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G = C42.137D10order 320 = 26·5

137th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.137D10, C10.872- 1+4, C10.702+ 1+4, C4.4D46D5, C422D57C2, (C2×Q8).81D10, D103Q827C2, (C4×Dic10)⋊43C2, (C2×D4).107D10, C22⋊C4.71D10, Dic5⋊Q820C2, Dic54D428C2, (C2×C10).213C24, (C2×C20).629C23, (C4×C20).183C22, Dic5⋊D4.5C2, D10.12D440C2, C2.72(D46D10), C23.35(C22×D5), Dic5.17(C4○D4), Dic5.5D437C2, (D4×C10).207C22, C23.D1036C2, C4⋊Dic5.232C22, (C22×C10).43C23, (Q8×C10).122C22, (C22×D5).93C23, C22.234(C23×D5), Dic5.14D437C2, C23.D5.50C22, D10⋊C4.59C22, C23.18D1024C2, C23.11D1016C2, C58(C22.36C24), (C4×Dic5).233C22, (C2×Dic5).260C23, C10.D4.82C22, C2.48(D4.10D10), (C2×Dic10).180C22, (C22×Dic5).138C22, C2.72(D5×C4○D4), (C5×C4.4D4)⋊7C2, C10.184(C2×C4○D4), (C2×C4×D5).128C22, (C2×C4).191(C22×D5), (C2×C5⋊D4).56C22, (C5×C22⋊C4).60C22, SmallGroup(320,1341)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.137D10
C1C5C10C2×C10C2×Dic5C22×Dic5C23.11D10 — C42.137D10
C5C2×C10 — C42.137D10
C1C22C4.4D4

Generators and relations for C42.137D10
 G = < a,b,c,d | a4=b4=c10=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b, dbd-1=a2b-1, dcd-1=c-1 >

Subgroups: 734 in 216 conjugacy classes, 93 normal (91 characteristic)
C1, C2 [×3], C2 [×3], C4 [×13], C22, C22 [×9], C5, C2×C4 [×5], C2×C4 [×11], D4 [×4], Q8 [×4], C23 [×2], C23, D5, C10 [×3], C10 [×2], C42, C42 [×3], C22⋊C4 [×4], C22⋊C4 [×8], C4⋊C4 [×10], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], Dic5 [×2], Dic5 [×6], C20 [×5], D10 [×3], C2×C10, C2×C10 [×6], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8 [×3], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C422C2 [×2], C4⋊Q8, Dic10 [×3], C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×3], C2×C20 [×5], C5×D4, C5×Q8, C22×D5, C22×C10 [×2], C22.36C24, C4×Dic5 [×3], C10.D4 [×8], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×4], C4×C20, C5×C22⋊C4 [×4], C2×Dic10 [×2], C2×C4×D5, C22×Dic5 [×2], C2×C5⋊D4 [×2], D4×C10, Q8×C10, C4×Dic10, C422D5, C23.11D10, Dic5.14D4 [×2], C23.D10, Dic54D4, D10.12D4, Dic5.5D4 [×2], C23.18D10, Dic5⋊D4, Dic5⋊Q8, D103Q8, C5×C4.4D4, C42.137D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4, 2- 1+4, C22×D5 [×7], C22.36C24, C23×D5, D46D10, D5×C4○D4, D4.10D10, C42.137D10

