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G = C4○D209C4order 320 = 26·5

3rd semidirect product of C4○D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D209C4, D2030(C2×C4), C4.60(C2×D20), (C2×C4).43D20, C4⋊C4.226D10, C20.140(C2×D4), (C2×C20).134D4, D206C424C2, Dic1027(C2×C4), C10.Q1624C2, (C22×C4).93D10, C10.83(C8⋊C22), C20.67(C22⋊C4), C20.119(C22×C4), (C2×C20).319C23, (C22×C10).184D4, C23.76(C5⋊D4), C54(C23.36D4), C4.37(D10⋊C4), C2.2(D4.D10), C2.2(C20.C23), (C2×D20).241C22, C10.83(C8.C22), C22.4(D10⋊C4), (C22×C20).134C22, (C2×Dic10).268C22, (C2×C4⋊C4)⋊2D5, (C10×C4⋊C4)⋊2C2, C4.49(C2×C4×D5), (C2×C4).39(C4×D5), (C2×C4○D20).6C2, (C2×C4.Dic5)⋊8C2, (C2×C20).251(C2×C4), (C2×C10).439(C2×D4), C10.80(C2×C22⋊C4), C22.58(C2×C5⋊D4), C2.12(C2×D10⋊C4), (C2×C4).182(C5⋊D4), (C5×C4⋊C4).257C22, (C2×C52C8).80C22, (C2×C4).419(C22×D5), (C2×C10).122(C22⋊C4), SmallGroup(320,593)

Series: Derived Chief Lower central Upper central

C1C20 — C4○D209C4
C1C5C10C2×C10C2×C20C2×D20C2×C4○D20 — C4○D209C4
C5C10C20 — C4○D209C4
C1C22C22×C4C2×C4⋊C4

Generators and relations for C4○D209C4
 G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, dad-1=a-1, cbc=a2b9, dbd-1=a2b, dcd-1=a2b5c >

Subgroups: 606 in 162 conjugacy classes, 63 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10 [×3], C10 [×2], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], C20 [×2], D10 [×4], C2×C10, C2×C10 [×2], C2×C10 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], C2×C4⋊C4, C2×M4(2), C2×C4○D4, C52C8 [×2], Dic10 [×2], Dic10, C4×D5 [×4], D20 [×2], D20, C2×Dic5, C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×D5, C22×C10, C23.36D4, C2×C52C8 [×2], C4.Dic5 [×2], C5×C4⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, C22×C20, D206C4 [×2], C10.Q16 [×2], C2×C4.Dic5, C10×C4⋊C4, C2×C4○D20, C4○D209C4
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], C2×C22⋊C4, C8⋊C22, C8.C22, C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.36D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.D10, C20.C23, C4○D209C4

Smallest permutation representation of C4○D209C4
On 160 points
Generators in S160
(1 100 11 90)(2 81 12 91)(3 82 13 92)(4 83 14 93)(5 84 15 94)(6 85 16 95)(7 86 17 96)(8 87 18 97)(9 88 19 98)(10 89 20 99)(21 103 31 113)(22 104 32 114)(23 105 33 115)(24 106 34 116)(25 107 35 117)(26 108 36 118)(27 109 37 119)(28 110 38 120)(29 111 39 101)(30 112 40 102)(41 135 51 125)(42 136 52 126)(43 137 53 127)(44 138 54 128)(45 139 55 129)(46 140 56 130)(47 121 57 131)(48 122 58 132)(49 123 59 133)(50 124 60 134)(61 150 71 160)(62 151 72 141)(63 152 73 142)(64 153 74 143)(65 154 75 144)(66 155 76 145)(67 156 77 146)(68 157 78 147)(69 158 79 148)(70 159 80 149)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 24)(22 23)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(61 69)(62 68)(63 67)(64 66)(70 80)(71 79)(72 78)(73 77)(74 76)(81 98)(82 97)(83 96)(84 95)(85 94)(86 93)(87 92)(88 91)(89 90)(99 100)(101 108)(102 107)(103 106)(104 105)(109 120)(110 119)(111 118)(112 117)(113 116)(114 115)(121 129)(122 128)(123 127)(124 126)(130 140)(131 139)(132 138)(133 137)(134 136)(141 147)(142 146)(143 145)(148 160)(149 159)(150 158)(151 157)(152 156)(153 155)
(1 73 33 49)(2 64 34 60)(3 75 35 51)(4 66 36 42)(5 77 37 53)(6 68 38 44)(7 79 39 55)(8 70 40 46)(9 61 21 57)(10 72 22 48)(11 63 23 59)(12 74 24 50)(13 65 25 41)(14 76 26 52)(15 67 27 43)(16 78 28 54)(17 69 29 45)(18 80 30 56)(19 71 31 47)(20 62 32 58)(81 143 116 124)(82 154 117 135)(83 145 118 126)(84 156 119 137)(85 147 120 128)(86 158 101 139)(87 149 102 130)(88 160 103 121)(89 151 104 132)(90 142 105 123)(91 153 106 134)(92 144 107 125)(93 155 108 136)(94 146 109 127)(95 157 110 138)(96 148 111 129)(97 159 112 140)(98 150 113 131)(99 141 114 122)(100 152 115 133)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D10A···10N20A···20X
order12222222444444444455888810···1020···20
size111122202022224444202022202020202···24···4

62 irreducible representations

dim11111112222222224444
type+++++++++++++-
imageC1C2C2C2C2C2C4D4D4D5D10D10C4×D5D20C5⋊D4C5⋊D4C8⋊C22C8.C22D4.D10C20.C23
kernelC4○D209C4D206C4C10.Q16C2×C4.Dic5C10×C4⋊C4C2×C4○D20C4○D20C2×C20C22×C10C2×C4⋊C4C4⋊C4C22×C4C2×C4C2×C4C2×C4C23C10C10C2C2
# reps12211183124288441144

Matrix representation of C4○D209C4 in GL6(𝔽41)

4000000
0400000
00236512
0035182936
0018351835
00623623
,
4000000
0400000
00040039
00135229
000101
00406406
,
4000000
910000
00040039
00400390
000001
000010
,
32390000
090000
002133413
0028392830
0015143928
002717132

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,23,35,18,6,0,0,6,18,35,23,0,0,5,29,18,6,0,0,12,36,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,0,40,0,0,40,35,1,6,0,0,0,2,0,40,0,0,39,29,1,6],[40,9,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,39,0,1,0,0,39,0,1,0],[32,0,0,0,0,0,39,9,0,0,0,0,0,0,2,28,15,27,0,0,13,39,14,17,0,0,34,28,39,13,0,0,13,30,28,2] >;

C4○D209C4 in GAP, Magma, Sage, TeX

C_4\circ D_{20}\rtimes_9C_4
% in TeX

G:=Group("C4oD20:9C4");
// GroupNames label

G:=SmallGroup(320,593);
// by ID

G=gap.SmallGroup(320,593);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,387,58,1684,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^2=d^4=1,b^10=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,c*b*c=a^2*b^9,d*b*d^-1=a^2*b,d*c*d^-1=a^2*b^5*c>;
// generators/relations

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