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G = (C2×C20).31D4order 320 = 26·5

5th non-split extension by C2×C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).31D4, (C2×C4).20D20, (C22×D5)⋊1Q8, C10.6C22≀C2, C51(C23⋊Q8), C22.42(Q8×D5), (C2×Dic5).20D4, C22.82(C2×D20), (C22×C4).72D10, C22.157(D4×D5), C2.9(C22⋊D20), C2.8(D102Q8), C2.7(C4.D20), (C22×Dic10)⋊1C2, C2.C4212D5, C10.26(C22⋊Q8), (C23×D5).5C22, C10.10C425C2, C2.10(D10⋊Q8), C10.20(C4.4D4), C22.90(C4○D20), (C22×C20).17C22, C23.361(C22×D5), C22.88(D42D5), (C22×C10).298C23, C2.10(Dic5.5D4), (C22×Dic5).20C22, (C2×C10).69(C2×Q8), (C2×C10).206(C2×D4), (C2×D10⋊C4).8C2, (C2×C10).60(C4○D4), (C5×C2.C42)⋊10C2, SmallGroup(320,300)

Series: Derived Chief Lower central Upper central

C1C22×C10 — (C2×C20).31D4
C1C5C10C2×C10C22×C10C23×D5C2×D10⋊C4 — (C2×C20).31D4
C5C22×C10 — (C2×C20).31D4
C1C23C2.C42

Generators and relations for (C2×C20).31D4
 G = < a,b,c,d | a2=b20=c4=1, d2=b10, cbc-1=ab=ba, ac=ca, ad=da, dbd-1=b-1, dcd-1=b10c-1 >

Subgroups: 886 in 202 conjugacy classes, 61 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×9], C22 [×3], C22 [×4], C22 [×10], C5, C2×C4 [×2], C2×C4 [×19], Q8 [×8], C23, C23 [×8], D5 [×2], C10 [×3], C10 [×4], C22⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×3], C2×Q8 [×6], C24, Dic5 [×5], C20 [×4], D10 [×10], C2×C10 [×3], C2×C10 [×4], C2.C42, C2.C42 [×2], C2×C22⋊C4 [×3], C22×Q8, Dic10 [×8], C2×Dic5 [×4], C2×Dic5 [×7], C2×C20 [×2], C2×C20 [×8], C22×D5 [×2], C22×D5 [×6], C22×C10, C23⋊Q8, D10⋊C4 [×6], C2×Dic10 [×6], C22×Dic5, C22×Dic5 [×2], C22×C20, C22×C20 [×2], C23×D5, C10.10C42 [×2], C5×C2.C42, C2×D10⋊C4, C2×D10⋊C4 [×2], C22×Dic10, (C2×C20).31D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×2], C23, D5, C2×D4 [×3], C2×Q8, C4○D4 [×3], D10 [×3], C22≀C2, C22⋊Q8 [×3], C4.4D4 [×3], D20 [×2], C22×D5, C23⋊Q8, C2×D20, C4○D20 [×2], D4×D5 [×2], D42D5, Q8×D5, C4.D20, C22⋊D20, Dic5.5D4 [×2], D10⋊Q8 [×2], D102Q8, (C2×C20).31D4

