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G = C206M4(2)  order 320 = 26·5

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C206M4(2), C42.204D10, C4⋊C814D5, C52C817D4, C56(C86D4), C41(C8⋊D5), (C4×D20).9C2, C4.208(D4×D5), C10.78(C4×D4), (C2×D20).25C4, C20.367(C2×D4), (C2×C8).183D10, C4⋊Dic5.31C4, D101C824C2, C10.53(C8○D4), (C4×C20).64C22, D10⋊C4.22C4, C20.336(C4○D4), C2.8(D208C4), (C2×C40).214C22, (C2×C20).835C23, C4.56(Q82D5), C10.43(C2×M4(2)), C2.13(D20.2C4), (C5×C4⋊C8)⋊19C2, (C4×C52C8)⋊5C2, (C2×C4).73(C4×D5), (C2×C8⋊D5)⋊22C2, C2.14(C2×C8⋊D5), C22.113(C2×C4×D5), (C2×C20).332(C2×C4), (C2×C4×D5).232C22, (C2×Dic5).25(C2×C4), (C22×D5).21(C2×C4), (C2×C4).777(C22×D5), (C2×C10).191(C22×C4), (C2×C52C8).315C22, SmallGroup(320,465)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C206M4(2)
C1C5C10C20C2×C20C2×C4×D5C4×D20 — C206M4(2)
C5C2×C10 — C206M4(2)
C1C2×C4C4⋊C8

Generators and relations for C206M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a9, cac=a-1, cbc=b5 >

Subgroups: 446 in 122 conjugacy classes, 53 normal (31 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×6], C5, C8 [×5], C2×C4 [×3], C2×C4 [×6], D4 [×2], C23 [×2], D5 [×2], C10 [×3], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×2], C2×D4, Dic5 [×2], C20 [×2], C20 [×2], C20, D10 [×6], C2×C10, C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C52C8 [×2], C52C8, C40 [×2], C4×D5 [×4], D20 [×2], C2×Dic5 [×2], C2×C20 [×3], C22×D5 [×2], C86D4, C8⋊D5 [×4], C2×C52C8 [×2], C4⋊Dic5, D10⋊C4 [×2], C4×C20, C2×C40 [×2], C2×C4×D5 [×2], C2×D20, C4×C52C8, D101C8 [×2], C5×C4⋊C8, C4×D20, C2×C8⋊D5 [×2], C206M4(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, D10 [×3], C4×D4, C2×M4(2), C8○D4, C4×D5 [×2], C22×D5, C86D4, C8⋊D5 [×2], C2×C4×D5, D4×D5, Q82D5, D208C4, C2×C8⋊D5, D20.2C4, C206M4(2)

Smallest permutation representation of C206M4(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)
(1 128 147 87 36 50 111 69)(2 137 148 96 37 59 112 78)(3 126 149 85 38 48 113 67)(4 135 150 94 39 57 114 76)(5 124 151 83 40 46 115 65)(6 133 152 92 21 55 116 74)(7 122 153 81 22 44 117 63)(8 131 154 90 23 53 118 72)(9 140 155 99 24 42 119 61)(10 129 156 88 25 51 120 70)(11 138 157 97 26 60 101 79)(12 127 158 86 27 49 102 68)(13 136 159 95 28 58 103 77)(14 125 160 84 29 47 104 66)(15 134 141 93 30 56 105 75)(16 123 142 82 31 45 106 64)(17 132 143 91 32 54 107 73)(18 121 144 100 33 43 108 62)(19 130 145 89 34 52 109 71)(20 139 146 98 35 41 110 80)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 31)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)(41 137)(42 136)(43 135)(44 134)(45 133)(46 132)(47 131)(48 130)(49 129)(50 128)(51 127)(52 126)(53 125)(54 124)(55 123)(56 122)(57 121)(58 140)(59 139)(60 138)(61 95)(62 94)(63 93)(64 92)(65 91)(66 90)(67 89)(68 88)(69 87)(70 86)(71 85)(72 84)(73 83)(74 82)(75 81)(76 100)(77 99)(78 98)(79 97)(80 96)(102 120)(103 119)(104 118)(105 117)(106 116)(107 115)(108 114)(109 113)(110 112)(141 153)(142 152)(143 151)(144 150)(145 149)(146 148)(154 160)(155 159)(156 158)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J5A5B8A8B8C8D8E···8L10A···10F20A···20H20I···20P40A···40P
order12222244444444445588888···810···1020···2020···2040···40
size1111202011112222202022444410···102···22···24···44···4

68 irreducible representations

dim111111111222222222444
type++++++++++++
imageC1C2C2C2C2C2C4C4C4D4D5M4(2)C4○D4D10D10C8○D4C4×D5C8⋊D5D4×D5Q82D5D20.2C4
kernelC206M4(2)C4×C52C8D101C8C5×C4⋊C8C4×D20C2×C8⋊D5C4⋊Dic5D10⋊C4C2×D20C52C8C4⋊C8C20C20C42C2×C8C10C2×C4C4C4C4C2
# reps1121122422242244816224

Matrix representation of C206M4(2) in GL6(𝔽41)

110000
560000
00211800
00212000
000010
000001
,
670000
36350000
0040000
0004000
000015
00001840
,
670000
36350000
001000
00254000
000010
00001640

G:=sub<GL(6,GF(41))| [1,5,0,0,0,0,1,6,0,0,0,0,0,0,21,21,0,0,0,0,18,20,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,18,0,0,0,0,5,40],[6,36,0,0,0,0,7,35,0,0,0,0,0,0,1,25,0,0,0,0,0,40,0,0,0,0,0,0,1,16,0,0,0,0,0,40] >;

C206M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes_6M_4(2)
% in TeX

G:=Group("C20:6M4(2)");
// GroupNames label

G:=SmallGroup(320,465);
// by ID

G=gap.SmallGroup(320,465);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,120,758,219,58,136,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=b^5>;
// generators/relations

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