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## G = Q8.1D20order 320 = 26·5

### 1st non-split extension by Q8 of D20 acting via D20/C20=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — Q8.1D20
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×D20 — C4.D20 — Q8.1D20
 Lower central C5 — C10 — C2×C20 — Q8.1D20
 Upper central C1 — C22 — C42 — C4×Q8

Generators and relations for Q8.1D20
G = < a,b,c,d | a4=c20=1, b2=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=a-1b, dcd-1=a2c-1 >

Subgroups: 454 in 112 conjugacy classes, 43 normal (39 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, D5, C10, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, SD16, Q16, C2×D4, C2×Q8, C2×Q8, Dic5, C20, C20, D10, C2×C10, D4⋊C4, Q8⋊C4, C4⋊C8, C4×Q8, C4.4D4, C2×SD16, C2×Q16, C52C8, Dic10, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C5×Q8, C22×D5, Q8.D4, C2×C52C8, D10⋊C4, Q8⋊D5, C5⋊Q16, C4×C20, C4×C20, C5×C4⋊C4, C5×C4⋊C4, C2×Dic10, C2×D20, Q8×C10, C203C8, D206C4, C10.Q16, C4.D20, C2×Q8⋊D5, C2×C5⋊Q16, Q8×C20, Q8.1D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, C4○D4, D10, C4⋊D4, C4○D8, C8.C22, D20, C5⋊D4, C22×D5, Q8.D4, C2×D20, C4○D20, C2×C5⋊D4, C207D4, C20.C23, D4.8D10, Q8.1D20

Smallest permutation representation of Q8.1D20
On 160 points
Generators in S160
(1 81 69 141)(2 82 70 142)(3 83 71 143)(4 84 72 144)(5 85 73 145)(6 86 74 146)(7 87 75 147)(8 88 76 148)(9 89 77 149)(10 90 78 150)(11 91 79 151)(12 92 80 152)(13 93 61 153)(14 94 62 154)(15 95 63 155)(16 96 64 156)(17 97 65 157)(18 98 66 158)(19 99 67 159)(20 100 68 160)(21 55 131 117)(22 56 132 118)(23 57 133 119)(24 58 134 120)(25 59 135 101)(26 60 136 102)(27 41 137 103)(28 42 138 104)(29 43 139 105)(30 44 140 106)(31 45 121 107)(32 46 122 108)(33 47 123 109)(34 48 124 110)(35 49 125 111)(36 50 126 112)(37 51 127 113)(38 52 128 114)(39 53 129 115)(40 54 130 116)
(1 47 69 109)(2 48 70 110)(3 49 71 111)(4 50 72 112)(5 51 73 113)(6 52 74 114)(7 53 75 115)(8 54 76 116)(9 55 77 117)(10 56 78 118)(11 57 79 119)(12 58 80 120)(13 59 61 101)(14 60 62 102)(15 41 63 103)(16 42 64 104)(17 43 65 105)(18 44 66 106)(19 45 67 107)(20 46 68 108)(21 149 131 89)(22 150 132 90)(23 151 133 91)(24 152 134 92)(25 153 135 93)(26 154 136 94)(27 155 137 95)(28 156 138 96)(29 157 139 97)(30 158 140 98)(31 159 121 99)(32 160 122 100)(33 141 123 81)(34 142 124 82)(35 143 125 83)(36 144 126 84)(37 145 127 85)(38 146 128 86)(39 147 129 87)(40 148 130 88)
(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 68 69 20)(2 19 70 67)(3 66 71 18)(4 17 72 65)(5 64 73 16)(6 15 74 63)(7 62 75 14)(8 13 76 61)(9 80 77 12)(10 11 78 79)(21 58 131 120)(22 119 132 57)(23 56 133 118)(24 117 134 55)(25 54 135 116)(26 115 136 53)(27 52 137 114)(28 113 138 51)(29 50 139 112)(30 111 140 49)(31 48 121 110)(32 109 122 47)(33 46 123 108)(34 107 124 45)(35 44 125 106)(36 105 126 43)(37 42 127 104)(38 103 128 41)(39 60 129 102)(40 101 130 59)(81 100 141 160)(82 159 142 99)(83 98 143 158)(84 157 144 97)(85 96 145 156)(86 155 146 95)(87 94 147 154)(88 153 148 93)(89 92 149 152)(90 151 150 91)

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

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

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

59 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E ··· 4I 4J 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20H 20I ··· 20AF order 1 2 2 2 2 4 4 4 4 4 ··· 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 40 2 2 2 2 4 ··· 4 40 2 2 20 20 20 20 2 ··· 2 2 ··· 2 4 ··· 4

59 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 D4 D4 D5 C4○D4 D10 D10 D10 C4○D8 C5⋊D4 D20 C4○D20 C8.C22 C20.C23 D4.8D10 kernel Q8.1D20 C20⋊3C8 D20⋊6C4 C10.Q16 C4.D20 C2×Q8⋊D5 C2×C5⋊Q16 Q8×C20 C2×C20 C5×Q8 C4×Q8 C20 C42 C4⋊C4 C2×Q8 C10 C2×C4 Q8 C4 C10 C2 C2 # reps 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 8 8 8 1 4 4

Matrix representation of Q8.1D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 39 0 0 0 0 1 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 11 30 0 0 0 0 26 30
,
 7 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 9 0 0 0 0 0 0 9
,
 40 7 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 9 0 0 0 0 0 9 32

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,11,26,0,0,0,0,30,30],[7,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9],[40,0,0,0,0,0,7,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,9,9,0,0,0,0,0,32] >;

Q8.1D20 in GAP, Magma, Sage, TeX

Q_8._1D_{20}
% in TeX

G:=Group("Q8.1D20");
// GroupNames label

G:=SmallGroup(320,655);
// by ID

G=gap.SmallGroup(320,655);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,344,254,184,1123,297,136,12550]);
// Polycyclic

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

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