Copied to
clipboard

G = Dic10.16D4order 320 = 26·5

16th non-split extension by Dic10 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10.16D4
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — C2×C4×D5 — C2×D4⋊2D5 — Dic10.16D4
 Lower central C5 — C10 — C2×C20 — Dic10.16D4
 Upper central C1 — C22 — C2×C4 — C2×SD16

Generators and relations for Dic10.16D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, dad=a9, cbc-1=a5b, dbd=a10b, dcd=a10c-1 >

Subgroups: 606 in 152 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, Q16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C22⋊C8, D4⋊C4, Q8⋊C4, C22⋊Q8, C2×SD16, C2×Q16, C2×C4○D4, C52C8, C40, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×D4, C5×Q8, C22×D5, C22×C10, D4.7D4, C2×C52C8, C10.D4, C4⋊Dic5, D10⋊C4, C5⋊Q16, C2×C40, C5×SD16, C2×Dic10, C2×C4×D5, D42D5, C22×Dic5, C2×C5⋊D4, D4×C10, Q8×C10, C20.44D4, D101C8, D4⋊Dic5, C2×C5⋊Q16, D103Q8, C10×SD16, C2×D42D5, Dic10.16D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C22≀C2, C4○D8, C8.C22, C5⋊D4, C22×D5, D4.7D4, D4×D5, C2×C5⋊D4, SD16⋊D5, SD163D5, C23⋊D10, Dic10.16D4

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

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 4 4 20 2 2 8 10 10 20 20 40 2 2 4 4 20 20 2 ··· 2 8 8 8 8 4 4 4 4 8 8 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 2 4 4 4 4 4 type + + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D4 D4 D5 D10 D10 D10 C4○D8 C5⋊D4 C8.C22 D4×D5 D4×D5 SD16⋊D5 SD16⋊3D5 kernel Dic10.16D4 C20.44D4 D10⋊1C8 D4⋊Dic5 C2×C5⋊Q16 D10⋊3Q8 C10×SD16 C2×D4⋊2D5 Dic10 C2×Dic5 C5×D4 C22×D5 C2×SD16 C2×C8 C2×D4 C2×Q8 C10 D4 C10 C4 C22 C2 C2 # reps 1 1 1 1 1 1 1 1 2 1 2 1 2 2 2 2 4 8 1 2 2 4 4

Matrix representation of Dic10.16D4 in GL6(𝔽41)

 40 1 0 0 0 0 33 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 32 0 0 0 0 0 0 9
,
 34 1 0 0 0 0 34 7 0 0 0 0 0 0 1 0 0 0 0 0 40 40 0 0 0 0 0 0 0 27 0 0 0 0 3 0
,
 34 1 0 0 0 0 34 7 0 0 0 0 0 0 25 9 0 0 0 0 17 16 0 0 0 0 0 0 0 9 0 0 0 0 32 0
,
 34 1 0 0 0 0 34 7 0 0 0 0 0 0 40 0 0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,33,0,0,0,0,1,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,0,0,0,0,0,0,9],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,1,40,0,0,0,0,0,40,0,0,0,0,0,0,0,3,0,0,0,0,27,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,25,17,0,0,0,0,9,16,0,0,0,0,0,0,0,32,0,0,0,0,9,0],[34,34,0,0,0,0,1,7,0,0,0,0,0,0,40,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40] >;

Dic10.16D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}._{16}D_4
% in TeX

G:=Group("Dic10.16D4");
// GroupNames label

G:=SmallGroup(320,800);
// by ID

G=gap.SmallGroup(320,800);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,254,219,184,851,438,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^4=d^2=1,b^2=a^10,b*a*b^-1=c*a*c^-1=a^-1,d*a*d=a^9,c*b*c^-1=a^5*b,d*b*d=a^10*b,d*c*d=a^10*c^-1>;
// generators/relations

׿
×
𝔽