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G = C5×D4.D4order 320 = 26·5

Direct product of C5 and D4.D4

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

Aliases: C5×D4.D4, C2012SD16, C4⋊C89C10, C4⋊Q82C10, D4.1(C5×D4), C42(C5×SD16), (C4×D4).6C10, (C5×D4).26D4, C4.33(D4×C10), (D4×C20).21C2, C20.394(C2×D4), (C2×C20).323D4, C2.8(C10×SD16), Q8⋊C411C10, C42.16(C2×C10), (C2×SD16).3C10, (C10×SD16).8C2, C10.88(C2×SD16), C22.85(D4×C10), C20.343(C4○D4), (C2×C20).920C23, (C2×C40).300C22, (C4×C20).258C22, C10.144(C4⋊D4), (D4×C10).297C22, (Q8×C10).159C22, C10.134(C8.C22), (C5×C4⋊C8)⋊28C2, (C5×C4⋊Q8)⋊23C2, C4.42(C5×C4○D4), C4⋊C4.53(C2×C10), (C2×C8).37(C2×C10), (C2×C4).129(C5×D4), C2.13(C5×C4⋊D4), (C2×Q8).5(C2×C10), C2.9(C5×C8.C22), (C5×Q8⋊C4)⋊33C2, (C2×D4).56(C2×C10), (C2×C10).641(C2×D4), (C5×C4⋊C4).374C22, (C2×C4).95(C22×C10), SmallGroup(320,962)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×D4.D4
C1C2C4C2×C4C2×C20Q8×C10C5×C4⋊Q8 — C5×D4.D4
C1C2C2×C4 — C5×D4.D4
C1C2×C10C4×C20 — C5×D4.D4

Generators and relations for C5×D4.D4
 G = < a,b,c,d,e | a5=b4=c2=d4=1, e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=bc, ede-1=d-1 >

Subgroups: 218 in 120 conjugacy classes, 58 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×4], C5, C8 [×2], C2×C4 [×3], C2×C4 [×5], D4 [×2], D4, Q8 [×4], C23, C10 [×3], C10 [×2], C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8 [×2], C20 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×4], Q8⋊C4 [×2], C4⋊C8, C4×D4, C4⋊Q8, C2×SD16 [×2], C40 [×2], C2×C20 [×3], C2×C20 [×5], C5×D4 [×2], C5×D4, C5×Q8 [×4], C22×C10, D4.D4, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4 [×2], C2×C40 [×2], C5×SD16 [×4], C22×C20, D4×C10, Q8×C10 [×2], C5×Q8⋊C4 [×2], C5×C4⋊C8, D4×C20, C5×C4⋊Q8, C10×SD16 [×2], C5×D4.D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×4], C23, C10 [×7], SD16 [×2], C2×D4 [×2], C4○D4, C2×C10 [×7], C4⋊D4, C2×SD16, C8.C22, C5×D4 [×4], C22×C10, D4.D4, C5×SD16 [×2], D4×C10 [×2], C5×C4○D4, C5×C4⋊D4, C10×SD16, C5×C8.C22, C5×D4.D4

