35 31)(8 52 48 36 32)(9 155 150 139 134)(10 156 151 140 135)(11 157 152 141 136)(12 158 145 142 129)(13 159 146 143 130)(14 160 147 144 131)(15 153 148 137 132)(16 154 149 138 133)(17 128 115 112 99)(18 121 116 105 100)(19 122 117 106 101)(20 123 118 107 102)(21 124 119 108 103)(22 125 120 109 104)(23 126 113 110 97)(24 127 114 111 98)(57 89 83 73 69)(58 90 84 74 70)(59 91 85 75 71)(60 92 86 76 72)(61 93 87 77 65)(62 94 88 78 66)(63 95 81 79 67)(64 96 82 80 68)
(1 17 132 62)(2 63 133 18)(3 19 134 64)(4 57 135 20)(5 21 136 58)(6 59 129 22)(7 23 130 60)(8 61 131 24)(9 96 55 122)(10 123 56 89)(11 90 49 124)(12 125 50 91)(13 92 51 126)(14 127 52 93)(15 94 53 128)(16 121 54 95)(25 99 137 66)(26 67 138 100)(27 101 139 68)(28 69 140 102)(29 103 141 70)(30 71 142 104)(31 97 143 72)(32 65 144 98)(33 108 152 74)(34 75 145 109)(35 110 146 76)(36 77 147 111)(37 112 148 78)(38 79 149 105)(39 106 150 80)(40 73 151 107)(41 115 153 88)(42 81 154 116)(43 117 155 82)(44 83 156 118)(45 119 157 84)(46 85 158 120)(47 113 159 86)(48 87 160 114)
(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)
(1 131 17 24 132 8 62 61)(2 60 63 7 133 23 18 130)(3 129 19 22 134 6 64 59)(4 58 57 5 135 21 20 136)(9 50 96 91 55 12 122 125)(10 124 123 11 56 90 89 49)(13 54 92 95 51 16 126 121)(14 128 127 15 52 94 93 53)(25 144 99 98 137 32 66 65)(26 72 67 31 138 97 100 143)(27 142 101 104 139 30 68 71)(28 70 69 29 140 103 102 141)(33 151 108 107 152 40 74 73)(34 80 75 39 145 106 109 150)(35 149 110 105 146 38 76 79)(36 78 77 37 147 112 111 148)(41 160 115 114 153 48 88 87)(42 86 81 47 154 113 116 159)(43 158 117 120 155 46 82 85)(44 84 83 45 156 119 118 157)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E5A5B5C5D8A···8H10A···10L10M···10T20A···20P20Q20R20S20T40A···40AF
order1222224444455558···810···1010···1020···202020202040···40
size1111882222411114···41···18···82···244444···4

95 irreducible representations

dim1111111122222244
type++++++
imageC1C2C2C4C5C10C10C20D4D8SD16C5×D4C5×D8C5×SD16C4.D4C5×C4.D4
kernelC5×C4.D8C5×C4⋊C8C5×C41D4D4×C10C4.D8C4⋊C8C41D4C2×D4C2×C20C20C20C2×C4C4C4C10C2
# reps1214484162448161614

Matrix representation of C5×C4.D8 in GL4(𝔽41) generated by

18000
01800
0010
0001
,
0100
40000
00400
00040
,
122900
292900
001526
001515
,
121200
291200
002615
001515
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[12,29,0,0,29,29,0,0,0,0,15,15,0,0,26,15],[12,29,0,0,12,12,0,0,0,0,26,15,0,0,15,15] >;

C5×C4.D8 in GAP, Magma, Sage, TeX

C_5\times C_4.D_8
% in TeX

G:=Group("C5xC4.D8");
// GroupNames label

G:=SmallGroup(320,136);
// by ID

G=gap.SmallGroup(320,136);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2530,248,4911,242]);
// Polycyclic

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

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