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## G = Dic10⋊14D4order 320 = 26·5

### 2nd semidirect product of Dic10 and D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Dic10⋊14D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×Dic10 — C22×Dic10 — Dic10⋊14D4
 Lower central C5 — C10 — C2×C20 — Dic10⋊14D4
 Upper central C1 — C22 — C22×C4 — C22⋊C8

Generators and relations for Dic1014D4
G = < a,b,c,d | a20=c4=d2=1, b2=a10, bab-1=cac-1=a-1, ad=da, cbc-1=a5b, bd=db, dcd=c-1 >

Subgroups: 734 in 158 conjugacy classes, 47 normal (25 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×6], C22, C22 [×2], C22 [×5], C5, C8 [×2], C2×C4 [×2], C2×C4 [×9], D4 [×4], Q8 [×10], C23, C23, D5, C10 [×3], C10 [×2], C22⋊C4, C4⋊C4, C2×C8 [×2], SD16 [×4], C22×C4, C22×C4, C2×D4 [×2], C2×Q8 [×7], Dic5 [×5], C20 [×2], C20, D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊C8, Q8⋊C4 [×2], C4⋊D4, C2×SD16 [×2], C22×Q8, C40 [×2], Dic10 [×4], Dic10 [×6], D20 [×2], C2×Dic5 [×7], C5⋊D4 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, Q8⋊D4, C40⋊C2 [×4], C4⋊Dic5, D10⋊C4, C2×C40 [×2], C2×Dic10 [×2], C2×Dic10 [×5], C2×D20, C22×Dic5, C2×C5⋊D4, C22×C20, C20.44D4 [×2], C5×C22⋊C8, C2×C40⋊C2 [×2], C207D4, C22×Dic10, Dic1014D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, D5, SD16 [×2], C2×D4 [×3], D10 [×3], C22≀C2, C2×SD16, C8.C22, D20 [×2], C22×D5, Q8⋊D4, C40⋊C2 [×2], C2×D20, D4×D5 [×2], C22⋊D20, C2×C40⋊C2, C8.D10, Dic1014D4

Smallest permutation representation of Dic1014D4
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)
(1 155 11 145)(2 154 12 144)(3 153 13 143)(4 152 14 142)(5 151 15 141)(6 150 16 160)(7 149 17 159)(8 148 18 158)(9 147 19 157)(10 146 20 156)(21 46 31 56)(22 45 32 55)(23 44 33 54)(24 43 34 53)(25 42 35 52)(26 41 36 51)(27 60 37 50)(28 59 38 49)(29 58 39 48)(30 57 40 47)(61 105 71 115)(62 104 72 114)(63 103 73 113)(64 102 74 112)(65 101 75 111)(66 120 76 110)(67 119 77 109)(68 118 78 108)(69 117 79 107)(70 116 80 106)(81 132 91 122)(82 131 92 121)(83 130 93 140)(84 129 94 139)(85 128 95 138)(86 127 96 137)(87 126 97 136)(88 125 98 135)(89 124 99 134)(90 123 100 133)
(1 36 107 81)(2 35 108 100)(3 34 109 99)(4 33 110 98)(5 32 111 97)(6 31 112 96)(7 30 113 95)(8 29 114 94)(9 28 115 93)(10 27 116 92)(11 26 117 91)(12 25 118 90)(13 24 119 89)(14 23 120 88)(15 22 101 87)(16 21 102 86)(17 40 103 85)(18 39 104 84)(19 38 105 83)(20 37 106 82)(41 74 122 160)(42 73 123 159)(43 72 124 158)(44 71 125 157)(45 70 126 156)(46 69 127 155)(47 68 128 154)(48 67 129 153)(49 66 130 152)(50 65 131 151)(51 64 132 150)(52 63 133 149)(53 62 134 148)(54 61 135 147)(55 80 136 146)(56 79 137 145)(57 78 138 144)(58 77 139 143)(59 76 140 142)(60 75 121 141)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)(17 32)(18 33)(19 34)(20 35)(41 145)(42 146)(43 147)(44 148)(45 149)(46 150)(47 151)(48 152)(49 153)(50 154)(51 155)(52 156)(53 157)(54 158)(55 159)(56 160)(57 141)(58 142)(59 143)(60 144)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 121)(79 122)(80 123)(81 107)(82 108)(83 109)(84 110)(85 111)(86 112)(87 113)(88 114)(89 115)(90 116)(91 117)(92 118)(93 119)(94 120)(95 101)(96 102)(97 103)(98 104)(99 105)(100 106)

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

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

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

59 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 ··· 20H 20I 20J 20K 20L 40A ··· 40P 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 40 ··· 40 size 1 1 1 1 2 2 40 2 2 4 20 20 20 20 40 2 2 4 4 4 4 2 ··· 2 4 4 4 4 2 ··· 2 4 4 4 4 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + - + - image C1 C2 C2 C2 C2 C2 D4 D4 D4 D5 SD16 D10 D10 D20 D20 C40⋊C2 C8.C22 D4×D5 C8.D10 kernel Dic10⋊14D4 C20.44D4 C5×C22⋊C8 C2×C40⋊C2 C20⋊7D4 C22×Dic10 Dic10 C2×C20 C22×C10 C22⋊C8 C2×C10 C2×C8 C22×C4 C2×C4 C23 C22 C10 C4 C2 # reps 1 2 1 2 1 1 4 1 1 2 4 4 2 4 4 16 1 4 4

Matrix representation of Dic1014D4 in GL4(𝔽41) generated by

 13 2 0 0 39 25 0 0 0 0 40 0 0 0 0 40
,
 15 31 0 0 39 26 0 0 0 0 0 40 0 0 40 0
,
 39 28 0 0 16 2 0 0 0 0 0 1 0 0 40 0
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 40 0
G:=sub<GL(4,GF(41))| [13,39,0,0,2,25,0,0,0,0,40,0,0,0,0,40],[15,39,0,0,31,26,0,0,0,0,0,40,0,0,40,0],[39,16,0,0,28,2,0,0,0,0,0,40,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,40,0] >;

Dic1014D4 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_{14}D_4
% in TeX

G:=Group("Dic10:14D4");
// GroupNames label

G:=SmallGroup(320,365);
// by ID

G=gap.SmallGroup(320,365);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,254,219,58,1123,136,12550]);
// 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,a*d=d*a,c*b*c^-1=a^5*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

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