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G = C2×Q8.10D10  order 320 = 26·5

Direct product of C2 and Q8.10D10

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×Q8.10D10, C20.45C24, C10.10C25, D10.5C24, D20.36C23, C10⋊12- (1+4), Dic5.5C24, Dic10.36C23, (C2×Q8)⋊35D10, (Q8×D5)⋊13C22, (C22×Q8)⋊10D5, C4.45(C23×D5), C2.11(D5×C24), C5⋊D4.6C23, C5⋊1(C2×2- (1+4)), C4○D20⋊23C22, (Q8×C10)⋊44C22, (C4×D5).18C23, Q8.29(C22×D5), (C5×Q8).29C23, C22.9(C23×D5), (C2×C10).330C24, (C2×C20).566C23, Q8⋊2D5⋊12C22, (C22×C4).291D10, (C2×D20).291C22, C23.242(C22×D5), (C22×C20).302C22, (C22×C10).437C23, (C2×Dic5).308C23, (C22×D5).258C23, (C2×Dic10).320C22, (C2×Q8×D5)⋊20C2, (Q8×C2×C10)⋊11C2, (C2×C4○D20)⋊35C2, (C2×Q8⋊2D5)⋊20C2, (C2×C4×D5).181C22, (C2×C4).252(C22×D5), (C2×C5⋊D4).162C22, SmallGroup(320,1617)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×Q8.10D10
Chief series
C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — C2×Q8×D5 — C2×Q8.10D10
Lower central
C5 — C10 — C2×Q8.10D10

Subgroups: 2190 in 794 conjugacy classes, 447 normal (11 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×12], C4 [×8], C22, C22 [×2], C22 [×18], C5, C2×C4 [×18], C2×C4 [×52], D4 [×40], Q8 [×16], Q8 [×24], C23, C23 [×4], D5 [×8], C10, C10 [×2], C10 [×2], C22×C4 [×3], C22×C4 [×12], C2×D4 [×10], C2×Q8 [×12], C2×Q8 [×38], C4○D4 [×80], Dic5 [×8], C20 [×12], D10 [×8], D10 [×8], C2×C10, C2×C10 [×2], C2×C10 [×2], C22×Q8, C22×Q8 [×4], C2×C4○D4 [×10], 2- (1+4) [×16], Dic10 [×24], C4×D5 [×48], D20 [×24], C2×Dic5 [×4], C5⋊D4 [×16], C2×C20 [×18], C5×Q8 [×16], C22×D5 [×4], C22×C10, C2×2- (1+4), C2×Dic10 [×6], C2×C4×D5 [×12], C2×D20 [×6], C4○D20 [×48], Q8×D5 [×32], Q8⋊2D5 [×32], C2×C5⋊D4 [×4], C22×C20 [×3], Q8×C10 [×12], C2×C4○D20 [×6], C2×Q8×D5 [×4], C2×Q8⋊2D5 [×4], Q8.10D10 [×16], Q8×C2×C10, C2×Q8.10D10

Quotients:
C1, C2 [×31], C22 [×155], C23 [×155], D5, C24 [×31], D10 [×15], 2- (1+4) [×2], C25, C22×D5 [×35], C2×2- (1+4), C23×D5 [×15], Q8.10D10 [×2], D5×C24, C2×Q8.10D10

Generators and relations
 G = < a,b,c,d,e | a2=b4=1, c2=d10=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=ece-1=b2c, ede-1=d9 >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽41)

4000000
0400000
001000
000100
000010
000001
,
100000
010000
00440270
00837027
00270371
00027334
,
4000000
0400000
000010
000001
0040000
0004000
,
1350000
660000
00331134
003301527
00134840
00152780
,
1350000
0400000
00242300
0071700
00001718
00003424

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,8,27,0,0,0,40,37,0,27,0,0,27,0,37,33,0,0,0,27,1,4],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,0,0,0,0,0,0,1,0,0],[1,6,0,0,0,0,35,6,0,0,0,0,0,0,33,33,1,15,0,0,1,0,34,27,0,0,1,15,8,8,0,0,34,27,40,0],[1,0,0,0,0,0,35,40,0,0,0,0,0,0,24,7,0,0,0,0,23,17,0,0,0,0,0,0,17,34,0,0,0,0,18,24] >;
 

74 conjugacy classes

class 1 2A2B2C2D2E2F···2M4A···4L4M···4T5A5B10A···10N20A···20X
order1222222···24···44···45510···1020···20
size11112210···102···210···10222···24···4

74 irreducible representations

dim11111122244
type+++++++++-
imageC1C2C2C2C2C2D5D10D102- (1+4)Q8.10D10
kernelC2×Q8.10D10C2×C4○D20C2×Q8×D5C2×Q8⋊2D5Q8.10D10Q8×C2×C10C22×Q8C22×C4C2×Q8C10C2
# reps1644161262428

In GAP, Magma, Sage, TeX 

C_2\times Q_8._{10}D_{10}
 
% in TeX
 
G:=Group("C2xQ8.10D10");
 
// GroupNames label
 
G:=SmallGroup(320,1617);
 
// by ID
 
G=gap.SmallGroup(320,1617);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,297,136,1684,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^10=e^2=b^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=e*c*e^-1=b^2*c,e*d*e^-1=d^9>;
 
// generators/relations
 

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