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G = C20.30M4(2)  order 320 = 26·5

5th non-split extension by C20 of M4(2) acting via M4(2)/C22=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.30M4(2), C20⋊C816C2, C23.46(C2×F5), (C22×C4).23F5, (C22×C20).25C4, C54(C42.6C4), (C4×Dic5).33C4, (C2×C10).9M4(2), C22.6(C4.F5), C4.8(C22.F5), C10.C428C2, C10.29(C2×M4(2)), C23.2F5.5C2, Dic5.32(C4○D4), C22.88(C22×F5), (C22×Dic5).35C4, C10.16(C42⋊C2), (C4×Dic5).349C22, (C2×Dic5).350C23, (C22×Dic5).278C22, C2.17(D10.C23), (C2×C5⋊C8).9C22, C2.12(C2×C4.F5), (C2×C4).111(C2×F5), (C2×C4×Dic5).48C2, (C2×C20).132(C2×C4), C2.8(C2×C22.F5), (C22×C10).66(C2×C4), (C2×C10).66(C22×C4), (C2×Dic5).188(C2×C4), SmallGroup(320,1097)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.30M4(2)
C1C5C10Dic5C2×Dic5C2×C5⋊C8C10.C42 — C20.30M4(2)
C5C2×C10 — C20.30M4(2)
C1C22C22×C4

Generators and relations for C20.30M4(2)
 G = < a,b,c | a20=b8=1, c2=a10, bab-1=a3, ac=ca, cbc-1=b5 >

Subgroups: 330 in 110 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×10], C23, C10 [×3], C10 [×2], C42 [×4], C2×C8 [×4], C22×C4, C22×C4 [×2], Dic5 [×2], Dic5 [×3], C20 [×2], C20, C2×C10, C2×C10 [×2], C2×C10 [×2], C8⋊C4 [×2], C22⋊C8 [×2], C4⋊C8 [×2], C2×C42, C5⋊C8 [×4], C2×Dic5 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C22×C10, C42.6C4, C4×Dic5 [×4], C2×C5⋊C8 [×4], C22×Dic5 [×2], C22×C20, C20⋊C8 [×2], C10.C42 [×2], C23.2F5 [×2], C2×C4×Dic5, C20.30M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, M4(2) [×4], C22×C4, C4○D4 [×2], F5, C42⋊C2, C2×M4(2) [×2], C2×F5 [×3], C42.6C4, C4.F5 [×2], C22.F5 [×2], C22×F5, C2×C4.F5, D10.C23, C2×C22.F5, C20.30M4(2)

Smallest permutation representation of C20.30M4(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)
(1 103 22 157 135 46 78 97)(2 110 31 160 136 53 67 100)(3 117 40 143 137 60 76 83)(4 104 29 146 138 47 65 86)(5 111 38 149 139 54 74 89)(6 118 27 152 140 41 63 92)(7 105 36 155 121 48 72 95)(8 112 25 158 122 55 61 98)(9 119 34 141 123 42 70 81)(10 106 23 144 124 49 79 84)(11 113 32 147 125 56 68 87)(12 120 21 150 126 43 77 90)(13 107 30 153 127 50 66 93)(14 114 39 156 128 57 75 96)(15 101 28 159 129 44 64 99)(16 108 37 142 130 51 73 82)(17 115 26 145 131 58 62 85)(18 102 35 148 132 45 71 88)(19 109 24 151 133 52 80 91)(20 116 33 154 134 59 69 94)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 113 51 103)(42 114 52 104)(43 115 53 105)(44 116 54 106)(45 117 55 107)(46 118 56 108)(47 119 57 109)(48 120 58 110)(49 101 59 111)(50 102 60 112)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 156 91 146)(82 157 92 147)(83 158 93 148)(84 159 94 149)(85 160 95 150)(86 141 96 151)(87 142 97 152)(88 143 98 153)(89 144 99 154)(90 145 100 155)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N 5 8A···8H10A···10G20A···20H
order122222444444444···458···810···1020···20
size1111222222555510···10420···204···44···4

44 irreducible representations

dim11111111222444444
type++++++++-
imageC1C2C2C2C2C4C4C4C4○D4M4(2)M4(2)F5C2×F5C2×F5C22.F5C4.F5D10.C23
kernelC20.30M4(2)C20⋊C8C10.C42C23.2F5C2×C4×Dic5C4×Dic5C22×Dic5C22×C20Dic5C20C2×C10C22×C4C2×C4C23C4C22C2
# reps12221422444121444

Matrix representation of C20.30M4(2) in GL6(𝔽41)

1390000
1400000
0003200
0091900
00166919
001152219
,
31100000
36100000
00210230
003134023
001327200
002337107
,
4020000
4010000
009000
000900
000090
000009

G:=sub<GL(6,GF(41))| [1,1,0,0,0,0,39,40,0,0,0,0,0,0,0,9,16,11,0,0,32,19,6,5,0,0,0,0,9,22,0,0,0,0,19,19],[31,36,0,0,0,0,10,10,0,0,0,0,0,0,21,31,13,23,0,0,0,34,27,37,0,0,23,0,20,10,0,0,0,23,0,7],[40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,9] >;

C20.30M4(2) in GAP, Magma, Sage, TeX

C_{20}._{30}M_4(2)
% in TeX

G:=Group("C20.30M4(2)");
// GroupNames label

G:=SmallGroup(320,1097);
// by ID

G=gap.SmallGroup(320,1097);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,120,1094,136,6278,1595]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=1,c^2=a^10,b*a*b^-1=a^3,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

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