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G = (C2×C20)⋊17D4order 320 = 26·5

13rd semidirect product of C2×C20 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20)⋊17D4, C20⋊D433C2, C20.454(C2×D4), Dic54(C4○D4), (C2×D4).237D10, (C2×Q8).194D10, Dic5⋊D447C2, Dic5⋊Q834C2, C20.17D433C2, C20.23D434C2, (C2×C10).318C24, (C2×C20).889C23, (C22×C4).288D10, C10.168(C22×D4), (D4×C10).277C22, (C2×D20).290C22, C57(C22.26C24), (Q8×C10).244C22, C22.327(C23×D5), C23.139(C22×D5), (C22×C20).297C22, (C22×C10).244C23, (C2×Dic5).304C23, (C4×Dic5).292C22, (C22×D5).139C23, C23.D5.137C22, D10⋊C4.160C22, (C2×Dic10).319C22, C10.D4.172C22, (C22×Dic5).260C22, (C2×C4○D4)⋊10D5, (C4×C5⋊D4)⋊61C2, (C2×C4×Dic5)⋊15C2, (C2×C4○D20)⋊32C2, (C10×C4○D4)⋊10C2, (C2×C4)⋊11(C5⋊D4), C2.106(D5×C4○D4), (C2×C10).83(C2×D4), C4.146(C2×C5⋊D4), C22.1(C2×C5⋊D4), C10.218(C2×C4○D4), (C2×C4×D5).272C22, C2.41(C22×C5⋊D4), (C2×C4).832(C22×D5), (C2×C5⋊D4).150C22, SmallGroup(320,1504)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C20)⋊17D4
C1C5C10C2×C10C22×D5C2×C4×D5C2×C4○D20 — (C2×C20)⋊17D4
C5C2×C10 — (C2×C20)⋊17D4
C1C2×C4C2×C4○D4

Generators and relations for (C2×C20)⋊17D4
 G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, ac=ca, dad=ab10, cbc-1=dbd=b9, dcd=c-1 >

Subgroups: 1022 in 310 conjugacy classes, 115 normal (31 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×4], C4 [×10], C22, C22 [×2], C22 [×14], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×18], D4 [×20], Q8 [×4], C23, C23 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C10 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C22×C4, C22×C4 [×2], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×7], C2×Q8, C2×Q8, C4○D4 [×8], Dic5 [×4], Dic5 [×4], C20 [×4], C20 [×2], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C2×C4○D4, C2×C4○D4, Dic10 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×6], C2×Dic5 [×4], C5⋊D4 [×12], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×2], C22×C10, C22×C10 [×2], C22.26C24, C4×Dic5 [×2], C4×Dic5 [×2], C10.D4 [×4], D10⋊C4 [×4], C23.D5 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×6], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C2×C4×Dic5, C4×C5⋊D4 [×4], C20.17D4, Dic5⋊D4 [×4], C20⋊D4, Dic5⋊Q8, C20.23D4, C2×C4○D20, C10×C4○D4, (C2×C20)⋊17D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C22.26C24, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4 [×2], C22×C5⋊D4, (C2×C20)⋊17D4

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I···4P4Q4R5A5B10A···10F10G···10R20A···20H20I···20T
order1222222222444444444···4445510···1010···1020···2020···20
size1111224420201111224410···102020222···24···42···24···4

68 irreducible representations

dim111111111122222224
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10C5⋊D4D5×C4○D4
kernel(C2×C20)⋊17D4C2×C4×Dic5C4×C5⋊D4C20.17D4Dic5⋊D4C20⋊D4Dic5⋊Q8C20.23D4C2×C4○D20C10×C4○D4C2×C20C2×C4○D4Dic5C22×C4C2×D4C2×Q8C2×C4C2
# reps1141411111428662168

Matrix representation of (C2×C20)⋊17D4 in GL6(𝔽41)

100000
010000
009200
0013200
000010
000001
,
610000
510000
0032000
0003200
0000400
0000040
,
610000
6350000
001000
000100
0000040
000010
,
610000
6350000
001000
00324000
000010
0000040

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,1,0,0,0,0,2,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,5,0,0,0,0,1,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[6,6,0,0,0,0,1,35,0,0,0,0,0,0,1,32,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

(C2×C20)⋊17D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_{17}D_4
% in TeX

G:=Group("(C2xC20):17D4");
// GroupNames label

G:=SmallGroup(320,1504);
// by ID

G=gap.SmallGroup(320,1504);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=d^2=1,a*b=b*a,a*c=c*a,d*a*d=a*b^10,c*b*c^-1=d*b*d=b^9,d*c*d=c^-1>;
// generators/relations

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