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## G = Dic10.D4order 320 = 26·5

### 8th 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.D4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4×Dic5 — Dic5⋊3Q8 — Dic10.D4
 Lower central C5 — C10 — C2×C20 — Dic10.D4
 Upper central C1 — C22 — C2×C4 — D4⋊C4

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

Subgroups: 422 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, C10 [×3], C10, C42 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16 [×2], Q16 [×2], C2×D4, C2×Q8 [×2], Dic5 [×5], C20 [×2], C20, C2×C10, C2×C10 [×3], D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, C40, Dic10 [×2], Dic10 [×3], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20, C2×C20, C5×D4 [×2], C22×C10, Q8.D4, Dic20 [×2], C2×C52C8, C4×Dic5, C4×Dic5, C10.D4, D4.D5 [×2], C23.D5 [×2], C5×C4⋊C4, C2×C40, C2×Dic10 [×2], D4×C10, C10.Q16, C20.8Q8, C5×D4⋊C4, Dic53Q8, C2×Dic20, C2×D4.D5, C20.17D4, Dic10.D4
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, D83D5, SD16⋊D5, Dic10.D4

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

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

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

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

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 10G 10H 10I 10J 20A 20B 20C 20D 20E 20F 20G 20H 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 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 1 1 8 2 2 4 4 10 10 20 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 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 D8⋊3D5 SD16⋊D5 kernel Dic10.D4 C10.Q16 C20.8Q8 C5×D4⋊C4 Dic5⋊3Q8 C2×Dic20 C2×D4.D5 C20.17D4 Dic10 C2×Dic5 D4⋊C4 C20 C4⋊C4 C2×C8 C2×D4 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.D4 in GL4(𝔽41) generated by

 23 0 0 0 33 25 0 0 0 0 40 39 0 0 1 1
,
 10 18 0 0 15 31 0 0 0 0 30 30 0 0 26 11
,
 33 2 0 0 29 8 0 0 0 0 32 0 0 0 0 32
,
 33 2 0 0 30 8 0 0 0 0 32 23 0 0 0 9
G:=sub<GL(4,GF(41))| [23,33,0,0,0,25,0,0,0,0,40,1,0,0,39,1],[10,15,0,0,18,31,0,0,0,0,30,26,0,0,30,11],[33,29,0,0,2,8,0,0,0,0,32,0,0,0,0,32],[33,30,0,0,2,8,0,0,0,0,32,0,0,0,23,9] >;

Dic10.D4 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(320,394);
// by ID

G=gap.SmallGroup(320,394);
# by ID

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

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

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