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G = C5×C42.29C22order 320 = 26·5

Direct product of C5 and C42.29C22

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

Aliases: C5×C42.29C22, C8⋊C411C10, C41D4.5C10, (C2×C20).341D4, C42.C23C10, D4⋊C419C10, C42.27(C2×C10), C20.272(C4○D4), (C4×C20).269C22, (C2×C40).336C22, (C2×C20).946C23, C22.111(D4×C10), C10.146(C8⋊C22), C10.75(C4.4D4), (D4×C10).201C22, (C5×C8⋊C4)⋊25C2, C4.17(C5×C4○D4), (C2×C4).42(C5×D4), C4⋊C4.21(C2×C10), (C2×C8).57(C2×C10), C2.21(C5×C8⋊C22), (C5×D4⋊C4)⋊42C2, (C2×D4).24(C2×C10), (C2×C10).667(C2×D4), (C5×C41D4).12C2, (C5×C42.C2)⋊20C2, C2.13(C5×C4.4D4), (C5×C4⋊C4).241C22, (C2×C4).121(C22×C10), SmallGroup(320,991)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C42.29C22
C1C2C4C2×C4C2×C20D4×C10C5×D4⋊C4 — C5×C42.29C22
C1C2C2×C4 — C5×C42.29C22
C1C2×C10C4×C20 — C5×C42.29C22

Generators and relations for C5×C42.29C22
 G = < a,b,c,d,e | a5=b4=c4=d2=1, e2=c, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b-1, ebe-1=bc2, dcd=c-1, ce=ec, ede-1=b2c-1d >

Subgroups: 242 in 110 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×8], C23 [×2], C10, C10 [×2], C10 [×2], C42, C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×D4 [×2], C2×D4 [×2], C20 [×2], C20 [×4], C2×C10, C2×C10 [×6], C8⋊C4, D4⋊C4 [×4], C42.C2, C41D4, C40 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C5×D4 [×8], C22×C10 [×2], C42.29C22, C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40 [×2], D4×C10 [×2], D4×C10 [×2], C5×C8⋊C4, C5×D4⋊C4 [×4], C5×C42.C2, C5×C41D4, C5×C42.29C22
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], C23, C10 [×7], C2×D4, C4○D4 [×2], C2×C10 [×7], C4.4D4, C8⋊C22 [×2], C5×D4 [×2], C22×C10, C42.29C22, D4×C10, C5×C4○D4 [×2], C5×C4.4D4, C5×C8⋊C22 [×2], C5×C42.29C22

Smallest permutation representation of C5×C42.29C22
On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 46 118 38 90)(10 47 119 39 91)(11 48 120 40 92)(12 41 113 33 93)(13 42 114 34 94)(14 43 115 35 95)(15 44 116 36 96)(16 45 117 37 89)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 84 153 73 145)(66 85 154 74 146)(67 86 155 75 147)(68 87 156 76 148)(69 88 157 77 149)(70 81 158 78 150)(71 82 159 79 151)(72 83 160 80 152)
(1 68 125 40)(2 65 126 37)(3 70 127 34)(4 67 128 39)(5 72 121 36)(6 69 122 33)(7 66 123 38)(8 71 124 35)(9 31 154 59)(10 28 155 64)(11 25 156 61)(12 30 157 58)(13 27 158 63)(14 32 159 60)(15 29 160 57)(16 26 153 62)(17 148 53 120)(18 145 54 117)(19 150 55 114)(20 147 56 119)(21 152 49 116)(22 149 50 113)(23 146 51 118)(24 151 52 115)(41 102 77 130)(42 99 78 135)(43 104 79 132)(44 101 80 129)(45 98 73 134)(46 103 74 131)(47 100 75 136)(48 97 76 133)(81 143 94 107)(82 140 95 112)(83 137 96 109)(84 142 89 106)(85 139 90 111)(86 144 91 108)(87 141 92 105)(88 138 93 110)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)(33 35 37 39)(34 36 38 40)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)(65 67 69 71)(66 68 70 72)(73 75 77 79)(74 76 78 80)(81 83 85 87)(82 84 86 88)(89 91 93 95)(90 92 94 96)(97 99 101 103)(98 100 102 104)(105 107 109 111)(106 108 110 112)(113 115 117 119)(114 116 118 120)(121 123 125 127)(122 124 126 128)(129 131 133 135)(130 132 134 136)(137 139 141 143)(138 140 142 144)(145 147 149 151)(146 148 150 152)(153 155 157 159)(154 156 158 160)
(1 40)(2 71)(3 38)(4 69)(5 36)(6 67)(7 34)(8 65)(9 27)(10 58)(11 25)(12 64)(13 31)(14 62)(15 29)(16 60)(17 120)(18 151)(19 118)(20 149)(21 116)(22 147)(23 114)(24 145)(26 159)(28 157)(30 155)(32 153)(33 128)(35 126)(37 124)(39 122)(41 136)(42 103)(43 134)(44 101)(45 132)(46 99)(47 130)(48 97)(49 152)(50 119)(51 150)(52 117)(53 148)(54 115)(55 146)(56 113)(57 160)(59 158)(61 156)(63 154)(66 127)(68 125)(70 123)(72 121)(73 104)(74 135)(75 102)(76 133)(77 100)(78 131)(79 98)(80 129)(81 139)(82 106)(83 137)(84 112)(85 143)(86 110)(87 141)(88 108)(89 140)(90 107)(91 138)(92 105)(93 144)(94 111)(95 142)(96 109)
(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)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F5A5B5C5D8A8B8C8D10A···10L10M···10T20A···20H20I···20P20Q···20X40A···40P
order1222224444445555888810···1010···1020···2020···2020···2040···40
size111188224488111144441···18···82···24···48···84···4

80 irreducible representations

dim1111111111222244
type+++++++
imageC1C2C2C2C2C5C10C10C10C10D4C4○D4C5×D4C5×C4○D4C8⋊C22C5×C8⋊C22
kernelC5×C42.29C22C5×C8⋊C4C5×D4⋊C4C5×C42.C2C5×C41D4C42.29C22C8⋊C4D4⋊C4C42.C2C41D4C2×C20C20C2×C4C4C10C2
# reps114114416442481628

Matrix representation of C5×C42.29C22 in GL6(𝔽41)

100000
010000
0018000
0001800
0000180
0000018
,
18230000
34230000
002902021
000292020
002020120
002120012
,
100000
010000
0004000
001000
0000040
000010
,
18230000
2230000
002902021
000122121
002021029
002121290
,
2390000
22390000
002120012
002121290
002902021
000292020

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18,0,0,0,0,0,0,18],[18,34,0,0,0,0,23,23,0,0,0,0,0,0,29,0,20,21,0,0,0,29,20,20,0,0,20,20,12,0,0,0,21,20,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[18,2,0,0,0,0,23,23,0,0,0,0,0,0,29,0,20,21,0,0,0,12,21,21,0,0,20,21,0,29,0,0,21,21,29,0],[2,22,0,0,0,0,39,39,0,0,0,0,0,0,21,21,29,0,0,0,20,21,0,29,0,0,0,29,20,20,0,0,12,0,21,20] >;

C5×C42.29C22 in GAP, Magma, Sage, TeX

C_5\times C_4^2._{29}C_2^2
% in TeX

G:=Group("C5xC4^2.29C2^2");
// GroupNames label

G:=SmallGroup(320,991);
// by ID

G=gap.SmallGroup(320,991);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,1688,1766,1731,226,7004,172,10085,124]);
// Polycyclic

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

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