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G = C2×Q8.Dic5  order 320 = 26·5

Direct product of C2 and Q8.Dic5

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

Aliases: C2×Q8.Dic5, C20.75C24, C10⋊5(C8○D4), C4○D4.44D10, (D4×C10).25C4, (Q8×C10).22C4, C4○D4.4Dic5, Q8.9(C2×Dic5), D4.8(C2×Dic5), C4.74(C23×D5), C10.69(C23×C4), C5⋊2C8.44C23, (C2×Q8).10Dic5, (C2×D4).12Dic5, (C2×C20).553C23, C20.156(C22×C4), (C22×C4).386D10, C4.Dic5⋊34C22, C23.19(C2×Dic5), C2.10(C23×Dic5), C4.40(C22×Dic5), (C22×C20).288C22, C22.33(C22×Dic5), C5⋊7(C2×C8○D4), (C5×C4○D4).7C4, (C10×C4○D4).9C2, (C2×C4○D4).13D5, (C5×D4).39(C2×C4), (C5×Q8).42(C2×C4), (C2×C20).308(C2×C4), (C22×C5⋊2C8)⋊14C2, (C2×C5⋊2C8)⋊41C22, (C2×C4.Dic5)⋊28C2, (C2×C4).56(C2×Dic5), (C5×C4○D4).48C22, (C2×C4).831(C22×D5), (C22×C10).148(C2×C4), (C2×C10).129(C22×C4), SmallGroup(320,1490)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2×Q8.Dic5
Chief series
C1 — C5 — C10 — C20 — C5⋊2C8 — C2×C5⋊2C8 — C22×C5⋊2C8 — C2×Q8.Dic5
Lower central
C5 — C10 — C2×Q8.Dic5

Subgroups: 494 in 266 conjugacy classes, 191 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×6], C22, C22 [×6], C22 [×6], C5, C8 [×8], C2×C4, C2×C4 [×15], D4 [×12], Q8 [×4], C23 [×3], C10, C10 [×2], C10 [×6], C2×C8 [×16], M4(2) [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C20 [×2], C20 [×6], C2×C10, C2×C10 [×6], C2×C10 [×6], C22×C8 [×3], C2×M4(2) [×3], C8○D4 [×8], C2×C4○D4, C5⋊2C8 [×8], C2×C20, C2×C20 [×15], C5×D4 [×12], C5×Q8 [×4], C22×C10 [×3], C2×C8○D4, C2×C5⋊2C8, C2×C5⋊2C8 [×15], C4.Dic5 [×12], C22×C20 [×3], D4×C10 [×3], Q8×C10, C5×C4○D4 [×8], C22×C5⋊2C8 [×3], C2×C4.Dic5 [×3], Q8.Dic5 [×8], C10×C4○D4, C2×Q8.Dic5

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], D5, C22×C4 [×14], C24, Dic5 [×8], D10 [×7], C8○D4 [×2], C23×C4, C2×Dic5 [×28], C22×D5 [×7], C2×C8○D4, C22×Dic5 [×14], C23×D5, Q8.Dic5 [×2], C23×Dic5, C2×Q8.Dic5

Generators and relations
 G = < a,b,c,d,e | a2=b4=1, c2=d10=b2, e2=d5, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >

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

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

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

(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)

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

Matrix representation ►G ⊆ GL4(𝔽41) generated by

40000
04000
00400
00040
,
40000
04000
0009
0090
,
1000
0100
0090
00032
,
404000
8700
0090
0009
,
92500
53200
0030
0003
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,0,9,0,0,9,0],[1,0,0,0,0,1,0,0,0,0,9,0,0,0,0,32],[40,8,0,0,40,7,0,0,0,0,9,0,0,0,0,9],[9,5,0,0,25,32,0,0,0,0,3,0,0,0,0,3] >;
 

80 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4J5A5B8A···8H8I···8T10A···10F10G···10R20A···20H20I···20T
order12222···244444···4558···88···810···1010···1020···2020···20
size11112···211112···2225···510···102···24···42···24···4

80 irreducible representations

dim1111111122222224
type+++++++---+
imageC1C2C2C2C2C4C4C4D5D10Dic5Dic5Dic5D10C8○D4Q8.Dic5
kernelC2×Q8.Dic5C22×C5⋊2C8C2×C4.Dic5Q8.Dic5C10×C4○D4D4×C10Q8×C10C5×C4○D4C2×C4○D4C22×C4C2×D4C2×Q8C4○D4C4○D4C10C2
# reps1338162826628888

In GAP, Magma, Sage, TeX 

C_2\times Q_8.Dic_5
 
% in TeX
 
G:=Group("C2xQ8.Dic5");
 
// GroupNames label
 
G:=SmallGroup(320,1490);
 
// by ID
 
G=gap.SmallGroup(320,1490);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,297,102,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=1,c^2=d^10=b^2,e^2=d^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
 
// generators/relations
 

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