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## G = (C2×C20).Q8order 320 = 26·5

### 5th non-split extension by C2×C20 of Q8 acting via Q8/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C20).Q8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C22×C20 — C2×C4.Dic5 — (C2×C20).Q8
 Lower central C5 — C10 — C2×C10 — (C2×C20).Q8
 Upper central C1 — C22 — C22×C4 — C2×C4⋊C4

Generators and relations for (C2×C20).Q8
G = < a,b,c,d | a2=b20=c4=1, d2=ab5c2, ab=ba, ac=ca, dad-1=ab10, cbc-1=b11, dbd-1=b-1, dcd-1=ab10c-1 >

Subgroups: 246 in 98 conjugacy classes, 51 normal (25 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C4⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C4⋊C4, C2×M4(2), C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C22.C42, C2×C52C8, C4.Dic5, C4.Dic5, C5×C4⋊C4, C22×C20, C22×C20, C2×C4.Dic5, C10×C4⋊C4, (C2×C20).Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, D5, C42, C22⋊C4, C4⋊C4, Dic5, D10, C2.C42, C4.D4, C4.10D4, Dic10, C4×D5, D20, C2×Dic5, C5⋊D4, C22.C42, C4×Dic5, C10.D4, C4⋊Dic5, D10⋊C4, C23.D5, C10.10C42, C20.D4, C20.10D4, (C2×C20).Q8

Smallest permutation representation of (C2×C20).Q8
On 160 points
Generators in S160
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)(101 111)(102 112)(103 113)(104 114)(105 115)(106 116)(107 117)(108 118)(109 119)(110 120)(121 131)(122 132)(123 133)(124 134)(125 135)(126 136)(127 137)(128 138)(129 139)(130 140)
(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 56 145 97)(2 47 146 88)(3 58 147 99)(4 49 148 90)(5 60 149 81)(6 51 150 92)(7 42 151 83)(8 53 152 94)(9 44 153 85)(10 55 154 96)(11 46 155 87)(12 57 156 98)(13 48 157 89)(14 59 158 100)(15 50 159 91)(16 41 160 82)(17 52 141 93)(18 43 142 84)(19 54 143 95)(20 45 144 86)(21 121 64 104)(22 132 65 115)(23 123 66 106)(24 134 67 117)(25 125 68 108)(26 136 69 119)(27 127 70 110)(28 138 71 101)(29 129 72 112)(30 140 73 103)(31 131 74 114)(32 122 75 105)(33 133 76 116)(34 124 77 107)(35 135 78 118)(36 126 79 109)(37 137 80 120)(38 128 61 111)(39 139 62 102)(40 130 63 113)
(1 24 150 62 11 34 160 72)(2 23 151 61 12 33 141 71)(3 22 152 80 13 32 142 70)(4 21 153 79 14 31 143 69)(5 40 154 78 15 30 144 68)(6 39 155 77 16 29 145 67)(7 38 156 76 17 28 146 66)(8 37 157 75 18 27 147 65)(9 36 158 74 19 26 148 64)(10 35 159 73 20 25 149 63)(41 102 87 134 51 112 97 124)(42 101 88 133 52 111 98 123)(43 120 89 132 53 110 99 122)(44 119 90 131 54 109 100 121)(45 118 91 130 55 108 81 140)(46 117 92 129 56 107 82 139)(47 116 93 128 57 106 83 138)(48 115 94 127 58 105 84 137)(49 114 95 126 59 104 85 136)(50 113 96 125 60 103 86 135)

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

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

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

62 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A ··· 8H 10A ··· 10N 20A ··· 20X order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 2 2 20 ··· 20 2 ··· 2 4 ··· 4

62 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 C4 C4 D4 Q8 D5 Dic5 D10 Dic10 C4×D5 D20 C5⋊D4 C4.D4 C4.10D4 C20.D4 C20.10D4 kernel (C2×C20).Q8 C2×C4.Dic5 C10×C4⋊C4 C4.Dic5 C22×C20 C2×C20 C2×C20 C2×C4⋊C4 C22×C4 C22×C4 C2×C4 C2×C4 C2×C4 C2×C4 C10 C10 C2 C2 # reps 1 2 1 8 4 3 1 2 4 2 4 8 4 8 1 1 4 4

Matrix representation of (C2×C20).Q8 in GL6(𝔽41)

 40 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 38 33 40 0 0 0 0 0 0 40
,
 35 40 0 0 0 0 36 40 0 0 0 0 0 0 0 40 0 0 0 0 1 0 0 0 0 0 37 22 0 1 0 0 19 37 40 0
,
 32 0 0 0 0 0 0 32 0 0 0 0 0 0 30 1 0 0 0 0 1 11 0 0 0 0 29 19 11 40 0 0 19 37 40 30
,
 29 15 0 0 0 0 26 12 0 0 0 0 0 0 38 33 39 0 0 0 0 0 0 39 0 0 25 32 3 8 0 0 21 0 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,38,0,0,0,0,1,33,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,36,0,0,0,0,40,40,0,0,0,0,0,0,0,1,37,19,0,0,40,0,22,37,0,0,0,0,0,40,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,30,1,29,19,0,0,1,11,19,37,0,0,0,0,11,40,0,0,0,0,40,30],[29,26,0,0,0,0,15,12,0,0,0,0,0,0,38,0,25,21,0,0,33,0,32,0,0,0,39,0,3,0,0,0,0,39,8,0] >;

(C2×C20).Q8 in GAP, Magma, Sage, TeX

(C_2\times C_{20}).Q_8
% in TeX

G:=Group("(C2xC20).Q8");
// GroupNames label

G:=SmallGroup(320,88);
// by ID

G=gap.SmallGroup(320,88);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=c^4=1,d^2=a*b^5*c^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^10,c*b*c^-1=b^11,d*b*d^-1=b^-1,d*c*d^-1=a*b^10*c^-1>;
// generators/relations

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