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G = C4○D2010C4order 320 = 26·5

4th semidirect product of C4○D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4○D2010C4, C4.63(C2×D20), C4⋊C4.235D10, D20.40(C2×C4), (C2×C4).146D20, C20.143(C2×D4), (C2×C20).496D4, C42⋊C22D5, D206C443C2, C10.85(C4○D8), C10.Q1643C2, (C22×C10).75D4, (C2×C20).329C23, C20.124(C22×C4), Dic10.41(C2×C4), (C22×C4).339D10, C23.38(C5⋊D4), C54(C23.24D4), C4.52(D10⋊C4), C20.111(C22⋊C4), C2.2(D4.8D10), (C2×D20).243C22, C22.1(D10⋊C4), (C22×C20).151C22, (C2×Dic10).270C22, C4.52(C2×C4×D5), (C2×C4).81(C4×D5), (C22×C52C8)⋊2C2, (C2×C4○D20).7C2, (C2×C20).265(C2×C4), (C5×C42⋊C2)⋊2C2, (C2×C10).458(C2×D4), C10.86(C2×C22⋊C4), C22.73(C2×C5⋊D4), C2.18(C2×D10⋊C4), (C2×C4).274(C5⋊D4), (C5×C4⋊C4).266C22, (C2×C4).429(C22×D5), (C2×C10).79(C22⋊C4), (C2×C52C8).251C22, SmallGroup(320,629)

Series: Derived Chief Lower central Upper central

C1C20 — C4○D2010C4
C1C5C10C2×C10C2×C20C2×D20C2×C4○D20 — C4○D2010C4
C5C10C20 — C4○D2010C4
C1C2×C4C22×C4C42⋊C2

Generators and relations for C4○D2010C4
 G = < a,b,c,d | a4=c2=d4=1, b10=a2, ab=ba, ac=ca, ad=da, cbc=a2b9, dbd-1=a2b, dcd-1=a2b5c >

Subgroups: 590 in 158 conjugacy classes, 63 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×4], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×6], C5, C8 [×2], C2×C4 [×2], C2×C4 [×4], C2×C4 [×7], D4 [×7], Q8 [×3], C23, C23, D5 [×2], C10, C10 [×2], C10 [×2], C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×4], C22×C4, C22×C4, 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], C42⋊C2, C22×C8, 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 [×2], C22×D5, C22×C10, C23.24D4, C2×C52C8 [×2], C2×C52C8 [×2], C4×C20, C5×C22⋊C4, C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5, C2×D20, C4○D20 [×4], C4○D20 [×2], C2×C5⋊D4, C22×C20, D206C4 [×2], C10.Q16 [×2], C22×C52C8, C5×C42⋊C2, C2×C4○D20, C4○D2010C4
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, C4○D8 [×2], C4×D5 [×2], D20 [×2], C5⋊D4 [×2], C22×D5, C23.24D4, D10⋊C4 [×4], C2×C4×D5, C2×D20, C2×C5⋊D4, C2×D10⋊C4, D4.8D10 [×2], C4○D2010C4

Smallest permutation representation of C4○D2010C4
On 160 points
Generators in S160
(1 85 11 95)(2 86 12 96)(3 87 13 97)(4 88 14 98)(5 89 15 99)(6 90 16 100)(7 91 17 81)(8 92 18 82)(9 93 19 83)(10 94 20 84)(21 108 31 118)(22 109 32 119)(23 110 33 120)(24 111 34 101)(25 112 35 102)(26 113 36 103)(27 114 37 104)(28 115 38 105)(29 116 39 106)(30 117 40 107)(41 125 51 135)(42 126 52 136)(43 127 53 137)(44 128 54 138)(45 129 55 139)(46 130 56 140)(47 131 57 121)(48 132 58 122)(49 133 59 123)(50 134 60 124)(61 142 71 152)(62 143 72 153)(63 144 73 154)(64 145 74 155)(65 146 75 156)(66 147 76 157)(67 148 77 158)(68 149 78 159)(69 150 79 160)(70 151 80 141)
(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 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 47)(42 46)(43 45)(48 60)(49 59)(50 58)(51 57)(52 56)(53 55)(61 79)(62 78)(63 77)(64 76)(65 75)(66 74)(67 73)(68 72)(69 71)(81 88)(82 87)(83 86)(84 85)(89 100)(90 99)(91 98)(92 97)(93 96)(94 95)(101 104)(102 103)(105 120)(106 119)(107 118)(108 117)(109 116)(110 115)(111 114)(112 113)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)(142 160)(143 159)(144 158)(145 157)(146 156)(147 155)(148 154)(149 153)(150 152)
(1 78 36 52)(2 69 37 43)(3 80 38 54)(4 71 39 45)(5 62 40 56)(6 73 21 47)(7 64 22 58)(8 75 23 49)(9 66 24 60)(10 77 25 51)(11 68 26 42)(12 79 27 53)(13 70 28 44)(14 61 29 55)(15 72 30 46)(16 63 31 57)(17 74 32 48)(18 65 33 59)(19 76 34 50)(20 67 35 41)(81 155 119 132)(82 146 120 123)(83 157 101 134)(84 148 102 125)(85 159 103 136)(86 150 104 127)(87 141 105 138)(88 152 106 129)(89 143 107 140)(90 154 108 131)(91 145 109 122)(92 156 110 133)(93 147 111 124)(94 158 112 135)(95 149 113 126)(96 160 114 137)(97 151 115 128)(98 142 116 139)(99 153 117 130)(100 144 118 121)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L5A5B8A···8H10A···10F10G10H10I10J20A···20H20I···20AB
order12222222444444444444558···810···101010101020···2020···20
size1111222020111122444420202210···102···244442···24···4

68 irreducible representations

dim111111122222222224
type++++++++++++
imageC1C2C2C2C2C2C4D4D4D5D10D10C4○D8C4×D5D20C5⋊D4C5⋊D4D4.8D10
kernelC4○D2010C4D206C4C10.Q16C22×C52C8C5×C42⋊C2C2×C4○D20C4○D20C2×C20C22×C10C42⋊C2C4⋊C4C22×C4C10C2×C4C2×C4C2×C4C23C2
# reps122111831242888448

Matrix representation of C4○D2010C4 in GL4(𝔽41) generated by

1000
0100
00320
00032
,
354000
1000
004018
0091
,
403500
0100
004018
0001
,
391300
28200
00030
00260
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[35,1,0,0,40,0,0,0,0,0,40,9,0,0,18,1],[40,0,0,0,35,1,0,0,0,0,40,0,0,0,18,1],[39,28,0,0,13,2,0,0,0,0,0,26,0,0,30,0] >;

C4○D2010C4 in GAP, Magma, Sage, TeX

C_4\circ D_{20}\rtimes_{10}C_4
% in TeX

G:=Group("C4oD20:10C4");
// GroupNames label

G:=SmallGroup(320,629);
// by ID

G=gap.SmallGroup(320,629);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,422,58,1684,438,102,12550]);
// 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,a*d=d*a,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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×
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