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G = Dic10.37D4order 320 = 26·5

7th non-split extension by Dic10 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: Dic10.37D4, (C2×C10)⋊2Q16, C4⋊C4.69D10, (C2×C20).79D4, C4.103(D4×D5), C22⋊Q8.5D5, C20.156(C2×D4), (C2×Q8).31D10, C10.38(C2×Q16), C53(C22⋊Q16), C10.49C22≀C2, C10.Q1638C2, C222(C5⋊Q16), (C22×C10).96D4, (C2×C20).369C23, C20.55D4.9C2, (C22×C4).130D10, C23.63(C5⋊D4), (Q8×C10).49C22, C2.17(C23⋊D10), C2.15(D4.9D10), C10.117(C8.C22), (C22×C20).173C22, (C22×Dic10).14C2, (C2×Dic10).279C22, (C2×C5⋊Q16)⋊9C2, C2.9(C2×C5⋊Q16), (C5×C22⋊Q8).4C2, (C2×C10).500(C2×D4), (C2×C4).57(C5⋊D4), (C5×C4⋊C4).116C22, (C2×C4).469(C22×D5), C22.175(C2×C5⋊D4), (C2×C52C8).117C22, SmallGroup(320,677)

Series: Derived Chief Lower central Upper central

C1C2×C20 — Dic10.37D4
C1C5C10C20C2×C20C2×Dic10C22×Dic10 — Dic10.37D4
C5C10C2×C20 — Dic10.37D4
C1C22C22×C4C22⋊Q8

Generators and relations for Dic10.37D4
 G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, cac-1=a11, ad=da, cbc-1=a5b, bd=db, dcd=a10c-1 >

Subgroups: 526 in 148 conjugacy classes, 47 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×7], C22, C22 [×2], C22 [×2], C5, C8 [×2], C2×C4 [×2], C2×C4 [×10], Q8 [×12], C23, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C4⋊C4, C2×C8 [×2], Q16 [×4], C22×C4, C22×C4, C2×Q8, C2×Q8 [×7], Dic5 [×4], C20 [×2], C20 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C22⋊Q8, C2×Q16 [×2], C22×Q8, C52C8 [×2], Dic10 [×4], Dic10 [×6], C2×Dic5 [×6], C2×C20 [×2], C2×C20 [×4], C5×Q8 [×2], C22×C10, C22⋊Q16, C2×C52C8 [×2], C5⋊Q16 [×4], C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10 [×2], C2×Dic10 [×5], C22×Dic5, C22×C20, Q8×C10, C10.Q16 [×2], C20.55D4, C2×C5⋊Q16 [×2], C5×C22⋊Q8, C22×Dic10, Dic10.37D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, Q16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×Q16, C8.C22, C5⋊D4 [×2], C22×D5, C22⋊Q16, C5⋊Q16 [×2], D4×D5 [×2], C2×C5⋊D4, C23⋊D10, C2×C5⋊Q16, D4.9D10, Dic10.37D4

Smallest permutation representation of Dic10.37D4
On 160 points
Generators in S160
(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)
(1 143 11 153)(2 142 12 152)(3 141 13 151)(4 160 14 150)(5 159 15 149)(6 158 16 148)(7 157 17 147)(8 156 18 146)(9 155 19 145)(10 154 20 144)(21 125 31 135)(22 124 32 134)(23 123 33 133)(24 122 34 132)(25 121 35 131)(26 140 36 130)(27 139 37 129)(28 138 38 128)(29 137 39 127)(30 136 40 126)(41 62 51 72)(42 61 52 71)(43 80 53 70)(44 79 54 69)(45 78 55 68)(46 77 56 67)(47 76 57 66)(48 75 58 65)(49 74 59 64)(50 73 60 63)(81 116 91 106)(82 115 92 105)(83 114 93 104)(84 113 94 103)(85 112 95 102)(86 111 96 101)(87 110 97 120)(88 109 98 119)(89 108 99 118)(90 107 100 117)
(1 109 70 139)(2 120 71 130)(3 111 72 121)(4 102 73 132)(5 113 74 123)(6 104 75 134)(7 115 76 125)(8 106 77 136)(9 117 78 127)(10 108 79 138)(11 119 80 129)(12 110 61 140)(13 101 62 131)(14 112 63 122)(15 103 64 133)(16 114 65 124)(17 105 66 135)(18 116 67 126)(19 107 68 137)(20 118 69 128)(21 142 82 42)(22 153 83 53)(23 144 84 44)(24 155 85 55)(25 146 86 46)(26 157 87 57)(27 148 88 48)(28 159 89 59)(29 150 90 50)(30 141 91 41)(31 152 92 52)(32 143 93 43)(33 154 94 54)(34 145 95 45)(35 156 96 56)(36 147 97 47)(37 158 98 58)(38 149 99 49)(39 160 100 60)(40 151 81 51)
(21 92)(22 93)(23 94)(24 95)(25 96)(26 97)(27 98)(28 99)(29 100)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(101 121)(102 122)(103 123)(104 124)(105 125)(106 126)(107 127)(108 128)(109 129)(110 130)(111 131)(112 132)(113 133)(114 134)(115 135)(116 136)(117 137)(118 138)(119 139)(120 140)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I5A5B8A8B8C8D10A···10F10G10H10I10J20A···20H20I···20P
order12222244444444455888810···101010101020···2020···20
size111122224882020202022202020202···244444···48···8

47 irreducible representations

dim11111122222222224444
type++++++++++-+++-+--
imageC1C2C2C2C2C2D4D4D4D5Q16D10D10D10C5⋊D4C5⋊D4C8.C22D4×D5C5⋊Q16D4.9D10
kernelDic10.37D4C10.Q16C20.55D4C2×C5⋊Q16C5×C22⋊Q8C22×Dic10Dic10C2×C20C22×C10C22⋊Q8C2×C10C4⋊C4C22×C4C2×Q8C2×C4C23C10C4C22C2
# reps12121141124222441444

Matrix representation of Dic10.37D4 in GL6(𝔽41)

1600000
37180000
0032000
0028900
0000400
0000040
,
11150000
33300000
00371500
0018400
000010
0000040
,
100000
010000
00123700
00262900
0000040
000010
,
100000
010000
001000
000100
000010
0000040

G:=sub<GL(6,GF(41))| [16,37,0,0,0,0,0,18,0,0,0,0,0,0,32,28,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[11,33,0,0,0,0,15,30,0,0,0,0,0,0,37,18,0,0,0,0,15,4,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,26,0,0,0,0,37,29,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

Dic10.37D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{37}D_4
% in TeX

G:=Group("Dic10.37D4");
// GroupNames label

G:=SmallGroup(320,677);
// by ID

G=gap.SmallGroup(320,677);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,184,1123,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=a^-1,c*a*c^-1=a^11,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=a^10*c^-1>;
// generators/relations

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