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G = C24.44D10order 320 = 26·5

2nd non-split extension by C24 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.44D10, C23.9Dic10, C22.94(D4×D5), C23.49(C4×D5), (C22×C4).22D10, (C22×C10).60D4, (C22×C10).10Q8, C10.81(C4⋊D4), C53(C23.7Q8), Dic54(C22⋊C4), (C2×Dic5).227D4, (C22×Dic5)⋊11C4, C23.35(C5⋊D4), C10.15(C22⋊Q8), (C23×Dic5).2C2, C10.10C429C2, C2.1(Dic5⋊D4), C221(C10.D4), (C22×C20).21C22, (C23×C10).25C22, C22.23(C2×Dic10), C23.275(C22×D5), C10.43(C42⋊C2), C22.41(D42D5), (C22×C10).317C23, C2.5(Dic5.14D4), (C22×Dic5).34C22, C2.11(C23.11D10), (C2×C10)⋊4(C4⋊C4), C10.52(C2×C4⋊C4), (C2×C22⋊C4).4D5, (C2×C10).30(C2×Q8), C2.27(D5×C22⋊C4), C22.121(C2×C4×D5), (C2×C10).429(C2×D4), (C2×C10.D4)⋊6C2, (C10×C22⋊C4).5C2, C10.68(C2×C22⋊C4), C2.5(C2×C10.D4), C22.45(C2×C5⋊D4), (C2×C23.D5).4C2, (C2×C10).138(C4○D4), (C2×C10).204(C22×C4), (C22×C10).113(C2×C4), (C2×Dic5).147(C2×C4), SmallGroup(320,569)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.44D10
C1C5C10C2×C10C22×C10C22×Dic5C23×Dic5 — C24.44D10
C5C2×C10 — C24.44D10
C1C23C2×C22⋊C4

Generators and relations for C24.44D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=b, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce9 >

Subgroups: 734 in 234 conjugacy classes, 87 normal (25 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×10], C22 [×3], C22 [×8], C22 [×12], C5, C2×C4 [×30], C23, C23 [×6], C23 [×4], C10 [×3], C10 [×4], C10 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4 [×2], C22×C4 [×12], C24, Dic5 [×4], Dic5 [×4], C20 [×2], C2×C10 [×3], C2×C10 [×8], C2×C10 [×12], C2.C42 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4 [×2], C23×C4, C2×Dic5 [×8], C2×Dic5 [×16], C2×C20 [×6], C22×C10, C22×C10 [×6], C22×C10 [×4], C23.7Q8, C10.D4 [×4], C23.D5 [×2], C5×C22⋊C4 [×2], C22×Dic5 [×2], C22×Dic5 [×6], C22×Dic5 [×4], C22×C20 [×2], C23×C10, C10.10C42 [×2], C2×C10.D4 [×2], C2×C23.D5, C10×C22⋊C4, C23×Dic5, C24.44D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, D5, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], D10 [×3], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], Dic10 [×2], C4×D5 [×2], C5⋊D4 [×2], C22×D5, C23.7Q8, C10.D4 [×4], C2×Dic10, C2×C4×D5, D4×D5 [×2], D42D5 [×2], C2×C5⋊D4, C23.11D10, Dic5.14D4 [×2], D5×C22⋊C4, C2×C10.D4, Dic5⋊D4 [×2], C24.44D10

