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G = C42.100D10order 320 = 26·5

100th non-split extension by C42 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.100D10, C10.1002+ 1+4, (C4×D20)⋊12C2, C204D44C2, D10⋊D45C2, C207D442C2, C4⋊C4.275D10, C20.6Q85C2, C42⋊C219D5, (C2×C10).79C24, (C4×C20).30C22, D10.13D45C2, C4.121(C4○D20), C20.237(C4○D4), (C2×C20).152C23, C22⋊C4.103D10, (C2×D20).27C22, (C22×C4).200D10, C2.12(D48D10), C23.90(C22×D5), C4⋊Dic5.294C22, (C2×Dic5).32C23, C10.D4.4C22, (C22×D5).27C23, C22.108(C23×D5), D10⋊C4.64C22, (C22×C20).309C22, (C22×C10).149C23, C51(C22.34C24), C2.38(C2×C4○D20), C10.35(C2×C4○D4), (C2×C4×D5).247C22, (C5×C42⋊C2)⋊21C2, (C5×C4⋊C4).315C22, (C2×C4).280(C22×D5), (C2×C5⋊D4).12C22, (C5×C22⋊C4).118C22, SmallGroup(320,1207)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C42.100D10
C1C5C10C2×C10C22×D5C2×D20C4×D20 — C42.100D10
C5C2×C10 — C42.100D10
C1C22C42⋊C2

Generators and relations for C42.100D10
 G = < a,b,c,d | a4=b4=1, c10=a2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, cbc-1=dbd-1=a2b, dcd-1=b2c9 >

Subgroups: 1022 in 240 conjugacy classes, 95 normal (23 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], C23, C23 [×4], D5 [×4], C10, C10 [×2], C10, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4 [×10], Dic5 [×4], C20 [×2], C20 [×5], D10 [×12], C2×C10, C2×C10 [×3], C42⋊C2, C4×D4 [×2], C4⋊D4 [×6], C22.D4 [×4], C42.C2, C41D4, C4×D5 [×4], D20 [×8], C2×Dic5 [×4], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C22×D5 [×4], C22×C10, C22.34C24, C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×8], C4×C20 [×2], C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×C4×D5 [×4], C2×D20 [×6], C2×C5⋊D4 [×4], C22×C20, C20.6Q8, C4×D20 [×2], C204D4, D10⋊D4 [×4], D10.13D4 [×4], C207D4 [×2], C5×C42⋊C2, C42.100D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.34C24, C4○D20 [×2], C23×D5, C2×C4○D20, D48D10 [×2], C42.100D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4F4G4H4I4J4K4L4M5A5B10A···10F10G10H10I10J20A···20H20I···20AB
order1222222224···444444445510···101010101020···2020···20
size11114202020202···244420202020222···244442···24···4

62 irreducible representations

dim11111111222222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2D5C4○D4D10D10D10D10C4○D202+ 1+4D48D10
kernelC42.100D10C20.6Q8C4×D20C204D4D10⋊D4D10.13D4C207D4C5×C42⋊C2C42⋊C2C20C42C22⋊C4C4⋊C4C22×C4C4C10C2
# reps112144212444421628

Matrix representation of C42.100D10 in GL6(𝔽41)

100000
010000
0023200
00373900
0000232
00003739
,
900000
090000
0022030
0002203
0030190
0003019
,
190000
0400000
000067
0000350
00353400
006000
,
40320000
2310000
00001630
00001225
00251100
00291600

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,37,0,0,0,0,32,39,0,0,0,0,0,0,2,37,0,0,0,0,32,39],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,22,0,3,0,0,0,0,22,0,3,0,0,3,0,19,0,0,0,0,3,0,19],[1,0,0,0,0,0,9,40,0,0,0,0,0,0,0,0,35,6,0,0,0,0,34,0,0,0,6,35,0,0,0,0,7,0,0,0],[40,23,0,0,0,0,32,1,0,0,0,0,0,0,0,0,25,29,0,0,0,0,11,16,0,0,16,12,0,0,0,0,30,25,0,0] >;

C42.100D10 in GAP, Magma, Sage, TeX

C_4^2._{100}D_{10}
% in TeX

G:=Group("C4^2.100D10");
// GroupNames label

G:=SmallGroup(320,1207);
// by ID

G=gap.SmallGroup(320,1207);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,232,100,675,297,136,12550]);
// Polycyclic

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

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