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G = C22.F5⋊C4order 320 = 26·5

4th semidirect product of C22.F5 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.F54C4, (C22×C4).3F5, C22.3(C4×F5), (C22×C20).9C4, C22.4(C4⋊F5), C23.35(C2×F5), Dic5.8(C4⋊C4), (C2×C10).15C42, (C2×Dic5).12Q8, C2.2(C23.F5), C52(C22.C42), (C2×Dic5).104D4, C10.2(C4.D4), (C22×Dic5).8C4, C10.5(C4.10D4), Dic5.6(C22⋊C4), C22.40(C22⋊F5), C2.19(D10.3Q8), C2.3(Dic5.D4), C10.19(C2.C42), (C22×Dic5).174C22, (C2×C10).18(C4⋊C4), (C2×C22.F5).2C2, (C22×C10).48(C2×C4), (C2×Dic5).45(C2×C4), (C2×C10.D4).2C2, (C2×C10).32(C22⋊C4), SmallGroup(320,257)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C22.F5⋊C4
C1C5C10Dic5C2×Dic5C22×Dic5C2×C22.F5 — C22.F5⋊C4
C5C10C2×C10 — C22.F5⋊C4
C1C22C23C22×C4

Generators and relations for C22.F5⋊C4
 G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=b, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=abc, ede-1=d3 >

Subgroups: 354 in 98 conjugacy classes, 38 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×6], C22 [×3], C22 [×2], C5, C8 [×4], C2×C4 [×10], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], Dic5 [×4], Dic5, C20, C2×C10 [×3], C2×C10 [×2], C2×C4⋊C4, C2×M4(2) [×2], C5⋊C8 [×4], C2×Dic5 [×2], C2×Dic5 [×4], C2×Dic5 [×2], C2×C20 [×2], C22×C10, C22.C42, C10.D4 [×2], C2×C5⋊C8 [×2], C22.F5 [×4], C22.F5 [×2], C22×Dic5 [×2], C22×C20, C2×C10.D4, C2×C22.F5 [×2], C22.F5⋊C4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], F5, C2.C42, C4.D4, C4.10D4, C2×F5, C22.C42, C4×F5, C4⋊F5, C22⋊F5, Dic5.D4, D10.3Q8, C23.F5, C22.F5⋊C4

