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G = (C22×D5).Q8  order 320 = 26·5

1st non-split extension by C22×D5 of Q8 acting via Q8/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×D5).1Q8, C22.43(Q8×D5), C10.8(C4⋊D4), (C2×Dic5).22D4, (C22×C4).21D10, C22.160(D4×D5), C2.C42⋊5D5, C5⋊1(C23.Q8), C2.7(C42⋊2D5), C10.27(C22⋊Q8), C2.11(D10⋊D4), (C23×D5).8C22, C10.2(C42⋊2C2), C2.11(D10⋊Q8), C22.93(C4○D20), (C22×C20).20C22, C23.364(C22×D5), (C22×C10).301C23, (C22×Dic5).23C22, (C2×C10).70(C2×Q8), (C2×C10.D4)⋊4C2, (C2×C10).208(C2×D4), (C2×C10).61(C4○D4), (C5×C2.C42)⋊2C2, (C2×D10⋊C4).11C2, SmallGroup(320,303)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22×C10 — (C22×D5).Q8
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C23×D5 — C2×D10⋊C4 — (C22×D5).Q8
Lower central
C5 — C22×C10 — (C22×D5).Q8
Upper central
C1 — C23 — C2.C42

Generators and relations for (C22×D5).Q8
 G = < a,b,c,d,e,f | a2=b2=c5=d2=e4=1, f2=ae2, ab=ba, ac=ca, ede-1=ad=da, ae=ea, af=fa, bc=cb, fdf-1=bd=db, be=eb, bf=fb, dcd=c-1, ce=ec, cf=fc, fef-1=be-1 >

Subgroups: 790 in 186 conjugacy classes, 59 normal (12 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C2×C4⋊C4, C2×Dic5, C2×Dic5, C2×C20, C22×D5, C22×D5, C22×C10, C23.Q8, C10.D4, D10⋊C4, C22×Dic5, C22×C20, C23×D5, C5×C2.C42, C2×C10.D4, C2×D10⋊C4, (C22×D5).Q8
Quotients: C1, C2, C22, D4, Q8, C23, D5, C2×D4, C2×Q8, C4○D4, D10, C4⋊D4, C22⋊Q8, C42⋊2C2, C22×D5, C23.Q8, C4○D20, D4×D5, Q8×D5, C42⋊2D5, D10⋊D4, D10⋊Q8, (C22×D5).Q8

Smallest permutation representation of (C22×D5).Q8
►On 160 points
Generators in S160
(1 36)(2 37)(3 38)(4 39)(5 40)(6 31)(7 32)(8 33)(9 34)(10 35)(11 26)(12 27)(13 28)(14 29)(15 30)(16 21)(17 22)(18 23)(19 24)(20 25)(41 76)(42 77)(43 78)(44 79)(45 80)(46 71)(47 72)(48 73)(49 74)(50 75)(51 66)(52 67)(53 68)(54 69)(55 70)(56 61)(57 62)(58 63)(59 64)(60 65)(81 116)(82 117)(83 118)(84 119)(85 120)(86 111)(87 112)(88 113)(89 114)(90 115)(91 106)(92 107)(93 108)(94 109)(95 110)(96 101)(97 102)(98 103)(99 104)(100 105)(121 156)(122 157)(123 158)(124 159)(125 160)(126 151)(127 152)(128 153)(129 154)(130 155)(131 146)(132 147)(133 148)(134 149)(135 150)(136 141)(137 142)(138 143)(139 144)(140 145)

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

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

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

62 conjugacy classes

class 1 2A···2G2H2I4A···4F4G···4L5A5B10A···10N20A···20X
order12···2224···44···45510···1020···20
size11···120204···420···20222···24···4

62 irreducible representations

dim111122222244
type+++++-+++-
imageC1C2C2C2D4Q8D5C4○D4D10C4○D20D4×D5Q8×D5
kernel(C22×D5).Q8C5×C2.C42C2×C10.D4C2×D10⋊C4C2×Dic5C22×D5C2.C42C2×C10C22×C4C22C22C22
# reps1133622662462

Matrix representation of (C22×D5).Q8 ►in GL6(𝔽41)

4000000
0400000
001000
000100
0000400
0000040
,
4000000
0400000
0040000
0004000
0000400
0000040
,
100000
010000
001000
000100
0000640
000010
,
4000000
010000
0040000
000100
0000351
000066
,
010000
100000
001000
0004000
00003928
0000132
,
0400000
100000
000100
001000
00001835
0000623

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,35,6,0,0,0,0,1,6],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,39,13,0,0,0,0,28,2],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,18,6,0,0,0,0,35,23] >;
 

(C22×D5).Q8 in GAP, Magma, Sage, TeX 

(C_2^2\times D_5).Q_8
 
% in TeX
 
G:=Group("(C2^2xD5).Q8");
 
// GroupNames label
 
G:=SmallGroup(320,303);
 
// by ID
 
G=gap.SmallGroup(320,303);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,64,1262,387,268,12550]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^5=d^2=e^4=1,f^2=a*e^2,a*b=b*a,a*c=c*a,e*d*e^-1=a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=c^-1,c*e=e*c,c*f=f*c,f*e*f^-1=b*e^-1>;
 
// generators/relations
 

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