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G = C5×C83D4order 320 = 26·5

Direct product of C5 and C83D4

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

Aliases: C5×C83D4, C4021D4, C83(C5×D4), (C2×D8)⋊9C10, C8⋊C44C10, C4.5(D4×C10), (C10×D8)⋊23C2, C41D44C10, (C2×SD16)⋊3C10, C20.312(C2×D4), C4.4D45C10, (C2×C20).343D4, (C10×SD16)⋊14C2, C42.29(C2×C10), C10.46(C41D4), (C2×C40).275C22, (C2×C20).952C23, (C4×C20).271C22, C22.117(D4×C10), C10.147(C8⋊C22), (D4×C10).205C22, (Q8×C10).179C22, (C5×C8⋊C4)⋊13C2, (C2×C4).44(C5×D4), C2.9(C5×C41D4), (C5×C41D4)⋊14C2, (C2×C8).27(C2×C10), C2.22(C5×C8⋊C22), (C2×D4).28(C2×C10), (C5×C4.4D4)⋊25C2, (C2×C10).673(C2×D4), (C2×Q8).23(C2×C10), (C2×C4).127(C22×C10), SmallGroup(320,997)

Series: Derived Chief Lower central Upper central

C1C2×C4 — C5×C83D4
C1C2C22C2×C4C2×C20D4×C10C10×SD16 — C5×C83D4
C1C2C2×C4 — C5×C83D4
C1C2×C10C4×C20 — C5×C83D4

Generators and relations for C5×C83D4
 G = < a,b,c,d | a5=b8=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=b5, dbd=b-1, dcd=c-1 >

Subgroups: 322 in 144 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×3], C4 [×2], C4 [×3], C22, C22 [×9], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4, D4 [×10], Q8 [×2], C23 [×3], C10, C10 [×2], C10 [×3], C42, C22⋊C4 [×2], C2×C8 [×2], D8 [×4], SD16 [×4], C2×D4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C20 [×2], C20 [×3], C2×C10, C2×C10 [×9], C8⋊C4, C4.4D4, C41D4, C2×D8 [×2], C2×SD16 [×2], C40 [×4], C2×C20, C2×C20 [×2], C2×C20, C5×D4 [×10], C5×Q8 [×2], C22×C10 [×3], C83D4, C4×C20, C5×C22⋊C4 [×2], C2×C40 [×2], C5×D8 [×4], C5×SD16 [×4], D4×C10, D4×C10 [×2], D4×C10 [×2], Q8×C10, C5×C8⋊C4, C5×C4.4D4, C5×C41D4, C10×D8 [×2], C10×SD16 [×2], C5×C83D4
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×6], C23, C10 [×7], C2×D4 [×3], C2×C10 [×7], C41D4, C8⋊C22 [×2], C5×D4 [×6], C22×C10, C83D4, D4×C10 [×3], C5×C41D4, C5×C8⋊C22 [×2], C5×C83D4

Smallest permutation representation of C5×C83D4
On 160 points
Generators in S160
(1 39 111 31 103)(2 40 112 32 104)(3 33 105 25 97)(4 34 106 26 98)(5 35 107 27 99)(6 36 108 28 100)(7 37 109 29 101)(8 38 110 30 102)(9 22 46 118 93)(10 23 47 119 94)(11 24 48 120 95)(12 17 41 113 96)(13 18 42 114 89)(14 19 43 115 90)(15 20 44 116 91)(16 21 45 117 92)(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 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(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 71 127 119)(2 68 128 116)(3 65 121 113)(4 70 122 118)(5 67 123 115)(6 72 124 120)(7 69 125 117)(8 66 126 114)(9 106 158 58)(10 111 159 63)(11 108 160 60)(12 105 153 57)(13 110 154 62)(14 107 155 59)(15 112 156 64)(16 109 157 61)(17 25 73 129)(18 30 74 134)(19 27 75 131)(20 32 76 136)(21 29 77 133)(22 26 78 130)(23 31 79 135)(24 28 80 132)(33 87 137 96)(34 84 138 93)(35 81 139 90)(36 86 140 95)(37 83 141 92)(38 88 142 89)(39 85 143 94)(40 82 144 91)(41 97 145 49)(42 102 146 54)(43 99 147 51)(44 104 148 56)(45 101 149 53)(46 98 150 50)(47 103 151 55)(48 100 152 52)
(1 122)(2 121)(3 128)(4 127)(5 126)(6 125)(7 124)(8 123)(9 10)(11 16)(12 15)(13 14)(17 20)(18 19)(21 24)(22 23)(25 136)(26 135)(27 134)(28 133)(29 132)(30 131)(31 130)(32 129)(33 144)(34 143)(35 142)(36 141)(37 140)(38 139)(39 138)(40 137)(41 44)(42 43)(45 48)(46 47)(49 104)(50 103)(51 102)(52 101)(53 100)(54 99)(55 98)(56 97)(57 112)(58 111)(59 110)(60 109)(61 108)(62 107)(63 106)(64 105)(65 68)(66 67)(69 72)(70 71)(73 76)(74 75)(77 80)(78 79)(81 88)(82 87)(83 86)(84 85)(89 90)(91 96)(92 95)(93 94)(113 116)(114 115)(117 120)(118 119)(145 148)(146 147)(149 152)(150 151)(153 156)(154 155)(157 160)(158 159)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E5A5B5C5D8A8B8C8D10A···10L10M···10X20A···20H20I···20P20Q20R20S20T40A···40P
order1222222444445555888810···1010···1020···2020···202020202040···40
size111188822448111144441···18···82···24···488884···4

80 irreducible representations

dim111111111111222244
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4D4C5×D4C5×D4C8⋊C22C5×C8⋊C22
kernelC5×C83D4C5×C8⋊C4C5×C4.4D4C5×C41D4C10×D8C10×SD16C83D4C8⋊C4C4.4D4C41D4C2×D8C2×SD16C40C2×C20C8C2×C4C10C2
# reps1111224444884216828

Matrix representation of C5×C83D4 in GL6(𝔽41)

100000
010000
0010000
0001000
0000100
0000010
,
4000000
0400000
001938019
0022191922
003382516
001919319
,
22400000
34190000
000010
00140139
001000
000001
,
100000
3400000
00223022
0022191922
003382516
0030316

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,19,22,3,19,0,0,38,19,38,19,0,0,0,19,25,3,0,0,19,22,16,19],[22,34,0,0,0,0,40,19,0,0,0,0,0,0,0,1,1,0,0,0,0,40,0,0,0,0,1,1,0,0,0,0,0,39,0,1],[1,3,0,0,0,0,0,40,0,0,0,0,0,0,22,22,3,3,0,0,3,19,38,0,0,0,0,19,25,3,0,0,22,22,16,16] >;

C5×C83D4 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes_3D_4
% in TeX

G:=Group("C5xC8:3D4");
// GroupNames label

G:=SmallGroup(320,997);
// by ID

G=gap.SmallGroup(320,997);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,589,288,1766,1731,436,7004,172]);
// Polycyclic

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

׿
×
𝔽