Copied to
clipboard

G = Dic10⋊10D4order 320 = 26·5

3rd semidirect product of Dic10 and D4 acting via D4/C4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — Dic10⋊10D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — Dic10⋊10D4
 Lower central C5 — C2×C10 — Dic10⋊10D4
 Upper central C1 — C22 — C4.4D4

Generators and relations for Dic1010D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=a-1, ac=ca, dad=a9, cbc-1=dbd=a10b, dcd=c-1 >

Subgroups: 1046 in 290 conjugacy classes, 105 normal (43 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×12], C22, C22 [×13], C5, C2×C4 [×3], C2×C4 [×2], C2×C4 [×18], D4 [×12], Q8 [×10], C23 [×2], C23 [×2], D5 [×3], C10 [×3], C10 [×2], C42, C42 [×2], C22⋊C4 [×4], C22⋊C4 [×6], C4⋊C4 [×6], C22×C4 [×6], C2×D4, C2×D4 [×5], C2×Q8, C2×Q8 [×7], C4○D4 [×4], Dic5 [×4], Dic5 [×4], C20 [×2], C20 [×4], D10 [×2], D10 [×5], C2×C10, C2×C10 [×6], C4×D4 [×3], C4×Q8, C4⋊D4 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4 [×2], C22×Q8, C2×C4○D4, Dic10 [×4], Dic10 [×4], C4×D5 [×8], D20 [×2], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×4], C5⋊D4 [×8], C2×C20 [×3], C2×C20 [×2], C5×D4 [×2], C5×Q8 [×2], C22×D5 [×2], C22×C10 [×2], Q85D4, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], D10⋊C4 [×2], C23.D5 [×2], C4×C20, C5×C22⋊C4 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5 [×2], C2×C4×D5 [×2], C2×D20, D42D5 [×4], Q8×D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×4], D4×C10, Q8×C10, C4×Dic10, C4×D20, Dic5.14D4 [×2], Dic54D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C202D4, D103Q8, C5×C4.4D4, C2×D42D5, C2×Q8×D5, Dic1010D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2- 1+4, C22×D5 [×7], Q85D4, D4×D5 [×2], C23×D5, C2×D4×D5, D5×C4○D4, D4.10D10, Dic1010D4

Smallest permutation representation of Dic1010D4
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)
(1 119 11 109)(2 118 12 108)(3 117 13 107)(4 116 14 106)(5 115 15 105)(6 114 16 104)(7 113 17 103)(8 112 18 102)(9 111 19 101)(10 110 20 120)(21 54 31 44)(22 53 32 43)(23 52 33 42)(24 51 34 41)(25 50 35 60)(26 49 36 59)(27 48 37 58)(28 47 38 57)(29 46 39 56)(30 45 40 55)(61 95 71 85)(62 94 72 84)(63 93 73 83)(64 92 74 82)(65 91 75 81)(66 90 76 100)(67 89 77 99)(68 88 78 98)(69 87 79 97)(70 86 80 96)(121 142 131 152)(122 141 132 151)(123 160 133 150)(124 159 134 149)(125 158 135 148)(126 157 136 147)(127 156 137 146)(128 155 138 145)(129 154 139 144)(130 153 140 143)
(1 85 121 29)(2 86 122 30)(3 87 123 31)(4 88 124 32)(5 89 125 33)(6 90 126 34)(7 91 127 35)(8 92 128 36)(9 93 129 37)(10 94 130 38)(11 95 131 39)(12 96 132 40)(13 97 133 21)(14 98 134 22)(15 99 135 23)(16 100 136 24)(17 81 137 25)(18 82 138 26)(19 83 139 27)(20 84 140 28)(41 104 76 147)(42 105 77 148)(43 106 78 149)(44 107 79 150)(45 108 80 151)(46 109 61 152)(47 110 62 153)(48 111 63 154)(49 112 64 155)(50 113 65 156)(51 114 66 157)(52 115 67 158)(53 116 68 159)(54 117 69 160)(55 118 70 141)(56 119 71 142)(57 120 72 143)(58 101 73 144)(59 102 74 145)(60 103 75 146)
(1 121)(2 130)(3 139)(4 128)(5 137)(6 126)(7 135)(8 124)(9 133)(10 122)(11 131)(12 140)(13 129)(14 138)(15 127)(16 136)(17 125)(18 134)(19 123)(20 132)(21 37)(22 26)(23 35)(25 33)(27 31)(28 40)(30 38)(32 36)(41 51)(42 60)(43 49)(44 58)(45 47)(46 56)(48 54)(50 52)(53 59)(55 57)(61 71)(62 80)(63 69)(64 78)(65 67)(66 76)(68 74)(70 72)(73 79)(75 77)(81 89)(82 98)(83 87)(84 96)(86 94)(88 92)(91 99)(93 97)(101 150)(102 159)(103 148)(104 157)(105 146)(106 155)(107 144)(108 153)(109 142)(110 151)(111 160)(112 149)(113 158)(114 147)(115 156)(116 145)(117 154)(118 143)(119 152)(120 141)

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

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

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

53 conjugacy classes

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

53 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 D10 2- 1+4 D4×D5 D5×C4○D4 D4.10D10 kernel Dic10⋊10D4 C4×Dic10 C4×D20 Dic5.14D4 Dic5⋊4D4 D10⋊D4 Dic5.5D4 C20⋊2D4 D10⋊3Q8 C5×C4.4D4 C2×D4⋊2D5 C2×Q8×D5 Dic10 C4.4D4 D10 C42 C22⋊C4 C2×D4 C2×Q8 C10 C4 C2 C2 # reps 1 1 1 2 2 2 2 1 1 1 1 1 4 2 4 2 8 2 2 1 4 4 4

Matrix representation of Dic1010D4 in GL6(𝔽41)

 7 40 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 32 32
,
 34 7 0 0 0 0 40 7 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 39 0 0 0 0 1 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 1 0 0 0 0 0 40 40
,
 34 7 0 0 0 0 40 7 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 40 40

G:=sub<GL(6,GF(41))| [7,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,32,0,0,0,0,0,32],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,1,0,0,0,0,39,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[34,40,0,0,0,0,7,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,40,0,0,0,0,0,40] >;

Dic1010D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{10}D_4
% in TeX

G:=Group("Dic10:10D4");
// GroupNames label

G:=SmallGroup(320,1349);
// by ID

G=gap.SmallGroup(320,1349);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,219,100,1571,297,192,12550]);
// Polycyclic

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

׿
×
𝔽