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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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C20).5Q8, (C2×C4).12D20, C4.Dic58C4, C20.38(C4⋊C4), (C2×C20).103D4, (C22×C20).3C4, (C2×C4).1Dic10, (C2×C10).40C42, (C22×C4).56D10, (C22×C4).2Dic5, C22.9(C4×Dic5), C20.55(C22⋊C4), C53(C22.C42), C2.2(C20.D4), C4.7(C10.D4), C22.9(C4⋊Dic5), C4.32(D10⋊C4), C23.23(C2×Dic5), C2.2(C20.10D4), C10.14(C4.D4), C10.12(C4.10D4), (C22×C20).119C22, C22.27(C23.D5), C10.25(C2.C42), C2.7(C10.10C42), (C2×C4⋊C4).3D5, (C10×C4⋊C4).2C2, (C2×C4).18(C4×D5), (C2×C10).64(C4⋊C4), (C2×C20).228(C2×C4), (C2×C4.Dic5).6C2, (C2×C4).175(C5⋊D4), (C22×C10).193(C2×C4), (C2×C10).152(C22⋊C4), SmallGroup(320,88)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×C20).Q8
C1C5C10C20C2×C20C22×C20C2×C4.Dic5 — (C2×C20).Q8
C5C10C2×C10 — (C2×C20).Q8
C1C22C22×C4C2×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 [×3], C2 [×2], C4 [×4], C4 [×2], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×4], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C20 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2) [×2], C52C8 [×4], C2×C20 [×2], C2×C20 [×4], C2×C20 [×4], C22×C10, C22.C42, C2×C52C8 [×2], C4.Dic5 [×4], C4.Dic5 [×2], C5×C4⋊C4 [×2], C22×C20, C22×C20 [×2], C2×C4.Dic5 [×2], C10×C4⋊C4, (C2×C20).Q8
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, D5, C42, C22⋊C4 [×3], C4⋊C4 [×3], Dic5 [×2], D10, C2.C42, C4.D4, C4.10D4, Dic10, C4×D5 [×2], D20, C2×Dic5, C5⋊D4 [×2], C22.C42, C4×Dic5, C10.D4 [×2], C4⋊Dic5, D10⋊C4 [×2], 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)(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)(141 151)(142 152)(143 153)(144 154)(145 155)(146 156)(147 157)(148 158)(149 159)(150 160)
(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 91 47 61)(2 82 48 72)(3 93 49 63)(4 84 50 74)(5 95 51 65)(6 86 52 76)(7 97 53 67)(8 88 54 78)(9 99 55 69)(10 90 56 80)(11 81 57 71)(12 92 58 62)(13 83 59 73)(14 94 60 64)(15 85 41 75)(16 96 42 66)(17 87 43 77)(18 98 44 68)(19 89 45 79)(20 100 46 70)(21 117 142 132)(22 108 143 123)(23 119 144 134)(24 110 145 125)(25 101 146 136)(26 112 147 127)(27 103 148 138)(28 114 149 129)(29 105 150 140)(30 116 151 131)(31 107 152 122)(32 118 153 133)(33 109 154 124)(34 120 155 135)(35 111 156 126)(36 102 157 137)(37 113 158 128)(38 104 159 139)(39 115 160 130)(40 106 141 121)
(1 105 52 135 11 115 42 125)(2 104 53 134 12 114 43 124)(3 103 54 133 13 113 44 123)(4 102 55 132 14 112 45 122)(5 101 56 131 15 111 46 121)(6 120 57 130 16 110 47 140)(7 119 58 129 17 109 48 139)(8 118 59 128 18 108 49 138)(9 117 60 127 19 107 50 137)(10 116 41 126 20 106 51 136)(21 64 157 99 31 74 147 89)(22 63 158 98 32 73 148 88)(23 62 159 97 33 72 149 87)(24 61 160 96 34 71 150 86)(25 80 141 95 35 70 151 85)(26 79 142 94 36 69 152 84)(27 78 143 93 37 68 153 83)(28 77 144 92 38 67 154 82)(29 76 145 91 39 66 155 81)(30 75 146 90 40 65 156 100)

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B8A···8H10A···10N20A···20X
order12222244444444558···810···1020···20
size111122222244442220···202···24···4

62 irreducible representations

dim111112222222224444
type++++-+-+-++-
imageC1C2C2C4C4D4Q8D5Dic5D10Dic10C4×D5D20C5⋊D4C4.D4C4.10D4C20.D4C20.10D4
kernel(C2×C20).Q8C2×C4.Dic5C10×C4⋊C4C4.Dic5C22×C20C2×C20C2×C20C2×C4⋊C4C22×C4C22×C4C2×C4C2×C4C2×C4C2×C4C10C10C2C2
# reps121843124248481144

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

4000000
0400000
001000
000100
003833400
0000040
,
35400000
36400000
0004000
001000
00372201
001937400
,
3200000
0320000
0030100
0011100
0029191140
0019374030
,
29150000
26120000
003833390
0000039
00253238
0021000

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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