Smallest permutation representation of (C2×C20).31D4
On 160 points
Generators in S160
(1 69)(2 70)(3 71)(4 72)(5 73)(6 74)(7 75)(8 76)(9 77)(10 78)(11 79)(12 80)(13 61)(14 62)(15 63)(16 64)(17 65)(18 66)(19 67)(20 68)(21 153)(22 154)(23 155)(24 156)(25 157)(26 158)(27 159)(28 160)(29 141)(30 142)(31 143)(32 144)(33 145)(34 146)(35 147)(36 148)(37 149)(38 150)(39 151)(40 152)(41 101)(42 102)(43 103)(44 104)(45 105)(46 106)(47 107)(48 108)(49 109)(50 110)(51 111)(52 112)(53 113)(54 114)(55 115)(56 116)(57 117)(58 118)(59 119)(60 120)(81 124)(82 125)(83 126)(84 127)(85 128)(86 129)(87 130)(88 131)(89 132)(90 133)(91 134)(92 135)(93 136)(94 137)(95 138)(96 139)(97 140)(98 121)(99 122)(100 123)
(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)
(1 117 95 26)(2 58 96 159)(3 119 97 28)(4 60 98 141)(5 101 99 30)(6 42 100 143)(7 103 81 32)(8 44 82 145)(9 105 83 34)(10 46 84 147)(11 107 85 36)(12 48 86 149)(13 109 87 38)(14 50 88 151)(15 111 89 40)(16 52 90 153)(17 113 91 22)(18 54 92 155)(19 115 93 24)(20 56 94 157)(21 64 112 133)(23 66 114 135)(25 68 116 137)(27 70 118 139)(29 72 120 121)(31 74 102 123)(33 76 104 125)(35 78 106 127)(37 80 108 129)(39 62 110 131)(41 122 142 73)(43 124 144 75)(45 126 146 77)(47 128 148 79)(49 130 150 61)(51 132 152 63)(53 134 154 65)(55 136 156 67)(57 138 158 69)(59 140 160 71)
(1 36 11 26)(2 35 12 25)(3 34 13 24)(4 33 14 23)(5 32 15 22)(6 31 16 21)(7 30 17 40)(8 29 18 39)(9 28 19 38)(10 27 20 37)(41 134 51 124)(42 133 52 123)(43 132 53 122)(44 131 54 121)(45 130 55 140)(46 129 56 139)(47 128 57 138)(48 127 58 137)(49 126 59 136)(50 125 60 135)(61 156 71 146)(62 155 72 145)(63 154 73 144)(64 153 74 143)(65 152 75 142)(66 151 76 141)(67 150 77 160)(68 149 78 159)(69 148 79 158)(70 147 80 157)(81 101 91 111)(82 120 92 110)(83 119 93 109)(84 118 94 108)(85 117 95 107)(86 116 96 106)(87 115 97 105)(88 114 98 104)(89 113 99 103)(90 112 100 102)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim1111122222222444
type+++++++-++++--
imageC1C2C2C2C2D4D4Q8D5C4○D4D10D20C4○D20D4×D5D42D5Q8×D5
kernel(C2×C20).31D4C10.10C42C5×C2.C42C2×D10⋊C4C22×Dic10C2×Dic5C2×C20C22×D5C2.C42C2×C10C22×C4C2×C4C22C22C22C22
# reps12131422266816422

Matrix representation of (C2×C20).31D4 in GL6(𝔽41)

100000
010000
001000
000100
0000400
0000040
,
14300000
1190000
0034100
0033100
0000320
0000219
,
2410000
40170000
00112800
00223000
0000162
00001625
,
2410000
38170000
0014200
0052700
0000162
00001525

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[14,11,0,0,0,0,30,9,0,0,0,0,0,0,34,33,0,0,0,0,1,1,0,0,0,0,0,0,32,21,0,0,0,0,0,9],[24,40,0,0,0,0,1,17,0,0,0,0,0,0,11,22,0,0,0,0,28,30,0,0,0,0,0,0,16,16,0,0,0,0,2,25],[24,38,0,0,0,0,1,17,0,0,0,0,0,0,14,5,0,0,0,0,2,27,0,0,0,0,0,0,16,15,0,0,0,0,2,25] >;

(C2×C20).31D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})._{31}D_4
% in TeX

G:=Group("(C2xC20).31D4");
// GroupNames label

G:=SmallGroup(320,300);
// by ID

G=gap.SmallGroup(320,300);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,926,387,268,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=b^10,c*b*c^-1=a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b^-1,d*c*d^-1=b^10*c^-1>;
// generators/relations

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