Smallest permutation representation of C5×D4.D4
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 37 29 66)(2 38 30 67)(3 39 26 68)(4 40 27 69)(5 36 28 70)(6 20 136 156)(7 16 137 157)(8 17 138 158)(9 18 139 159)(10 19 140 160)(11 46 53 41)(12 47 54 42)(13 48 55 43)(14 49 51 44)(15 50 52 45)(21 35 142 151)(22 31 143 152)(23 32 144 153)(24 33 145 154)(25 34 141 155)(56 110 85 76)(57 106 81 77)(58 107 82 78)(59 108 83 79)(60 109 84 80)(61 89 91 75)(62 90 92 71)(63 86 93 72)(64 87 94 73)(65 88 95 74)(96 116 125 150)(97 117 121 146)(98 118 122 147)(99 119 123 148)(100 120 124 149)(101 115 131 129)(102 111 132 130)(103 112 133 126)(104 113 134 127)(105 114 135 128)
(1 66)(2 67)(3 68)(4 69)(5 70)(6 136)(7 137)(8 138)(9 139)(10 140)(11 41)(12 42)(13 43)(14 44)(15 45)(21 142)(22 143)(23 144)(24 145)(25 141)(26 39)(27 40)(28 36)(29 37)(30 38)(46 53)(47 54)(48 55)(49 51)(50 52)(56 76)(57 77)(58 78)(59 79)(60 80)(61 75)(62 71)(63 72)(64 73)(65 74)(81 106)(82 107)(83 108)(84 109)(85 110)(86 93)(87 94)(88 95)(89 91)(90 92)(96 125)(97 121)(98 122)(99 123)(100 124)(101 131)(102 132)(103 133)(104 134)(105 135)
(1 64 12 81)(2 65 13 82)(3 61 14 83)(4 62 15 84)(5 63 11 85)(6 103 143 125)(7 104 144 121)(8 105 145 122)(9 101 141 123)(10 102 142 124)(16 113 153 146)(17 114 154 147)(18 115 155 148)(19 111 151 149)(20 112 152 150)(21 100 140 132)(22 96 136 133)(23 97 137 134)(24 98 138 135)(25 99 139 131)(26 91 51 59)(27 92 52 60)(28 93 53 56)(29 94 54 57)(30 95 55 58)(31 116 156 126)(32 117 157 127)(33 118 158 128)(34 119 159 129)(35 120 160 130)(36 86 46 76)(37 87 47 77)(38 88 48 78)(39 89 49 79)(40 90 50 80)(41 110 70 72)(42 106 66 73)(43 107 67 74)(44 108 68 75)(45 109 69 71)
(1 23 29 144)(2 24 30 145)(3 25 26 141)(4 21 27 142)(5 22 28 143)(6 11 136 53)(7 12 137 54)(8 13 138 55)(9 14 139 51)(10 15 140 52)(16 42 157 47)(17 43 158 48)(18 44 159 49)(19 45 160 50)(20 41 156 46)(31 36 152 70)(32 37 153 66)(33 38 154 67)(34 39 155 68)(35 40 151 69)(56 125 85 96)(57 121 81 97)(58 122 82 98)(59 123 83 99)(60 124 84 100)(61 131 91 101)(62 132 92 102)(63 133 93 103)(64 134 94 104)(65 135 95 105)(71 130 90 111)(72 126 86 112)(73 127 87 113)(74 128 88 114)(75 129 89 115)(76 150 110 116)(77 146 106 117)(78 147 107 118)(79 148 108 119)(80 149 109 120)

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

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

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

95 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20P20Q···20AB20AC···20AJ40A···40P
order1222224444444445555888810···1010···1020···2020···2020···2040···40
size111144222244488111144441···14···42···24···48···84···4

95 irreducible representations

dim1111111111112222222244
type++++++++-
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4SD16C4○D4C5×D4C5×D4C5×SD16C5×C4○D4C8.C22C5×C8.C22
kernelC5×D4.D4C5×Q8⋊C4C5×C4⋊C8D4×C20C5×C4⋊Q8C10×SD16D4.D4Q8⋊C4C4⋊C8C4×D4C4⋊Q8C2×SD16C2×C20C5×D4C20C20C2×C4D4C4C4C10C2
# reps12111248444822428816814

Matrix representation of C5×D4.D4 in GL4(𝔽41) generated by

1000
0100
00160
00016
,
0100
40000
0010
0001
,
0100
1000
0010
0001
,
40000
04000
004039
0011
,
152600
262600
0010
004040
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[0,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,1,0,0,39,1],[15,26,0,0,26,26,0,0,0,0,1,40,0,0,0,40] >;

C5×D4.D4 in GAP, Magma, Sage, TeX

C_5\times D_4.D_4
% in TeX

G:=Group("C5xD4.D4");
// GroupNames label

G:=SmallGroup(320,962);
// by ID

G=gap.SmallGroup(320,962);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,1120,589,288,1766,10085,2539,124]);
// Polycyclic

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

׿
×
𝔽