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## G = Dic10.37D4order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10.37D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C22×Dic10 — Dic10.37D4
 Lower central C5 — C10 — C2×C20 — Dic10.37D4
 Upper central C1 — C22 — C22×C4 — C22⋊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 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20H 20I ··· 20P order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 4 8 8 20 20 20 20 2 2 20 20 20 20 2 ··· 2 4 4 4 4 4 ··· 4 8 ··· 8

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + - + + + - + - - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 Q16 D10 D10 D10 C5⋊D4 C5⋊D4 C8.C22 D4×D5 C5⋊Q16 D4.9D10 kernel Dic10.37D4 C10.Q16 C20.55D4 C2×C5⋊Q16 C5×C22⋊Q8 C22×Dic10 Dic10 C2×C20 C22×C10 C22⋊Q8 C2×C10 C4⋊C4 C22×C4 C2×Q8 C2×C4 C23 C10 C4 C22 C2 # reps 1 2 1 2 1 1 4 1 1 2 4 2 2 2 4 4 1 4 4 4

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

 16 0 0 0 0 0 37 18 0 0 0 0 0 0 32 0 0 0 0 0 28 9 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 11 15 0 0 0 0 33 30 0 0 0 0 0 0 37 15 0 0 0 0 18 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 37 0 0 0 0 26 29 0 0 0 0 0 0 0 40 0 0 0 0 1 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

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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