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G = C205M4(2)  order 320 = 26·5

1st semidirect product of C20 and M4(2) acting via M4(2)/C4=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C205M4(2), Dic54M4(2), C42.203D10, C4⋊C813D5, C4.55(Q8×D5), C43(C8⋊D5), C203C814C2, (C4×D5).72D4, C4.207(D4×D5), (C4×D5).18Q8, C20.366(C2×D4), (C2×C8).182D10, C20.113(C2×Q8), (D5×C42).4C2, D10.12(C4⋊C4), (C4×C20).63C22, C54(C4⋊M4(2)), C20.8Q823C2, (C4×Dic5).10C4, C2.18(D5×M4(2)), Dic5.13(C4⋊C4), (C2×C20).834C23, (C2×C40).213C22, C10.42(C2×M4(2)), (C4×Dic5).308C22, (C5×C4⋊C8)⋊18C2, C2.9(D5×C4⋊C4), (C2×C4×D5).11C4, C10.31(C2×C4⋊C4), (C2×C4).147(C4×D5), C2.13(C2×C8⋊D5), C22.112(C2×C4×D5), (C2×C20).244(C2×C4), (C2×C8⋊D5).12C2, (C2×C4×D5).350C22, (C2×C4).776(C22×D5), (C2×C10).190(C22×C4), (C2×C52C8).200C22, (C2×Dic5).145(C2×C4), (C22×D5).103(C2×C4), SmallGroup(320,464)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C205M4(2)
C1C5C10C20C2×C20C2×C4×D5D5×C42 — C205M4(2)
C5C2×C10 — C205M4(2)
C1C2×C4C4⋊C8

Generators and relations for C205M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a-1, cac=a9, cbc=b5 >

Subgroups: 398 in 126 conjugacy classes, 61 normal (37 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C4⋊C8, C4⋊C8 [×3], C2×C42, C2×M4(2) [×2], C52C8 [×2], C40 [×2], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4⋊M4(2), C8⋊D5 [×4], C2×C52C8 [×2], C4×Dic5 [×3], C4×C20, C2×C40 [×2], C2×C4×D5 [×3], C203C8, C20.8Q8 [×2], C5×C4⋊C8, D5×C42, C2×C8⋊D5 [×2], C205M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C2×M4(2) [×2], C4×D5 [×2], C22×D5, C4⋊M4(2), C8⋊D5 [×2], C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, C2×C8⋊D5, D5×M4(2), C205M4(2)

Smallest permutation representation of C205M4(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 66 81 129 33 106 57 159)(2 65 82 128 34 105 58 158)(3 64 83 127 35 104 59 157)(4 63 84 126 36 103 60 156)(5 62 85 125 37 102 41 155)(6 61 86 124 38 101 42 154)(7 80 87 123 39 120 43 153)(8 79 88 122 40 119 44 152)(9 78 89 121 21 118 45 151)(10 77 90 140 22 117 46 150)(11 76 91 139 23 116 47 149)(12 75 92 138 24 115 48 148)(13 74 93 137 25 114 49 147)(14 73 94 136 26 113 50 146)(15 72 95 135 27 112 51 145)(16 71 96 134 28 111 52 144)(17 70 97 133 29 110 53 143)(18 69 98 132 30 109 54 142)(19 68 99 131 31 108 55 141)(20 67 100 130 32 107 56 160)
(2 10)(3 19)(4 8)(5 17)(7 15)(9 13)(12 20)(14 18)(21 25)(22 34)(24 32)(26 30)(27 39)(29 37)(31 35)(36 40)(41 53)(43 51)(44 60)(45 49)(46 58)(48 56)(50 54)(55 59)(61 101)(62 110)(63 119)(64 108)(65 117)(66 106)(67 115)(68 104)(69 113)(70 102)(71 111)(72 120)(73 109)(74 118)(75 107)(76 116)(77 105)(78 114)(79 103)(80 112)(82 90)(83 99)(84 88)(85 97)(87 95)(89 93)(92 100)(94 98)(121 147)(122 156)(123 145)(124 154)(125 143)(126 152)(127 141)(128 150)(129 159)(130 148)(131 157)(132 146)(133 155)(134 144)(135 153)(136 142)(137 151)(138 160)(139 149)(140 158)

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

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

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

68 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B8A8B8C8D8E8F8G8H10A···10F20A···20H20I···20P40A···40P
order122222444444444···4558888888810···1020···2020···2040···40
size111110101111222210···10224444202020202···22···24···44···4

68 irreducible representations

dim11111111222222222444
type+++++++-++++-
imageC1C2C2C2C2C2C4C4D4Q8D5M4(2)M4(2)D10D10C4×D5C8⋊D5D4×D5Q8×D5D5×M4(2)
kernelC205M4(2)C203C8C20.8Q8C5×C4⋊C8D5×C42C2×C8⋊D5C4×Dic5C2×C4×D5C4×D5C4×D5C4⋊C8Dic5C20C42C2×C8C2×C4C4C4C4C2
# reps112112442224424816224

Matrix representation of C205M4(2) in GL4(𝔽41) generated by

7100
40000
00320
0009
,
81400
403300
00040
00320
,
40000
7100
0010
00040
G:=sub<GL(4,GF(41))| [7,40,0,0,1,0,0,0,0,0,32,0,0,0,0,9],[8,40,0,0,14,33,0,0,0,0,0,32,0,0,40,0],[40,7,0,0,0,1,0,0,0,0,1,0,0,0,0,40] >;

C205M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes_5M_4(2)
% in TeX

G:=Group("C20:5M4(2)");
// GroupNames label

G:=SmallGroup(320,464);
// by ID

G=gap.SmallGroup(320,464);
# by ID

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

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

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