Copied to
clipboard

## G = Dic10.11D4order 320 = 26·5

### 11st non-split extension by Dic10 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10.11D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Dic5⋊3Q8 — Dic10.11D4
 Lower central C5 — C10 — C2×C20 — Dic10.11D4
 Upper central C1 — C22 — C2×C4 — Q8⋊C4

Generators and relations for Dic10.11D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=dad=a-1, cac-1=a9, bc=cb, dbd=a15b, dcd=a10c-1 >

Subgroups: 486 in 112 conjugacy classes, 39 normal (37 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×6], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×5], D4 [×2], Q8 [×5], C23, D5, C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C2×D4, C2×Q8, C2×Q8, Dic5 [×4], C20 [×2], C20 [×2], D10 [×3], C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, Dic10 [×2], Dic10, D20 [×2], C2×Dic5 [×2], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, Q8.D4, C40⋊C2 [×2], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D10⋊C4 [×2], C5⋊Q16 [×2], C5×C4⋊C4, C2×C40, C2×Dic10, C2×D20, Q8×C10, D206C4, C20.8Q8, C5×Q8⋊C4, Dic53Q8, C2×C40⋊C2, C2×C5⋊Q16, C20.23D4, Dic10.11D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×4], C23, D5, C2×D4 [×2], C4○D4, D10 [×3], C4⋊D4, C4○D8, C8.C22, C22×D5, Q8.D4, C4○D20, D4×D5 [×2], D10⋊D4, SD163D5, Q16⋊D5, Dic10.11D4

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 2 2 4 4 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 1 1 40 2 2 4 4 8 10 10 20 20 20 2 2 4 4 20 20 2 ··· 2 4 4 4 4 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 type + + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C4○D20 C8.C22 D4×D5 D4×D5 SD16⋊3D5 Q16⋊D5 kernel Dic10.11D4 D20⋊6C4 C20.8Q8 C5×Q8⋊C4 Dic5⋊3Q8 C2×C40⋊C2 C2×C5⋊Q16 C20.23D4 Dic10 C2×Dic5 Q8⋊C4 C20 C4⋊C4 C2×C8 C2×Q8 C10 C4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of Dic10.11D4 in GL6(𝔽41)

 0 1 0 0 0 0 40 6 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 2 0 0 0 0 40 1
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 9 23 0 0 0 0 9 32 0 0 0 0 0 0 0 11 0 0 0 0 26 0
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 40 2 0 0 0 0 40 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 1 0 0 0 0 0 1 40 0 0 0 0 0 0 1 0 0 0 0 0 1 40

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,6,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,9,9,0,0,0,0,23,32,0,0,0,0,0,0,0,26,0,0,0,0,11,0],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,40,40,0,0,0,0,2,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,1,1,0,0,0,0,0,40] >;

Dic10.11D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{11}D_4
% in TeX

G:=Group("Dic10.11D4");
// GroupNames label

G:=SmallGroup(320,425);
// by ID

G=gap.SmallGroup(320,425);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,344,1094,135,184,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=d*a*d=a^-1,c*a*c^-1=a^9,b*c=c*b,d*b*d=a^15*b,d*c*d=a^10*c^-1>;
// generators/relations

׿
×
𝔽