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G = C207M4(2)  order 320 = 26·5

3rd semidirect product of C20 and M4(2) acting via M4(2)/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C207M4(2), C42.206D10, C52C818D4, C57(C86D4), (C4×D4).6D5, (D4×C20).7C2, C203C822C2, C4.216(D4×D5), C4⋊C4.8Dic5, C2.9(D4×Dic5), (D4×C10).22C4, C10.122(C4×D4), C20.375(C2×D4), C41(C4.Dic5), (C2×D4).6Dic5, C10.64(C8○D4), C20.55D43C2, (C4×C20).83C22, C22⋊C4.5Dic5, C20.308(C4○D4), C23.9(C2×Dic5), (C2×C20).850C23, (C22×C4).115D10, C10.75(C2×M4(2)), C2.7(D4.Dic5), C4.135(D42D5), (C22×C20).100C22, C22.46(C22×Dic5), (C4×C52C8)⋊7C2, (C5×C4⋊C4).25C4, (C2×C4.Dic5)⋊5C2, (C2×C20).337(C2×C4), C2.9(C2×C4.Dic5), (C5×C22⋊C4).14C4, (C2×C4).20(C2×Dic5), (C2×C4).792(C22×D5), (C2×C10).288(C22×C4), (C22×C10).129(C2×C4), (C2×C52C8).205C22, SmallGroup(320,639)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C207M4(2)
C1C5C10C20C2×C20C2×C52C8C2×C4.Dic5 — C207M4(2)
C5C2×C10 — C207M4(2)
C1C2×C4C4×D4

Generators and relations for C207M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a11, cbc=b5 >

Subgroups: 286 in 122 conjugacy classes, 61 normal (33 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×5], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×2], C23 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×4], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C52C8 [×2], C52C8 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×2], C22×C10 [×2], C86D4, C2×C52C8 [×2], C2×C52C8 [×2], C4.Dic5 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, C4×C52C8, C203C8, C20.55D4 [×2], C2×C4.Dic5 [×2], D4×C20, C207M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, M4(2) [×2], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C2×M4(2), C8○D4, C2×Dic5 [×6], C22×D5, C86D4, C4.Dic5 [×2], D4×D5, D42D5, C22×Dic5, C2×C4.Dic5, D4×Dic5, D4.Dic5, C207M4(2)

Smallest permutation representation of C207M4(2)
On 160 points
Generators in S160
(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)
(1 131 141 78 37 50 114 96)(2 130 142 77 38 49 115 95)(3 129 143 76 39 48 116 94)(4 128 144 75 40 47 117 93)(5 127 145 74 21 46 118 92)(6 126 146 73 22 45 119 91)(7 125 147 72 23 44 120 90)(8 124 148 71 24 43 101 89)(9 123 149 70 25 42 102 88)(10 122 150 69 26 41 103 87)(11 121 151 68 27 60 104 86)(12 140 152 67 28 59 105 85)(13 139 153 66 29 58 106 84)(14 138 154 65 30 57 107 83)(15 137 155 64 31 56 108 82)(16 136 156 63 32 55 109 81)(17 135 157 62 33 54 110 100)(18 134 158 61 34 53 111 99)(19 133 159 80 35 52 112 98)(20 132 160 79 36 51 113 97)
(1 22)(2 33)(3 24)(4 35)(5 26)(6 37)(7 28)(8 39)(9 30)(10 21)(11 32)(12 23)(13 34)(14 25)(15 36)(16 27)(17 38)(18 29)(19 40)(20 31)(41 46)(42 57)(43 48)(44 59)(45 50)(47 52)(49 54)(51 56)(53 58)(55 60)(61 66)(62 77)(63 68)(64 79)(65 70)(67 72)(69 74)(71 76)(73 78)(75 80)(81 86)(82 97)(83 88)(84 99)(85 90)(87 92)(89 94)(91 96)(93 98)(95 100)(101 143)(102 154)(103 145)(104 156)(105 147)(106 158)(107 149)(108 160)(109 151)(110 142)(111 153)(112 144)(113 155)(114 146)(115 157)(116 148)(117 159)(118 150)(119 141)(120 152)(121 136)(122 127)(123 138)(124 129)(125 140)(126 131)(128 133)(130 135)(132 137)(134 139)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A···8H8I8J8K8L10A···10F10G···10N20A···20H20I···20X
order1222224444444444558···8888810···1010···1020···2020···20
size11114411112222442210···10202020202···24···42···24···4

68 irreducible representations

dim11111111122222222222444
type+++++++++--+-+-
imageC1C2C2C2C2C2C4C4C4D4D5M4(2)C4○D4D10Dic5Dic5D10Dic5C8○D4C4.Dic5D4×D5D42D5D4.Dic5
kernelC207M4(2)C4×C52C8C203C8C20.55D4C2×C4.Dic5D4×C20C5×C22⋊C4C5×C4⋊C4D4×C10C52C8C4×D4C20C20C42C22⋊C4C4⋊C4C22×C4C2×D4C10C4C4C4C2
# reps111221422224224242416224

Matrix representation of C207M4(2) in GL4(𝔽41) generated by

10000
143700
00405
00161
,
301100
11100
00380
0073
,
40000
39100
00405
0001
G:=sub<GL(4,GF(41))| [10,14,0,0,0,37,0,0,0,0,40,16,0,0,5,1],[30,1,0,0,11,11,0,0,0,0,38,7,0,0,0,3],[40,39,0,0,0,1,0,0,0,0,40,0,0,0,5,1] >;

C207M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes_7M_4(2)
% in TeX

G:=Group("C20:7M4(2)");
// GroupNames label

G:=SmallGroup(320,639);
// by ID

G=gap.SmallGroup(320,639);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,758,219,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^-1,c*a*c=a^11,c*b*c=b^5>;
// generators/relations

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