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## G = C22×C22.F5order 320 = 26·5

### Direct product of C22 and C22.F5

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 [×6], C2 [×4], C4 [×8], C22 [×11], C22 [×12], C5, C8 [×8], C2×C4 [×28], C23, C23 [×6], C23 [×4], C10, C10 [×6], C10 [×4], C2×C8 [×12], M4(2) [×16], C22×C4 [×14], C24, Dic5, Dic5 [×7], C2×C10 [×11], C2×C10 [×12], C22×C8 [×2], C2×M4(2) [×12], C23×C4, C5⋊C8 [×8], C2×Dic5 [×28], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×M4(2), C2×C5⋊C8 [×12], C22.F5 [×16], C22×Dic5 [×2], C22×Dic5 [×12], C23×C10, C22×C5⋊C8 [×2], C2×C22.F5 [×12], C23×Dic5, C22×C22.F5
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], M4(2) [×4], C22×C4 [×14], C24, F5, C2×M4(2) [×6], C23×C4, C2×F5 [×7], C22×M4(2), C22.F5 [×4], C22×F5 [×7], C2×C22.F5 [×6], 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 ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 5 8A ··· 8P 10A ··· 10O order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 8 ··· 8 10 ··· 10 size 1 1 ··· 1 2 2 2 2 5 ··· 5 10 10 10 10 4 10 ··· 10 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 2 4 4 4 type + + + + + + - image C1 C2 C2 C2 C4 C4 M4(2) F5 C2×F5 C22.F5 kernel C22×C22.F5 C22×C5⋊C8 C2×C22.F5 C23×Dic5 C22×Dic5 C23×C10 C2×C10 C24 C23 C22 # reps 1 2 12 1 14 2 8 1 7 8

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

 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 0 0 0 0 0 0 0 0 0 9 1 0 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 29 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 1 0 0 0 0 0 0 0 0 33 7 0 0 0 0 0 0 0 0 0 0 34 6 0 0 0 0 0 0 0 0 34 0
,
 9 2 0 0 0 0 0 0 0 0 5 32 0 0 0 0 0 0 0 0 0 0 33 37 0 0 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0 0 0 0 17 37 0 0 0 0 0 0 0 0 29 24 0 0 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 15 23 0 0 0 0 0 0 0 0 17 26 0 0

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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