Copied to
clipboard

G = (C8×D5)⋊C4order 320 = 26·5

4th semidirect product of C8×D5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C8×D5)⋊4C4, C8.30(C4×D5), C4.24(Q8×D5), C4.Q814D5, C40.75(C2×C4), C406C425C2, (C4×D5).14Q8, C20.13(C2×Q8), C4⋊C4.161D10, (C2×C8).259D10, C22.84(D4×D5), D10.14(C4⋊C4), C10.54(C4○D8), C10.D814C2, (C22×D5).83D4, Dic5.38(C4⋊C4), (C2×C20).276C23, C20.102(C22×C4), (C2×C40).160C22, (C2×Dic5).274D4, C52(C23.25D4), C2.6(SD163D5), C4⋊Dic5.108C22, (D5×C2×C8).8C2, C4.77(C2×C4×D5), C2.12(D5×C4⋊C4), (C5×C4.Q8)⋊8C2, C10.34(C2×C4⋊C4), C52C8.38(C2×C4), C4⋊C47D5.4C2, (C4×D5).74(C2×C4), (C2×C10).281(C2×D4), (C5×C4⋊C4).69C22, (C2×C4×D5).302C22, (C2×C4).379(C22×D5), (C2×C52C8).235C22, SmallGroup(320,487)

Series: Derived Chief Lower central Upper central

C1C20 — (C8×D5)⋊C4
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — (C8×D5)⋊C4
C5C10C20 — (C8×D5)⋊C4
C1C22C2×C4C4.Q8

Generators and relations for (C8×D5)⋊C4
 G = < a,b,c,d | a8=b5=c2=d4=1, ab=ba, ac=ca, dad-1=a3, cbc=b-1, bd=db, dcd-1=a4c >

Subgroups: 382 in 114 conjugacy classes, 55 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10, C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8, C4.Q8, C2.D8 [×2], C42⋊C2 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.25D4, C8×D5 [×4], C2×C52C8, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C10.D8 [×2], C406C4, C5×C4.Q8, C4⋊C47D5 [×2], D5×C2×C8, (C8×D5)⋊C4
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, D10 [×3], C2×C4⋊C4, C4○D8 [×2], C4×D5 [×2], C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, SD163D5 [×2], (C8×D5)⋊C4

Smallest permutation representation of (C8×D5)⋊C4
On 160 points
Generators in S160
(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 99 133 21 113)(2 100 134 22 114)(3 101 135 23 115)(4 102 136 24 116)(5 103 129 17 117)(6 104 130 18 118)(7 97 131 19 119)(8 98 132 20 120)(9 124 81 93 49)(10 125 82 94 50)(11 126 83 95 51)(12 127 84 96 52)(13 128 85 89 53)(14 121 86 90 54)(15 122 87 91 55)(16 123 88 92 56)(25 137 146 107 70)(26 138 147 108 71)(27 139 148 109 72)(28 140 149 110 65)(29 141 150 111 66)(30 142 151 112 67)(31 143 152 105 68)(32 144 145 106 69)(33 45 160 58 74)(34 46 153 59 75)(35 47 154 60 76)(36 48 155 61 77)(37 41 156 62 78)(38 42 157 63 79)(39 43 158 64 80)(40 44 159 57 73)
(1 148)(2 149)(3 150)(4 151)(5 152)(6 145)(7 146)(8 147)(9 38)(10 39)(11 40)(12 33)(13 34)(14 35)(15 36)(16 37)(17 68)(18 69)(19 70)(20 71)(21 72)(22 65)(23 66)(24 67)(25 131)(26 132)(27 133)(28 134)(29 135)(30 136)(31 129)(32 130)(41 56)(42 49)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(57 83)(58 84)(59 85)(60 86)(61 87)(62 88)(63 81)(64 82)(73 126)(74 127)(75 128)(76 121)(77 122)(78 123)(79 124)(80 125)(89 153)(90 154)(91 155)(92 156)(93 157)(94 158)(95 159)(96 160)(97 137)(98 138)(99 139)(100 140)(101 141)(102 142)(103 143)(104 144)(105 117)(106 118)(107 119)(108 120)(109 113)(110 114)(111 115)(112 116)
(1 55 105 155)(2 50 106 158)(3 53 107 153)(4 56 108 156)(5 51 109 159)(6 54 110 154)(7 49 111 157)(8 52 112 160)(9 66 63 97)(10 69 64 100)(11 72 57 103)(12 67 58 98)(13 70 59 101)(14 65 60 104)(15 68 61 99)(16 71 62 102)(17 83 139 40)(18 86 140 35)(19 81 141 38)(20 84 142 33)(21 87 143 36)(22 82 144 39)(23 85 137 34)(24 88 138 37)(25 75 135 128)(26 78 136 123)(27 73 129 126)(28 76 130 121)(29 79 131 124)(30 74 132 127)(31 77 133 122)(32 80 134 125)(41 116 92 147)(42 119 93 150)(43 114 94 145)(44 117 95 148)(45 120 96 151)(46 115 89 146)(47 118 90 149)(48 113 91 152)

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444444558888888810···102020202020···2040···40
size11111010224444555520202020222222101010102···244448···84···4

56 irreducible representations

dim111111122222222444
type++++++-+++++-+
imageC1C2C2C2C2C2C4Q8D4D4D5D10D10C4○D8C4×D5Q8×D5D4×D5SD163D5
kernel(C8×D5)⋊C4C10.D8C406C4C5×C4.Q8C4⋊C47D5D5×C2×C8C8×D5C4×D5C2×Dic5C22×D5C4.Q8C4⋊C4C2×C8C10C8C4C22C2
# reps121121821124288228

Matrix representation of (C8×D5)⋊C4 in GL4(𝔽41) generated by

40000
04000
00270
00293
,
354000
364000
0010
0001
,
0700
6000
00400
0011
,
32000
03200
004039
0001
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,27,29,0,0,0,3],[35,36,0,0,40,40,0,0,0,0,1,0,0,0,0,1],[0,6,0,0,7,0,0,0,0,0,40,1,0,0,0,1],[32,0,0,0,0,32,0,0,0,0,40,0,0,0,39,1] >;

(C8×D5)⋊C4 in GAP, Magma, Sage, TeX

(C_8\times D_5)\rtimes C_4
% in TeX

G:=Group("(C8xD5):C4");
// GroupNames label

G:=SmallGroup(320,487);
// by ID

G=gap.SmallGroup(320,487);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,555,58,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^5=c^2=d^4=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^3,c*b*c=b^-1,b*d=d*b,d*c*d^-1=a^4*c>;
// generators/relations

׿
×
𝔽