Smallest permutation representation of C24.44D10
On 160 points
Generators in S160
(1 11)(2 35)(3 13)(4 37)(5 15)(6 39)(7 17)(8 21)(9 19)(10 23)(12 25)(14 27)(16 29)(18 31)(20 33)(22 32)(24 34)(26 36)(28 38)(30 40)(41 97)(42 52)(43 99)(44 54)(45 81)(46 56)(47 83)(48 58)(49 85)(50 60)(51 87)(53 89)(55 91)(57 93)(59 95)(61 71)(62 135)(63 73)(64 137)(65 75)(66 139)(67 77)(68 121)(69 79)(70 123)(72 125)(74 127)(76 129)(78 131)(80 133)(82 92)(84 94)(86 96)(88 98)(90 100)(101 111)(102 149)(103 113)(104 151)(105 115)(106 153)(107 117)(108 155)(109 119)(110 157)(112 159)(114 141)(116 143)(118 145)(120 147)(122 132)(124 134)(126 136)(128 138)(130 140)(142 152)(144 154)(146 156)(148 158)(150 160)
(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 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 61)(10 62)(11 63)(12 64)(13 65)(14 66)(15 67)(16 68)(17 69)(18 70)(19 71)(20 72)(21 133)(22 134)(23 135)(24 136)(25 137)(26 138)(27 139)(28 140)(29 121)(30 122)(31 123)(32 124)(33 125)(34 126)(35 127)(36 128)(37 129)(38 130)(39 131)(40 132)(41 153)(42 154)(43 155)(44 156)(45 157)(46 158)(47 159)(48 160)(49 141)(50 142)(51 143)(52 144)(53 145)(54 146)(55 147)(56 148)(57 149)(58 150)(59 151)(60 152)(81 110)(82 111)(83 112)(84 113)(85 114)(86 115)(87 116)(88 117)(89 118)(90 119)(91 120)(92 101)(93 102)(94 103)(95 104)(96 105)(97 106)(98 107)(99 108)(100 109)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 32)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 40)(18 21)(19 22)(20 23)(41 87)(42 88)(43 89)(44 90)(45 91)(46 92)(47 93)(48 94)(49 95)(50 96)(51 97)(52 98)(53 99)(54 100)(55 81)(56 82)(57 83)(58 84)(59 85)(60 86)(61 124)(62 125)(63 126)(64 127)(65 128)(66 129)(67 130)(68 131)(69 132)(70 133)(71 134)(72 135)(73 136)(74 137)(75 138)(76 139)(77 140)(78 121)(79 122)(80 123)(101 158)(102 159)(103 160)(104 141)(105 142)(106 143)(107 144)(108 145)(109 146)(110 147)(111 148)(112 149)(113 150)(114 151)(115 152)(116 153)(117 154)(118 155)(119 156)(120 157)
(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 85 11 95)(2 103 12 113)(3 83 13 93)(4 101 14 111)(5 81 15 91)(6 119 16 109)(7 99 17 89)(8 117 18 107)(9 97 19 87)(10 115 20 105)(21 144 31 154)(22 41 32 51)(23 142 33 152)(24 59 34 49)(25 160 35 150)(26 57 36 47)(27 158 37 148)(28 55 38 45)(29 156 39 146)(30 53 40 43)(42 133 52 123)(44 131 54 121)(46 129 56 139)(48 127 58 137)(50 125 60 135)(61 106 71 116)(62 86 72 96)(63 104 73 114)(64 84 74 94)(65 102 75 112)(66 82 76 92)(67 120 77 110)(68 100 78 90)(69 118 79 108)(70 98 80 88)(122 145 132 155)(124 143 134 153)(126 141 136 151)(128 159 138 149)(130 157 140 147)

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

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

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

68 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4L4M4N4O4P5A5B10A···10N10O···10V20A···20P
order12···2222244444···444445510···1010···1020···20
size11···12222444410···1020202020222···24···44···4

68 irreducible representations

dim1111111222222222244
type++++++++-+++-+-
imageC1C2C2C2C2C2C4D4D4Q8D5C4○D4D10D10Dic10C4×D5C5⋊D4D4×D5D42D5
kernelC24.44D10C10.10C42C2×C10.D4C2×C23.D5C10×C22⋊C4C23×Dic5C22×Dic5C2×Dic5C22×C10C22×C10C2×C22⋊C4C2×C10C22×C4C24C23C23C23C22C22
# reps1221118422244288844

Matrix representation of C24.44D10 in GL5(𝔽41)

10000
01000
004000
00010
00001
,
400000
01000
00100
00010
00001
,
10000
040000
004000
000400
000040
,
10000
040000
004000
00010
00001
,
320000
00100
01000
00034
0002040
,
90000
003200
09000
00078
0003534

G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,40],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,3,20,0,0,0,4,40],[9,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,7,35,0,0,0,8,34] >;

C24.44D10 in GAP, Magma, Sage, TeX

C_2^4._{44}D_{10}
% in TeX

G:=Group("C2^4.44D10");
// GroupNames label

G:=SmallGroup(320,569);
// by ID

G=gap.SmallGroup(320,569);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,422,387,58,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=b,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^9>;
// generators/relations

׿
×
𝔽