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G = C202M4(2)  order 320 = 26·5

2nd semidirect product of C20 and M4(2) acting via M4(2)/C2=C2×C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C202M4(2), C5⋊C83D4, C53(C86D4), C20⋊C86C2, (C2×D4).6F5, C2.28(D4×F5), (D4×C10).9C4, C10.28(C4×D4), C4⋊Dic5.14C4, C41(C22.F5), C23.12(C2×F5), C23.D5.7C4, C10.16(C8○D4), (D4×Dic5).18C2, Dic5.80(C2×D4), C2.16(D4.F5), C10.31(C2×M4(2)), C23.2F510C2, Dic5.59(C4○D4), C22.92(C22×F5), (C4×Dic5).195C22, (C2×Dic5).353C23, (C22×Dic5).186C22, (C4×C5⋊C8)⋊6C2, (C2×C4).81(C2×F5), (C2×C20).55(C2×C4), (C2×C5⋊C8).10C22, (C2×C22.F5)⋊5C2, C2.10(C2×C22.F5), (C22×C10).25(C2×C4), (C2×C10).77(C22×C4), (C2×Dic5).72(C2×C4), SmallGroup(320,1112)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C202M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C2×C22.F5 — C202M4(2)
C5C2×C10 — C202M4(2)
C1C22C2×D4

Generators and relations for C202M4(2)
 G = < a,b,c | a20=b8=c2=1, bab-1=a3, cac=a11, cbc=b5 >

Subgroups: 394 in 122 conjugacy classes, 48 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×6], C5, C8 [×5], C2×C4, C2×C4 [×8], 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, Dic5 [×2], Dic5 [×3], C20 [×2], C2×C10, C2×C10 [×6], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C2×M4(2) [×2], C5⋊C8 [×2], C5⋊C8 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×C20, C5×D4 [×2], C22×C10 [×2], C86D4, C4×Dic5, C4⋊Dic5, C23.D5 [×2], C2×C5⋊C8 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22×Dic5 [×2], D4×C10, C4×C5⋊C8, C20⋊C8, C23.2F5 [×2], D4×Dic5, C2×C22.F5 [×2], C202M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, M4(2) [×2], C22×C4, C2×D4, C4○D4, F5, C4×D4, C2×M4(2), C8○D4, C2×F5 [×3], C86D4, C22.F5 [×2], C22×F5, D4.F5, D4×F5, C2×C22.F5, C202M4(2)

Smallest permutation representation of C202M4(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)
(1 101 133 50 91 155 63 39)(2 108 122 53 92 142 72 22)(3 115 131 56 93 149 61 25)(4 102 140 59 94 156 70 28)(5 109 129 42 95 143 79 31)(6 116 138 45 96 150 68 34)(7 103 127 48 97 157 77 37)(8 110 136 51 98 144 66 40)(9 117 125 54 99 151 75 23)(10 104 134 57 100 158 64 26)(11 111 123 60 81 145 73 29)(12 118 132 43 82 152 62 32)(13 105 121 46 83 159 71 35)(14 112 130 49 84 146 80 38)(15 119 139 52 85 153 69 21)(16 106 128 55 86 160 78 24)(17 113 137 58 87 147 67 27)(18 120 126 41 88 154 76 30)(19 107 135 44 89 141 65 33)(20 114 124 47 90 148 74 36)
(1 6)(2 17)(3 8)(4 19)(5 10)(7 12)(9 14)(11 16)(13 18)(15 20)(21 47)(22 58)(23 49)(24 60)(25 51)(26 42)(27 53)(28 44)(29 55)(30 46)(31 57)(32 48)(33 59)(34 50)(35 41)(36 52)(37 43)(38 54)(39 45)(40 56)(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)(101 150)(102 141)(103 152)(104 143)(105 154)(106 145)(107 156)(108 147)(109 158)(110 149)(111 160)(112 151)(113 142)(114 153)(115 144)(116 155)(117 146)(118 157)(119 148)(120 159)(121 126)(122 137)(123 128)(124 139)(125 130)(127 132)(129 134)(131 136)(133 138)(135 140)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J 5 8A···8H8I8J8K8L10A10B10C10D10E10F10G20A20B
order122222444444444458···88888101010101010102020
size11114422555510102020410···1020202020444888888

38 irreducible representations

dim1111111112222444488
type++++++++++--+
imageC1C2C2C2C2C2C4C4C4D4C4○D4M4(2)C8○D4F5C2×F5C2×F5C22.F5D4.F5D4×F5
kernelC202M4(2)C4×C5⋊C8C20⋊C8C23.2F5D4×Dic5C2×C22.F5C4⋊Dic5C23.D5D4×C10C5⋊C8Dic5C20C10C2×D4C2×C4C23C4C2C2
# reps1112122422244112411

Matrix representation of C202M4(2) in GL8(𝔽41)

139000000
140000000
00100000
00010000
00000001
000040001
000004001
000000401
,
402000000
01000000
0034190000
002570000
0000371442
00000162939
0000251222
00003916427
,
402000000
01000000
00100000
0018400000
000040000
000004000
000000400
000000040

G:=sub<GL(8,GF(41))| [1,1,0,0,0,0,0,0,39,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,1,1,1],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,34,25,0,0,0,0,0,0,19,7,0,0,0,0,0,0,0,0,37,0,25,39,0,0,0,0,14,16,12,16,0,0,0,0,4,29,2,4,0,0,0,0,2,39,2,27],[40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40] >;

C202M4(2) in GAP, Magma, Sage, TeX

C_{20}\rtimes_2M_4(2)
% in TeX

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

G:=SmallGroup(320,1112);
// by ID

G=gap.SmallGroup(320,1112);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,758,219,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,b*a*b^-1=a^3,c*a*c=a^11,c*b*c=b^5>;
// generators/relations

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