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## G = Q16.Dic5order 320 = 26·5

### 2nd non-split extension by Q16 of Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C40 — Q16.Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C40 — C40.6C4 — Q16.Dic5
 Lower central C5 — C10 — C20 — C40 — Q16.Dic5
 Upper central C1 — C2 — C2×C4 — C2×C8 — C2×Q16

Generators and relations for Q16.Dic5
G = < a,b,c,d | a8=1, b2=c10=a4, d2=c5, bab-1=a-1, ac=ca, dad-1=a3, cbc-1=a4b, dbd-1=a3b, dcd-1=c9 >

Smallest permutation representation of Q16.Dic5
On 160 points
Generators in S160
(1 24 16 39 11 34 6 29)(2 25 17 40 12 35 7 30)(3 26 18 21 13 36 8 31)(4 27 19 22 14 37 9 32)(5 28 20 23 15 38 10 33)(41 70 56 65 51 80 46 75)(42 71 57 66 52 61 47 76)(43 72 58 67 53 62 48 77)(44 73 59 68 54 63 49 78)(45 74 60 69 55 64 50 79)(81 111 86 116 91 101 96 106)(82 112 87 117 92 102 97 107)(83 113 88 118 93 103 98 108)(84 114 89 119 94 104 99 109)(85 115 90 120 95 105 100 110)(121 141 126 146 131 151 136 156)(122 142 127 147 132 152 137 157)(123 143 128 148 133 153 138 158)(124 144 129 149 134 154 139 159)(125 145 130 150 135 155 140 160)
(1 57 11 47)(2 48 12 58)(3 59 13 49)(4 50 14 60)(5 41 15 51)(6 52 16 42)(7 43 17 53)(8 54 18 44)(9 45 19 55)(10 56 20 46)(21 78 31 68)(22 69 32 79)(23 80 33 70)(24 71 34 61)(25 62 35 72)(26 73 36 63)(27 64 37 74)(28 75 38 65)(29 66 39 76)(30 77 40 67)(81 146 91 156)(82 157 92 147)(83 148 93 158)(84 159 94 149)(85 150 95 160)(86 141 96 151)(87 152 97 142)(88 143 98 153)(89 154 99 144)(90 145 100 155)(101 136 111 126)(102 127 112 137)(103 138 113 128)(104 129 114 139)(105 140 115 130)(106 131 116 121)(107 122 117 132)(108 133 118 123)(109 124 119 134)(110 135 120 125)
(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 81 6 86 11 91 16 96)(2 90 7 95 12 100 17 85)(3 99 8 84 13 89 18 94)(4 88 9 93 14 98 19 83)(5 97 10 82 15 87 20 92)(21 109 26 114 31 119 36 104)(22 118 27 103 32 108 37 113)(23 107 28 112 33 117 38 102)(24 116 29 101 34 106 39 111)(25 105 30 110 35 115 40 120)(41 127 46 132 51 137 56 122)(42 136 47 121 52 126 57 131)(43 125 48 130 53 135 58 140)(44 134 49 139 54 124 59 129)(45 123 50 128 55 133 60 138)(61 151 66 156 71 141 76 146)(62 160 67 145 72 150 77 155)(63 149 68 154 73 159 78 144)(64 158 69 143 74 148 79 153)(65 147 70 152 75 157 80 142)

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

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

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

44 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 5A 5B 8A 8B 8C 8D 8E 10A ··· 10F 16A 16B 16C 16D 20A 20B 20C 20D 20E ··· 20L 40A ··· 40H order 1 2 2 4 4 4 4 5 5 8 8 8 8 8 10 ··· 10 16 16 16 16 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 2 2 2 8 8 2 2 2 2 4 40 40 2 ··· 2 20 20 20 20 4 4 4 4 8 ··· 8 4 ··· 4

44 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + - - + - image C1 C2 C2 C2 C4 D4 D4 D5 D8 SD16 D10 Dic5 C5⋊D4 C5⋊D4 C8.17D4 D4⋊D5 D4.D5 Q16.Dic5 kernel Q16.Dic5 C20.4C8 C40.6C4 C10×Q16 C5×Q16 C40 C2×C20 C2×Q16 C20 C2×C10 C2×C8 Q16 C8 C2×C4 C5 C4 C22 C1 # reps 1 1 1 1 4 1 1 2 2 2 2 4 4 4 2 2 2 8

Matrix representation of Q16.Dic5 in GL4(𝔽241) generated by

 11 11 0 0 230 11 0 0 0 0 230 11 0 0 230 230
,
 52 123 0 0 123 189 0 0 0 0 238 58 0 0 58 3
,
 0 36 0 0 205 0 0 0 0 0 0 154 0 0 87 0
,
 0 0 1 0 0 0 0 1 0 240 0 0 1 0 0 0
G:=sub<GL(4,GF(241))| [11,230,0,0,11,11,0,0,0,0,230,230,0,0,11,230],[52,123,0,0,123,189,0,0,0,0,238,58,0,0,58,3],[0,205,0,0,36,0,0,0,0,0,0,87,0,0,154,0],[0,0,0,1,0,0,240,0,1,0,0,0,0,1,0,0] >;

Q16.Dic5 in GAP, Magma, Sage, TeX

Q_{16}.{\rm Dic}_5
% in TeX

G:=Group("Q16.Dic5");
// GroupNames label

G:=SmallGroup(320,123);
// by ID

G=gap.SmallGroup(320,123);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,387,184,675,794,80,1684,851,102,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^8=1,b^2=c^10=a^4,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^3,c*b*c^-1=a^4*b,d*b*d^-1=a^3*b,d*c*d^-1=c^9>;
// generators/relations

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