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G = C42.216D10order 320 = 26·5

36th non-split extension by C42 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.216D10, C4⋊C4.77D10, C42.C23D5, (C2×C20).276D4, C20.72(C4○D4), C10.Q1641C2, C4.D20.9C2, D206C4.13C2, C10.110(C4○D8), (C2×C20).386C23, (C4×C20).116C22, C4.14(Q82D5), C10.56(C4.4D4), C2.9(C20.23D4), (C2×D20).108C22, C2.29(D4.8D10), C55(C42.78C22), (C2×Dic10).113C22, (C4×C52C8)⋊13C2, (C5×C42.C2)⋊3C2, (C2×C10).517(C2×D4), (C2×C4).112(C5⋊D4), (C5×C4⋊C4).124C22, (C2×C4).484(C22×D5), C22.190(C2×C5⋊D4), (C2×C52C8).263C22, SmallGroup(320,695)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.216D10
C1C5C10C20C2×C20C2×D20C4.D20 — C42.216D10
C5C10C2×C20 — C42.216D10
C1C22C42C42.C2

Generators and relations for C42.216D10
 G = < a,b,c,d | a4=b4=1, c10=a2b2, d2=a2b, ab=ba, cac-1=a-1b2, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 398 in 96 conjugacy classes, 39 normal (17 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×5], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×3], D4 [×2], Q8 [×2], C23, D5, C10, C10 [×2], C42, C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×2], C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×4], D10 [×3], C2×C10, C4×C8, D4⋊C4 [×2], Q8⋊C4 [×2], C4.4D4, C42.C2, C52C8 [×2], Dic10 [×2], D20 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C2×C20 [×2], C22×D5, C42.78C22, C2×C52C8 [×2], D10⋊C4 [×2], C4×C20, C5×C4⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×D20, C4×C52C8, D206C4 [×2], C10.Q16 [×2], C4.D20, C5×C42.C2, C42.216D10
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C4○D8 [×2], C5⋊D4 [×2], C22×D5, C42.78C22, Q82D5 [×2], C2×C5⋊D4, C20.23D4, D4.8D10 [×2], C42.216D10

Smallest permutation representation of C42.216D10
On 160 points
Generators in S160
(1 103 63 155)(2 114 64 146)(3 105 65 157)(4 116 66 148)(5 107 67 159)(6 118 68 150)(7 109 69 141)(8 120 70 152)(9 111 71 143)(10 102 72 154)(11 113 73 145)(12 104 74 156)(13 115 75 147)(14 106 76 158)(15 117 77 149)(16 108 78 160)(17 119 79 151)(18 110 80 142)(19 101 61 153)(20 112 62 144)(21 86 121 42)(22 97 122 53)(23 88 123 44)(24 99 124 55)(25 90 125 46)(26 81 126 57)(27 92 127 48)(28 83 128 59)(29 94 129 50)(30 85 130 41)(31 96 131 52)(32 87 132 43)(33 98 133 54)(34 89 134 45)(35 100 135 56)(36 91 136 47)(37 82 137 58)(38 93 138 49)(39 84 139 60)(40 95 140 51)
(1 24 73 134)(2 135 74 25)(3 26 75 136)(4 137 76 27)(5 28 77 138)(6 139 78 29)(7 30 79 140)(8 121 80 31)(9 32 61 122)(10 123 62 33)(11 34 63 124)(12 125 64 35)(13 36 65 126)(14 127 66 37)(15 38 67 128)(16 129 68 39)(17 40 69 130)(18 131 70 21)(19 22 71 132)(20 133 72 23)(41 119 95 141)(42 142 96 120)(43 101 97 143)(44 144 98 102)(45 103 99 145)(46 146 100 104)(47 105 81 147)(48 148 82 106)(49 107 83 149)(50 150 84 108)(51 109 85 151)(52 152 86 110)(53 111 87 153)(54 154 88 112)(55 113 89 155)(56 156 90 114)(57 115 91 157)(58 158 92 116)(59 117 93 159)(60 160 94 118)
(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 10 124 23 73 62 34 133)(2 132 35 61 74 22 125 9)(3 8 126 21 75 80 36 131)(4 130 37 79 76 40 127 7)(5 6 128 39 77 78 38 129)(11 20 134 33 63 72 24 123)(12 122 25 71 64 32 135 19)(13 18 136 31 65 70 26 121)(14 140 27 69 66 30 137 17)(15 16 138 29 67 68 28 139)(41 82 151 158 95 48 109 116)(42 115 110 47 96 157 152 81)(43 100 153 156 97 46 111 114)(44 113 112 45 98 155 154 99)(49 94 159 150 83 60 117 108)(50 107 118 59 84 149 160 93)(51 92 141 148 85 58 119 106)(52 105 120 57 86 147 142 91)(53 90 143 146 87 56 101 104)(54 103 102 55 88 145 144 89)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D4A···4F4G4H4I5A5B8A···8H10A···10F20A···20L20M···20T
order122224···4444558···810···1020···2020···20
size1111402···288402210···102···24···48···8

50 irreducible representations

dim111111222222244
type+++++++++++
imageC1C2C2C2C2C2D4D5C4○D4D10D10C4○D8C5⋊D4Q82D5D4.8D10
kernelC42.216D10C4×C52C8D206C4C10.Q16C4.D20C5×C42.C2C2×C20C42.C2C20C42C4⋊C4C10C2×C4C4C2
# reps112211224248848

Matrix representation of C42.216D10 in GL6(𝔽41)

3200000
0320000
0040000
0004000
00003225
000059
,
010000
4000000
0040000
0004000
00004021
0000371
,
12120000
12290000
00243800
003300
0000028
0000190
,
12290000
12120000
00244000
0031700
00003013
0000190

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,5,0,0,0,0,25,9],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,37,0,0,0,0,21,1],[12,12,0,0,0,0,12,29,0,0,0,0,0,0,24,3,0,0,0,0,38,3,0,0,0,0,0,0,0,19,0,0,0,0,28,0],[12,12,0,0,0,0,29,12,0,0,0,0,0,0,24,3,0,0,0,0,40,17,0,0,0,0,0,0,30,19,0,0,0,0,13,0] >;

C42.216D10 in GAP, Magma, Sage, TeX

C_4^2._{216}D_{10}
% in TeX

G:=Group("C4^2.216D10");
// GroupNames label

G:=SmallGroup(320,695);
// by ID

G=gap.SmallGroup(320,695);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,219,100,1123,297,136,12550]);
// Polycyclic

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

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