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G = C22×C22.F5  order 320 = 26·5

Direct product of C22 and C22.F5

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

Aliases: C22×C22.F5, C24.5F5, Dic5.24C24, C5⋊C8⋊3C23, (C2×C10)⋊9M4(2), C10⋊3(C2×M4(2)), C5⋊3(C22×M4(2)), C23.55(C2×F5), C2.20(C23×F5), C10.20(C23×C4), (C23×C10).10C4, C22.61(C22×F5), (C22×Dic5).39C4, (C23×Dic5).15C2, Dic5.51(C22×C4), (C2×Dic5).367C23, (C22×Dic5).285C22, (C22×C5⋊C8)⋊10C2, (C2×C5⋊C8)⋊13C22, (C22×C10).84(C2×C4), (C2×C10).106(C22×C4), (C2×Dic5).201(C2×C4), SmallGroup(320,1606)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C22×C22.F5
Chief series
C1 — C5 — C10 — Dic5 — C5⋊C8 — C2×C5⋊C8 — C22×C5⋊C8 — C22×C22.F5
Lower central
C5 — C10 — C22×C22.F5
Upper central
C1 — C23 — C24

Generators and relations for C22×C22.F5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=1, f4=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 746 in 298 conjugacy classes, 156 normal (13 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C24, Dic5, Dic5, C2×C10, C2×C10, C22×C8, C2×M4(2), C23×C4, C5⋊C8, C2×Dic5, C22×C10, C22×C10, C22×C10, C22×M4(2), C2×C5⋊C8, C22.F5, C22×Dic5, C22×Dic5, C23×C10, C22×C5⋊C8, C2×C22.F5, C23×Dic5, C22×C22.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, M4(2), C22×C4, C24, F5, C2×M4(2), C23×C4, C2×F5, C22×M4(2), C22.F5, C22×F5, C2×C22.F5, C23×F5, C22×C22.F5

Smallest permutation representation of C22×C22.F5
►On 160 points
Generators in S160
(1 21)(2 22)(3 23)(4 24)(5 17)(6 18)(7 19)(8 20)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 25)(16 26)(33 117)(34 118)(35 119)(36 120)(37 113)(38 114)(39 115)(40 116)(41 100)(42 101)(43 102)(44 103)(45 104)(46 97)(47 98)(48 99)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)(65 85)(66 86)(67 87)(68 88)(69 81)(70 82)(71 83)(72 84)(73 93)(74 94)(75 95)(76 96)(77 89)(78 90)(79 91)(80 92)(105 160)(106 153)(107 154)(108 155)(109 156)(110 157)(111 158)(112 159)(121 141)(122 142)(123 143)(124 144)(125 137)(126 138)(127 139)(128 140)(129 147)(130 148)(131 149)(132 150)(133 151)(134 152)(135 145)(136 146)

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

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

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

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I4J4K4L 5 8A···8P10A···10O
order12···222224···4444458···810···10
size11···122225···510101010410···104···4

56 irreducible representations

dim1111112444
type++++++-
imageC1C2C2C2C4C4M4(2)F5C2×F5C22.F5
kernelC22×C22.F5C22×C5⋊C8C2×C22.F5C23×Dic5C22×Dic5C23×C10C2×C10C24C23C22
# reps121211428178

Matrix representation of C22×C22.F5 ►in GL10(𝔽41)

1000000000
0100000000
00400000000
00040000000
00004000000
00000400000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
00004000000
00000400000
0000001000
0000000100
0000000010
0000000001
,
40000000000
9100000000
00400000000
0041000000
0000100000
000029400000
0000001000
0000000100
0000000010
0000000001
,
40000000000
04000000000
00400000000
00040000000
00004000000
00000400000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
00000040100
00000033700
00000000346
00000000340
,
9200000000
53200000000
003337000000
0088000000
000017370000
000029240000
0000000010
0000000001
000000152300
000000172600

G:=sub<GL(10,GF(41))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,29,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,33,0,0,0,0,0,0,0,0,1,7,0,0,0,0,0,0,0,0,0,0,34,34,0,0,0,0,0,0,0,0,6,0],[9,5,0,0,0,0,0,0,0,0,2,32,0,0,0,0,0,0,0,0,0,0,33,8,0,0,0,0,0,0,0,0,37,8,0,0,0,0,0,0,0,0,0,0,17,29,0,0,0,0,0,0,0,0,37,24,0,0,0,0,0,0,0,0,0,0,0,0,15,17,0,0,0,0,0,0,0,0,23,26,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
 

C22×C22.F5 in GAP, Magma, Sage, TeX 

C_2^2\times C_2^2.F_5
 
% in TeX
 
G:=Group("C2^2xC2^2.F5");
 
// GroupNames label
 
G:=SmallGroup(320,1606);
 
// by ID
 
G=gap.SmallGroup(320,1606);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,112,1123,102,6278,818]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=1,f^4=d,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
 
// generators/relations
 

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