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G = C42.126D10order 320 = 26·5

126th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.126D10, C10.102- 1+4, C10.1092+ 1+4, (C4×Q8)⋊7D5, Q89(C4×D5), (Q8×C20)⋊9C2, (C4×D20)⋊37C2, D2025(C2×C4), Q82D55C4, (Q8×Dic5)⋊9C2, C4⋊C4.325D10, D208C417C2, C42⋊D516C2, C20.72(C22×C4), C10.48(C23×C4), (C2×Q8).202D10, C2.4(D48D10), (C2×C20).497C23, (C4×C20).170C22, (C2×C10).118C24, D10.20(C22×C4), C22.37(C23×D5), (C2×D20).270C22, C4⋊Dic5.368C22, (Q8×C10).218C22, (C4×Dic5).93C22, Dic5.41(C22×C4), C2.3(Q8.10D10), C55(C23.33C23), (C2×Dic5).224C23, (C22×D5).187C23, D10⋊C4.163C22, C10.D4.138C22, C4.37(C2×C4×D5), (D5×C4⋊C4)⋊17C2, (C4×D5)⋊5(C2×C4), (C5×Q8)⋊21(C2×C4), C2.29(D5×C22×C4), (C2×C4×D5).79C22, (C2×Q82D5).6C2, (C5×C4⋊C4).346C22, (C2×C4).654(C22×D5), SmallGroup(320,1246)

Series: Derived Chief Lower central Upper central

C1C10 — C42.126D10
C1C5C10C2×C10C22×D5C2×C4×D5D5×C4⋊C4 — C42.126D10
C5C10 — C42.126D10
C1C22C4×Q8

Generators and relations for C42.126D10
 G = < a,b,c,d | a4=b4=1, c10=d2=a2b2, ab=ba, cac-1=dad-1=a-1, bc=cb, dbd-1=a2b, dcd-1=c9 >

Subgroups: 958 in 294 conjugacy classes, 151 normal (22 characteristic)
C1, C2 [×3], C2 [×6], C4 [×6], C4 [×10], C22, C22 [×12], C5, C2×C4, C2×C4 [×6], C2×C4 [×23], D4 [×12], Q8 [×4], C23 [×3], D5 [×6], C10 [×3], C42 [×3], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×7], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], Dic5 [×2], Dic5 [×4], C20 [×6], C20 [×4], D10 [×6], D10 [×6], C2×C10, C2×C4⋊C4 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8, C4×Q8, C2×C4○D4, C4×D5 [×12], C4×D5 [×6], D20 [×12], C2×Dic5 [×2], C2×Dic5 [×3], C2×C20, C2×C20 [×6], C5×Q8 [×4], C22×D5 [×3], C23.33C23, C4×Dic5 [×3], C10.D4, C10.D4 [×3], C4⋊Dic5 [×3], D10⋊C4 [×6], C4×C20 [×3], C5×C4⋊C4 [×3], C2×C4×D5 [×9], C2×D20 [×3], Q82D5 [×8], Q8×C10, C42⋊D5 [×3], C4×D20 [×3], D5×C4⋊C4 [×3], D208C4 [×3], Q8×Dic5, Q8×C20, C2×Q82D5, C42.126D10
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, D10 [×7], C23×C4, 2+ 1+4, 2- 1+4, C4×D5 [×4], C22×D5 [×7], C23.33C23, C2×C4×D5 [×6], C23×D5, D5×C22×C4, Q8.10D10, D48D10, C42.126D10

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

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

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

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

74 conjugacy classes

class 1 2A2B2C2D···2I4A···4N4O···4X5A5B10A···10F20A···20H20I···20AF
order12222···24···44···45510···1020···2020···20
size111110···102···210···10222···22···24···4

74 irreducible representations

dim111111111222224444
type+++++++++++++-+
imageC1C2C2C2C2C2C2C2C4D5D10D10D10C4×D52+ 1+42- 1+4Q8.10D10D48D10
kernelC42.126D10C42⋊D5C4×D20D5×C4⋊C4D208C4Q8×Dic5Q8×C20C2×Q82D5Q82D5C4×Q8C42C4⋊C4C2×Q8Q8C10C10C2C2
# reps13333111162662161144

Matrix representation of C42.126D10 in GL6(𝔽41)

4000000
0400000
0038353529
0073146
00352936
001463438
,
3200000
0320000
0023500
0011800
0000235
0000118
,
22130000
1900000
000066
0000340
00353500
007000
,
2290000
19190000
00203800
00382100
00002038
00003821

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,38,7,35,14,0,0,35,3,29,6,0,0,35,14,3,34,0,0,29,6,6,38],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,23,1,0,0,0,0,5,18,0,0,0,0,0,0,23,1,0,0,0,0,5,18],[22,19,0,0,0,0,13,0,0,0,0,0,0,0,0,0,35,7,0,0,0,0,35,0,0,0,6,34,0,0,0,0,6,0,0,0],[22,19,0,0,0,0,9,19,0,0,0,0,0,0,20,38,0,0,0,0,38,21,0,0,0,0,0,0,20,38,0,0,0,0,38,21] >;

C42.126D10 in GAP, Magma, Sage, TeX

C_4^2._{126}D_{10}
% in TeX

G:=Group("C4^2.126D10");
// GroupNames label

G:=SmallGroup(320,1246);
// by ID

G=gap.SmallGroup(320,1246);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,268,675,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=d^2=a^2*b^2,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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