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G = C42.56D10order 320 = 26·5

56th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.56D10, Q8⋊D59C4, (C4×Q8)⋊2D5, Q84(C4×D5), (Q8×C20)⋊2C2, C4⋊C4.252D10, (C4×D20).15C2, D20.31(C2×C4), (C2×C20).258D4, C10.104(C4×D4), C20.59(C4○D4), C4.41(C4○D20), Q8⋊Dic511C2, C10.D833C2, C57(SD16⋊C4), (C4×C20).97C22, C20.61(C22×C4), (C2×Q8).159D10, C42.D57C2, D206C4.10C2, C2.4(D4⋊D10), (C2×C20).346C23, C10.111(C8⋊C22), C2.3(C20.C23), (C2×D20).247C22, C10.87(C8.C22), C4⋊Dic5.331C22, (Q8×C10).194C22, C4.26(C2×C4×D5), C52C810(C2×C4), (C5×Q8)⋊18(C2×C4), C2.20(C4×C5⋊D4), (C2×Q8⋊D5).4C2, (C2×C10).477(C2×D4), C22.80(C2×C5⋊D4), (C2×C4).221(C5⋊D4), (C5×C4⋊C4).283C22, (C2×C4).446(C22×D5), (C2×C52C8).100C22, SmallGroup(320,653)

Series: Derived Chief Lower central Upper central

C1C20 — C42.56D10
C1C5C10C2×C10C2×C20C2×D20C2×Q8⋊D5 — C42.56D10
C5C10C20 — C42.56D10
C1C22C42C4×Q8

Generators and relations for C42.56D10
 G = < a,b,c,d | a4=b4=1, c10=b2, d2=b, ab=ba, cac-1=dad-1=ab2, cbc-1=b-1, bd=db, dcd-1=b-1c9 >

Subgroups: 454 in 120 conjugacy classes, 51 normal (39 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×3], C2×C4 [×3], C2×C4 [×5], D4 [×3], Q8 [×2], Q8, C23, D5 [×2], C10 [×3], C42, C42, C22⋊C4, C4⋊C4, C4⋊C4 [×2], C2×C8 [×2], SD16 [×4], C22×C4, C2×D4, C2×Q8, Dic5, C20 [×2], C20 [×5], D10 [×4], C2×C10, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C52C8 [×2], C52C8, C4×D5 [×2], D20 [×2], D20, C2×Dic5, C2×C20 [×3], C2×C20 [×2], C5×Q8 [×2], C5×Q8, C22×D5, SD16⋊C4, C2×C52C8 [×2], C4⋊Dic5, D10⋊C4, Q8⋊D5 [×4], C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×C4×D5, C2×D20, Q8×C10, C42.D5, C10.D8, D206C4, Q8⋊Dic5, C4×D20, C2×Q8⋊D5, Q8×C20, C42.56D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C8⋊C22, C8.C22, C4×D5 [×2], C5⋊D4 [×2], C22×D5, SD16⋊C4, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4, C20.C23, D4⋊D10, C42.56D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A···4F4G4H4I4J4K4L5A5B8A8B8C8D10A···10F20A···20H20I···20AF
order1222224···444444455888810···1020···2020···20
size111120202···24444202022202020202···22···24···4

62 irreducible representations

dim1111111112222222224444
type++++++++++++++-+
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10D10C5⋊D4C4×D5C4○D20C8⋊C22C8.C22C20.C23D4⋊D10
kernelC42.56D10C42.D5C10.D8D206C4Q8⋊Dic5C4×D20C2×Q8⋊D5Q8×C20Q8⋊D5C2×C20C4×Q8C20C42C4⋊C4C2×Q8C2×C4Q8C4C10C10C2C2
# reps1111111182222228881144

Matrix representation of C42.56D10 in GL6(𝔽41)

3200000
0320000
0010210
00612839
0000400
00602240
,
100000
010000
001900
00184000
00260135
00127728
,
160000
3560000
0030273533
0035351120
0021154026
001039018
,
100000
35400000
001714118
00663021
0037263615
00021323

G:=sub<GL(6,GF(41))| [32,0,0,0,0,0,0,32,0,0,0,0,0,0,1,6,0,6,0,0,0,1,0,0,0,0,21,28,40,22,0,0,0,39,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,18,26,1,0,0,9,40,0,27,0,0,0,0,13,7,0,0,0,0,5,28],[1,35,0,0,0,0,6,6,0,0,0,0,0,0,30,35,21,10,0,0,27,35,15,39,0,0,35,11,40,0,0,0,33,20,26,18],[1,35,0,0,0,0,0,40,0,0,0,0,0,0,17,6,37,0,0,0,14,6,26,2,0,0,11,30,36,13,0,0,8,21,15,23] >;

C42.56D10 in GAP, Magma, Sage, TeX

C_4^2._{56}D_{10}
% in TeX

G:=Group("C4^2.56D10");
// GroupNames label

G:=SmallGroup(320,653);
// by ID

G=gap.SmallGroup(320,653);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,232,387,58,1684,851,102,12550]);
// Polycyclic

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

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