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G = C2×C20.47D4order 320 = 26·5

Direct product of C2 and C20.47D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C20.47D4, M4(2).32D10, C4.67(C2×D20), (C2×C4).53D20, (C2×C20).176D4, C20.422(C2×D4), C23.56(C4×D5), C102(C4.10D4), (C2×C20).418C23, (C2×Dic10).28C4, (C22×C4).144D10, (C2×M4(2)).17D5, C4.29(D10⋊C4), (C22×Dic5).5C4, C20.101(C22⋊C4), (C10×M4(2)).28C2, C4.Dic5.43C22, (C22×C20).191C22, (C22×Dic10).16C2, (C5×M4(2)).35C22, C22.51(D10⋊C4), (C2×Dic10).287C22, (C2×C4).55(C4×D5), C54(C2×C4.10D4), C22.22(C2×C4×D5), C4.115(C2×C5⋊D4), (C2×C20).284(C2×C4), (C2×Dic5).6(C2×C4), C2.33(C2×D10⋊C4), (C2×C4).257(C5⋊D4), C10.102(C2×C22⋊C4), (C2×C4).122(C22×D5), (C2×C4.Dic5).26C2, (C22×C10).140(C2×C4), (C2×C10).117(C22×C4), (C2×C10).131(C22⋊C4), SmallGroup(320,763)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C20.47D4
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — C2×C20.47D4
C5C10C2×C10 — C2×C20.47D4
C1C22C22×C4C2×M4(2)

Generators and relations for C2×C20.47D4
 G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b5c3 >

Subgroups: 478 in 146 conjugacy classes, 63 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×6], C2×C4 [×8], Q8 [×8], C23, C10, C10 [×2], C10 [×2], C2×C8 [×2], M4(2) [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C2×Q8 [×8], Dic5 [×4], C20 [×4], C2×C10 [×3], C2×C10 [×2], C4.10D4 [×4], C2×M4(2), C2×M4(2), C22×Q8, C52C8 [×2], C40 [×2], Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×6], C22×C10, C2×C4.10D4, C2×C52C8, C4.Dic5 [×2], C4.Dic5, C2×C40, C5×M4(2) [×2], C5×M4(2), C2×Dic10 [×4], C2×Dic10 [×4], C22×Dic5 [×2], C22×C20, C20.47D4 [×4], C2×C4.Dic5, C10×M4(2), C22×Dic10, C2×C20.47D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, D5, C22⋊C4 [×4], C22×C4, C2×D4 [×2], D10 [×3], C4.10D4 [×2], C2×C22⋊C4, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C2×C4.10D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C20.47D4 [×2], C2×D10⋊C4, C2×C20.47D4

Smallest permutation representation of C2×C20.47D4
On 160 points
Generators in S160
(1 40)(2 21)(3 22)(4 23)(5 24)(6 25)(7 26)(8 27)(9 28)(10 29)(11 30)(12 31)(13 32)(14 33)(15 34)(16 35)(17 36)(18 37)(19 38)(20 39)(41 103)(42 104)(43 105)(44 106)(45 107)(46 108)(47 109)(48 110)(49 111)(50 112)(51 113)(52 114)(53 115)(54 116)(55 117)(56 118)(57 119)(58 120)(59 101)(60 102)(61 145)(62 146)(63 147)(64 148)(65 149)(66 150)(67 151)(68 152)(69 153)(70 154)(71 155)(72 156)(73 157)(74 158)(75 159)(76 160)(77 141)(78 142)(79 143)(80 144)(81 129)(82 130)(83 131)(84 132)(85 133)(86 134)(87 135)(88 136)(89 137)(90 138)(91 139)(92 140)(93 121)(94 122)(95 123)(96 124)(97 125)(98 126)(99 127)(100 128)
(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 44 25 101 11 54 35 111)(2 43 26 120 12 53 36 110)(3 42 27 119 13 52 37 109)(4 41 28 118 14 51 38 108)(5 60 29 117 15 50 39 107)(6 59 30 116 16 49 40 106)(7 58 31 115 17 48 21 105)(8 57 32 114 18 47 22 104)(9 56 33 113 19 46 23 103)(10 55 34 112 20 45 24 102)(61 121 160 98 71 131 150 88)(62 140 141 97 72 130 151 87)(63 139 142 96 73 129 152 86)(64 138 143 95 74 128 153 85)(65 137 144 94 75 127 154 84)(66 136 145 93 76 126 155 83)(67 135 146 92 77 125 156 82)(68 134 147 91 78 124 157 81)(69 133 148 90 79 123 158 100)(70 132 149 89 80 122 159 99)
(1 129 11 139)(2 128 12 138)(3 127 13 137)(4 126 14 136)(5 125 15 135)(6 124 16 134)(7 123 17 133)(8 122 18 132)(9 121 19 131)(10 140 20 130)(21 100 31 90)(22 99 32 89)(23 98 33 88)(24 97 34 87)(25 96 35 86)(26 95 36 85)(27 94 37 84)(28 93 38 83)(29 92 39 82)(30 91 40 81)(41 61 51 71)(42 80 52 70)(43 79 53 69)(44 78 54 68)(45 77 55 67)(46 76 56 66)(47 75 57 65)(48 74 58 64)(49 73 59 63)(50 72 60 62)(101 147 111 157)(102 146 112 156)(103 145 113 155)(104 144 114 154)(105 143 115 153)(106 142 116 152)(107 141 117 151)(108 160 118 150)(109 159 119 149)(110 158 120 148)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A8B8C8D8E8F8G8H10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order12222244444444558888888810···101010101020···202020202040···40
size111122222220202020224444202020202···244442···244444···4

62 irreducible representations

dim11111112222222244
type++++++++++--
imageC1C2C2C2C2C4C4D4D5D10D10C4×D5D20C5⋊D4C4×D5C4.10D4C20.47D4
kernelC2×C20.47D4C20.47D4C2×C4.Dic5C10×M4(2)C22×Dic10C2×Dic10C22×Dic5C2×C20C2×M4(2)M4(2)C22×C4C2×C4C2×C4C2×C4C23C10C2
# reps14111444242488428

Matrix representation of C2×C20.47D4 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
1400000
3660000
0091100
00301400
00003911
00001425
,
3530000
1560000
00124439
0017402739
00817231
00109318
,
28270000
12130000
0003200
0032000
0000320
000049

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,36,0,0,0,0,40,6,0,0,0,0,0,0,9,30,0,0,0,0,11,14,0,0,0,0,0,0,39,14,0,0,0,0,11,25],[35,15,0,0,0,0,3,6,0,0,0,0,0,0,1,17,8,10,0,0,24,40,17,9,0,0,4,27,23,3,0,0,39,39,1,18],[28,12,0,0,0,0,27,13,0,0,0,0,0,0,0,32,0,0,0,0,32,0,0,0,0,0,0,0,32,4,0,0,0,0,0,9] >;

C2×C20.47D4 in GAP, Magma, Sage, TeX

C_2\times C_{20}._{47}D_4
% in TeX

G:=Group("C2xC20.47D4");
// GroupNames label

G:=SmallGroup(320,763);
// by ID

G=gap.SmallGroup(320,763);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,422,58,1123,136,438,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^5*c^3>;
// generators/relations

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×
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