Smallest permutation representation of C42.137D10
On 160 points
Generators in S160
(1 111 91 38)(2 71 92 144)(3 113 93 40)(4 73 94 146)(5 115 95 32)(6 75 96 148)(7 117 97 34)(8 77 98 150)(9 119 99 36)(10 79 100 142)(11 69 47 132)(12 101 48 28)(13 61 49 134)(14 103 50 30)(15 63 41 136)(16 105 42 22)(17 65 43 138)(18 107 44 24)(19 67 45 140)(20 109 46 26)(21 89 104 156)(23 81 106 158)(25 83 108 160)(27 85 110 152)(29 87 102 154)(31 126 114 53)(33 128 116 55)(35 130 118 57)(37 122 120 59)(39 124 112 51)(52 145 125 72)(54 147 127 74)(56 149 129 76)(58 141 121 78)(60 143 123 80)(62 155 135 88)(64 157 137 90)(66 159 139 82)(68 151 131 84)(70 153 133 86)
(1 133 123 28)(2 61 124 102)(3 135 125 30)(4 63 126 104)(5 137 127 22)(6 65 128 106)(7 139 129 24)(8 67 130 108)(9 131 121 26)(10 69 122 110)(11 37 85 142)(12 111 86 80)(13 39 87 144)(14 113 88 72)(15 31 89 146)(16 115 90 74)(17 33 81 148)(18 117 82 76)(19 35 83 150)(20 119 84 78)(21 94 136 53)(23 96 138 55)(25 98 140 57)(27 100 132 59)(29 92 134 51)(32 157 147 42)(34 159 149 44)(36 151 141 46)(38 153 143 48)(40 155 145 50)(41 114 156 73)(43 116 158 75)(45 118 160 77)(47 120 152 79)(49 112 154 71)(52 103 93 62)(54 105 95 64)(56 107 97 66)(58 109 99 68)(60 101 91 70)
(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 42 123 157)(2 41 124 156)(3 50 125 155)(4 49 126 154)(5 48 127 153)(6 47 128 152)(7 46 129 151)(8 45 130 160)(9 44 121 159)(10 43 122 158)(11 55 85 96)(12 54 86 95)(13 53 87 94)(14 52 88 93)(15 51 89 92)(16 60 90 91)(17 59 81 100)(18 58 82 99)(19 57 83 98)(20 56 84 97)(21 112 136 71)(22 111 137 80)(23 120 138 79)(24 119 139 78)(25 118 140 77)(26 117 131 76)(27 116 132 75)(28 115 133 74)(29 114 134 73)(30 113 135 72)(31 61 146 102)(32 70 147 101)(33 69 148 110)(34 68 149 109)(35 67 150 108)(36 66 141 107)(37 65 142 106)(38 64 143 105)(39 63 144 104)(40 62 145 103)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K···4O5A5B10A···10F10G10H10I10J20A···20L20M20N20O20P
order122222244444444444···45510···101010101020···2020202020
size111144202244441010101020···20222···288884···48888

50 irreducible representations

dim1111111111111122222244444
type++++++++++++++++++++--
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102+ 1+42- 1+4D46D10D5×C4○D4D4.10D10
kernelC42.137D10C4×Dic10C422D5C23.11D10Dic5.14D4C23.D10Dic54D4D10.12D4Dic5.5D4C23.18D10Dic5⋊D4Dic5⋊Q8D103Q8C5×C4.4D4C4.4D4Dic5C42C22⋊C4C2×D4C2×Q8C10C10C2C2C2
# reps1111211121111124282211444

Matrix representation of C42.137D10 in GL6(𝔽41)

900000
090000
0018351028
006231023
003712246
0016253417
,
010000
100000
00103838
000130
00013400
002828040
,
0320000
900000
002121030
002024611
007333720
0001540
,
090000
3200000
00202100
00182100
0000323
00003738

G:=sub<GL(6,GF(41))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,18,6,37,16,0,0,35,23,12,25,0,0,10,10,24,34,0,0,28,23,6,17],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,28,0,0,0,1,13,28,0,0,38,3,40,0,0,0,38,0,0,40],[0,9,0,0,0,0,32,0,0,0,0,0,0,0,21,20,7,0,0,0,21,24,33,15,0,0,0,6,37,4,0,0,30,11,20,0],[0,32,0,0,0,0,9,0,0,0,0,0,0,0,20,18,0,0,0,0,21,21,0,0,0,0,0,0,3,37,0,0,0,0,23,38] >;

C42.137D10 in GAP, Magma, Sage, TeX

C_4^2._{137}D_{10}
% in TeX

G:=Group("C4^2.137D10");
// GroupNames label

G:=SmallGroup(320,1341);
// by ID

G=gap.SmallGroup(320,1341);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,387,100,1123,346,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^-1>;
// generators/relations

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