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G = C8.27(C4×D5)  order 320 = 26·5

4th non-split extension by C8 of C4×D5 acting via C4×D5/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C8×D5)⋊2C4, C8.27(C4×D5), C4.30(Q8×D5), C2.D814D5, C40.58(C2×C4), C405C421C2, (C4×D5).16Q8, C20.21(C2×Q8), C4⋊C4.170D10, (C2×C8).230D10, C22.90(D4×D5), D10.16(C4⋊C4), C10.28(C4○D8), C2.4(D83D5), (C2×C40).82C22, C20.Q819C2, (C22×D5).85D4, C2.4(Q8.D10), Dic5.39(C4⋊C4), (C2×C20).296C23, C20.108(C22×C4), (C2×Dic5).277D4, C53(C23.25D4), C4⋊Dic5.122C22, (D5×C2×C8).3C2, C4.80(C2×C4×D5), C2.15(D5×C4⋊C4), (C5×C2.D8)⋊4C2, C10.37(C2×C4⋊C4), C52C8.39(C2×C4), C4⋊C47D5.7C2, (C4×D5).76(C2×C4), (C2×C10).301(C2×D4), (C5×C4⋊C4).89C22, (C2×C4×D5).305C22, (C2×C4).399(C22×D5), (C2×C52C8).242C22, SmallGroup(320,507)

Series: Derived Chief Lower central Upper central

C1C20 — C8.27(C4×D5)
C1C5C10C2×C10C2×C20C2×C4×D5D5×C2×C8 — C8.27(C4×D5)
C5C10C20 — C8.27(C4×D5)
C1C22C2×C4C2.D8

Generators and relations for C8.27(C4×D5)
 G = < a,b,c,d | a8=b4=c5=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 382 in 114 conjugacy classes, 55 normal (27 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×2], C8 [×2], C2×C4, C2×C4 [×9], C23, D5 [×2], C10 [×3], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8, C2×C8 [×5], C22×C4, Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.Q8 [×2], C2.D8, C2.D8, C42⋊C2 [×2], C22×C8, C52C8 [×2], C40 [×2], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C22×D5, C23.25D4, C8×D5 [×4], C2×C52C8, C4×Dic5 [×2], C4⋊Dic5 [×2], D10⋊C4 [×2], C5×C4⋊C4 [×2], C2×C40, C2×C4×D5, C20.Q8 [×2], C405C4, C5×C2.D8, C4⋊C47D5 [×2], D5×C2×C8, C8.27(C4×D5)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, D5, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, D10 [×3], C2×C4⋊C4, C4○D8 [×2], C4×D5 [×2], C22×D5, C23.25D4, C2×C4×D5, D4×D5, Q8×D5, D5×C4⋊C4, D83D5, Q8.D10, C8.27(C4×D5)

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

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

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

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

56 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N5A5B8A8B8C8D8E8F8G8H10A···10F20A20B20C20D20E···20L40A···40H
order12222244444444444444558888888810···102020202020···2040···40
size11111010224444555520202020222222101010102···244448···84···4

56 irreducible representations

dim1111111222222224444
type++++++-+++++-+-+
imageC1C2C2C2C2C2C4Q8D4D4D5D10D10C4○D8C4×D5Q8×D5D4×D5D83D5Q8.D10
kernelC8.27(C4×D5)C20.Q8C405C4C5×C2.D8C4⋊C47D5D5×C2×C8C8×D5C4×D5C2×Dic5C22×D5C2.D8C4⋊C4C2×C8C10C8C4C22C2C2
# reps1211218211242882244

Matrix representation of C8.27(C4×D5) in GL4(𝔽41) generated by

32000
01400
00400
00040
,
303000
371100
0090
0009
,
1000
0100
00040
00134
,
11500
04000
00341
00347
G:=sub<GL(4,GF(41))| [3,0,0,0,20,14,0,0,0,0,40,0,0,0,0,40],[30,37,0,0,30,11,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,34],[1,0,0,0,15,40,0,0,0,0,34,34,0,0,1,7] >;

C8.27(C4×D5) in GAP, Magma, Sage, TeX

C_8._{27}(C_4\times D_5)
% in TeX

G:=Group("C8.27(C4xD5)");
// GroupNames label

G:=SmallGroup(320,507);
// by ID

G=gap.SmallGroup(320,507);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,120,758,219,58,438,102,12550]);
// Polycyclic

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

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