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G = C20.37C42order 320 = 26·5

7th non-split extension by C20 of C42 acting via C42/C22=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.37C42, M4(2)⋊5Dic5, C40.81(C2×C4), C408C427C2, C4.7(C4×Dic5), (C8×Dic5)⋊30C2, (C2×C8).275D10, (C5×M4(2))⋊9C4, C4⋊Dic5.33C4, C8.12(C2×Dic5), C23.28(C4×D5), C10.55(C8○D4), C59(C82M4(2)), (C2×C10).29C42, C10.46(C2×C42), C23.D5.18C4, C22.7(C4×Dic5), C20.236(C22×C4), (C2×C40).233C22, (C2×C20).865C23, (C22×C4).347D10, (C10×M4(2)).6C2, (C2×M4(2)).19D5, C4.35(C22×Dic5), C2.5(D20.2C4), (C22×C20).181C22, (C4×Dic5).316C22, C23.21D10.18C2, C4.115(C2×C4×D5), (C2×C52C8)⋊11C4, (C2×C4).83(C4×D5), C2.14(C2×C4×Dic5), C22.63(C2×C4×D5), C52C8.46(C2×C4), (C2×C20).274(C2×C4), (C2×C4).47(C2×Dic5), (C22×C52C8).11C2, (C2×C4).807(C22×D5), (C2×C10).236(C22×C4), (C22×C10).133(C2×C4), (C2×C52C8).357C22, (C2×Dic5).112(C2×C4), SmallGroup(320,749)

Series: Derived Chief Lower central Upper central

C1C10 — C20.37C42
C1C5C10C20C2×C20C2×C52C8C22×C52C8 — C20.37C42
C5C10 — C20.37C42
C1C2×C4C2×M4(2)

Generators and relations for C20.37C42
 G = < a,b,c | a20=1, b4=c4=a10, bab-1=a9, cac-1=a11, bc=cb >

Subgroups: 286 in 130 conjugacy classes, 87 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C5, C8 [×4], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×C8 [×6], M4(2) [×4], C22×C4, Dic5 [×4], C20 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×C8 [×2], C8⋊C4 [×2], C42⋊C2, C22×C8, C2×M4(2), C52C8 [×4], C40 [×4], C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×4], C22×C10, C82M4(2), C2×C52C8 [×2], C2×C52C8 [×4], C4×Dic5 [×2], C4⋊Dic5 [×2], C23.D5 [×2], C2×C40 [×2], C5×M4(2) [×4], C22×C20, C8×Dic5 [×2], C408C4 [×2], C22×C52C8, C23.21D10, C10×M4(2), C20.37C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, D5, C42 [×4], C22×C4 [×3], Dic5 [×4], D10 [×3], C2×C42, C8○D4 [×2], C4×D5 [×4], C2×Dic5 [×6], C22×D5, C82M4(2), C4×Dic5 [×4], C2×C4×D5 [×2], C22×Dic5, D20.2C4 [×2], C2×C4×Dic5, C20.37C42

Smallest permutation representation of C20.37C42
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 51 6 56 11 41 16 46)(2 60 7 45 12 50 17 55)(3 49 8 54 13 59 18 44)(4 58 9 43 14 48 19 53)(5 47 10 52 15 57 20 42)(21 64 36 79 31 74 26 69)(22 73 37 68 32 63 27 78)(23 62 38 77 33 72 28 67)(24 71 39 66 34 61 29 76)(25 80 40 75 35 70 30 65)(81 113 86 118 91 103 96 108)(82 102 87 107 92 112 97 117)(83 111 88 116 93 101 98 106)(84 120 89 105 94 110 99 115)(85 109 90 114 95 119 100 104)(121 148 136 143 131 158 126 153)(122 157 137 152 132 147 127 142)(123 146 138 141 133 156 128 151)(124 155 139 150 134 145 129 160)(125 144 140 159 135 154 130 149)
(1 73 95 131 11 63 85 121)(2 64 96 122 12 74 86 132)(3 75 97 133 13 65 87 123)(4 66 98 124 14 76 88 134)(5 77 99 135 15 67 89 125)(6 68 100 126 16 78 90 136)(7 79 81 137 17 69 91 127)(8 70 82 128 18 80 92 138)(9 61 83 139 19 71 93 129)(10 72 84 130 20 62 94 140)(21 103 142 45 31 113 152 55)(22 114 143 56 32 104 153 46)(23 105 144 47 33 115 154 57)(24 116 145 58 34 106 155 48)(25 107 146 49 35 117 156 59)(26 118 147 60 36 108 157 50)(27 109 148 51 37 119 158 41)(28 120 149 42 38 110 159 52)(29 111 150 53 39 101 160 43)(30 102 151 44 40 112 141 54)

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

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

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

80 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4N5A5B8A···8H8I···8P8Q8R8S8T10A···10F10G10H10I10J20A···20H20I20J20K20L40A···40P
order1222224444444···4558···88···8888810···101010101020···202020202040···40
size11112211112210···10222···25···5101010102···244442···244444···4

80 irreducible representations

dim111111111122222224
type++++++++-+
imageC1C2C2C2C2C2C4C4C4C4D5D10Dic5D10C8○D4C4×D5C4×D5D20.2C4
kernelC20.37C42C8×Dic5C408C4C22×C52C8C23.21D10C10×M4(2)C2×C52C8C4⋊Dic5C23.D5C5×M4(2)C2×M4(2)C2×C8M4(2)C22×C4C10C2×C4C23C2
# reps1221118448248281248

Matrix representation of C20.37C42 in GL5(𝔽41)

10000
032000
00900
000351
000400
,
320000
014000
001400
000152
0001026
,
400000
00100
032000
000320
000032

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,35,40,0,0,0,1,0],[32,0,0,0,0,0,14,0,0,0,0,0,14,0,0,0,0,0,15,10,0,0,0,2,26],[40,0,0,0,0,0,0,32,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,32] >;

C20.37C42 in GAP, Magma, Sage, TeX

C_{20}._{37}C_4^2
% in TeX

G:=Group("C20.37C4^2");
// GroupNames label

G:=SmallGroup(320,749);
// by ID

G=gap.SmallGroup(320,749);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,387,100,136,102,12550]);
// Polycyclic

G:=Group<a,b,c|a^20=1,b^4=c^4=a^10,b*a*b^-1=a^9,c*a*c^-1=a^11,b*c=c*b>;
// generators/relations

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