Smallest permutation representation of C22.F5⋊C4
On 160 points
Generators in S160
(1 75)(2 80)(3 77)(4 74)(5 79)(6 76)(7 73)(8 78)(9 66)(10 71)(11 68)(12 65)(13 70)(14 67)(15 72)(16 69)(17 123)(18 128)(19 125)(20 122)(21 127)(22 124)(23 121)(24 126)(25 150)(26 147)(27 152)(28 149)(29 146)(30 151)(31 148)(32 145)(33 131)(34 136)(35 133)(36 130)(37 135)(38 132)(39 129)(40 134)(41 105)(42 110)(43 107)(44 112)(45 109)(46 106)(47 111)(48 108)(49 120)(50 117)(51 114)(52 119)(53 116)(54 113)(55 118)(56 115)(57 96)(58 93)(59 90)(60 95)(61 92)(62 89)(63 94)(64 91)(81 139)(82 144)(83 141)(84 138)(85 143)(86 140)(87 137)(88 142)(97 154)(98 159)(99 156)(100 153)(101 158)(102 155)(103 160)(104 157)
(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 48 75 108)(2 105 76 45)(3 46 77 106)(4 111 78 43)(5 44 79 112)(6 109 80 41)(7 42 73 110)(8 107 74 47)(9 151 70 26)(10 27 71 152)(11 149 72 32)(12 25 65 150)(13 147 66 30)(14 31 67 148)(15 145 68 28)(16 29 69 146)(17 86 127 144)(18 137 128 87)(19 84 121 142)(20 143 122 85)(21 82 123 140)(22 141 124 83)(23 88 125 138)(24 139 126 81)(33 61 135 96)(34 89 136 62)(35 59 129 94)(36 95 130 60)(37 57 131 92)(38 93 132 58)(39 63 133 90)(40 91 134 64)(49 160 116 99)(50 100 117 153)(51 158 118 97)(52 98 119 159)(53 156 120 103)(54 104 113 157)(55 154 114 101)(56 102 115 155)
(1 159 36 87 65)(2 88 160 66 37)(3 67 81 38 153)(4 39 68 154 82)(5 155 40 83 69)(6 84 156 70 33)(7 71 85 34 157)(8 35 72 158 86)(9 135 80 142 103)(10 143 136 104 73)(11 97 144 74 129)(12 75 98 130 137)(13 131 76 138 99)(14 139 132 100 77)(15 101 140 78 133)(16 79 102 134 141)(17 47 94 149 51)(18 150 48 52 95)(19 53 151 96 41)(20 89 54 42 152)(21 43 90 145 55)(22 146 44 56 91)(23 49 147 92 45)(24 93 50 46 148)(25 108 119 60 128)(26 61 109 121 120)(27 122 62 113 110)(28 114 123 111 63)(29 112 115 64 124)(30 57 105 125 116)(31 126 58 117 106)(32 118 127 107 59)
(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)| (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,66)(10,71)(11,68)(12,65)(13,70)(14,67)(15,72)(16,69)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,131)(34,136)(35,133)(36,130)(37,135)(38,132)(39,129)(40,134)(41,105)(42,110)(43,107)(44,112)(45,109)(46,106)(47,111)(48,108)(49,120)(50,117)(51,114)(52,119)(53,116)(54,113)(55,118)(56,115)(57,96)(58,93)(59,90)(60,95)(61,92)(62,89)(63,94)(64,91)(81,139)(82,144)(83,141)(84,138)(85,143)(86,140)(87,137)(88,142)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (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), (1,48,75,108)(2,105,76,45)(3,46,77,106)(4,111,78,43)(5,44,79,112)(6,109,80,41)(7,42,73,110)(8,107,74,47)(9,151,70,26)(10,27,71,152)(11,149,72,32)(12,25,65,150)(13,147,66,30)(14,31,67,148)(15,145,68,28)(16,29,69,146)(17,86,127,144)(18,137,128,87)(19,84,121,142)(20,143,122,85)(21,82,123,140)(22,141,124,83)(23,88,125,138)(24,139,126,81)(33,61,135,96)(34,89,136,62)(35,59,129,94)(36,95,130,60)(37,57,131,92)(38,93,132,58)(39,63,133,90)(40,91,134,64)(49,160,116,99)(50,100,117,153)(51,158,118,97)(52,98,119,159)(53,156,120,103)(54,104,113,157)(55,154,114,101)(56,102,115,155), (1,159,36,87,65)(2,88,160,66,37)(3,67,81,38,153)(4,39,68,154,82)(5,155,40,83,69)(6,84,156,70,33)(7,71,85,34,157)(8,35,72,158,86)(9,135,80,142,103)(10,143,136,104,73)(11,97,144,74,129)(12,75,98,130,137)(13,131,76,138,99)(14,139,132,100,77)(15,101,140,78,133)(16,79,102,134,141)(17,47,94,149,51)(18,150,48,52,95)(19,53,151,96,41)(20,89,54,42,152)(21,43,90,145,55)(22,146,44,56,91)(23,49,147,92,45)(24,93,50,46,148)(25,108,119,60,128)(26,61,109,121,120)(27,122,62,113,110)(28,114,123,111,63)(29,112,115,64,124)(30,57,105,125,116)(31,126,58,117,106)(32,118,127,107,59), (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( (1,75)(2,80)(3,77)(4,74)(5,79)(6,76)(7,73)(8,78)(9,66)(10,71)(11,68)(12,65)(13,70)(14,67)(15,72)(16,69)(17,123)(18,128)(19,125)(20,122)(21,127)(22,124)(23,121)(24,126)(25,150)(26,147)(27,152)(28,149)(29,146)(30,151)(31,148)(32,145)(33,131)(34,136)(35,133)(36,130)(37,135)(38,132)(39,129)(40,134)(41,105)(42,110)(43,107)(44,112)(45,109)(46,106)(47,111)(48,108)(49,120)(50,117)(51,114)(52,119)(53,116)(54,113)(55,118)(56,115)(57,96)(58,93)(59,90)(60,95)(61,92)(62,89)(63,94)(64,91)(81,139)(82,144)(83,141)(84,138)(85,143)(86,140)(87,137)(88,142)(97,154)(98,159)(99,156)(100,153)(101,158)(102,155)(103,160)(104,157), (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), (1,48,75,108)(2,105,76,45)(3,46,77,106)(4,111,78,43)(5,44,79,112)(6,109,80,41)(7,42,73,110)(8,107,74,47)(9,151,70,26)(10,27,71,152)(11,149,72,32)(12,25,65,150)(13,147,66,30)(14,31,67,148)(15,145,68,28)(16,29,69,146)(17,86,127,144)(18,137,128,87)(19,84,121,142)(20,143,122,85)(21,82,123,140)(22,141,124,83)(23,88,125,138)(24,139,126,81)(33,61,135,96)(34,89,136,62)(35,59,129,94)(36,95,130,60)(37,57,131,92)(38,93,132,58)(39,63,133,90)(40,91,134,64)(49,160,116,99)(50,100,117,153)(51,158,118,97)(52,98,119,159)(53,156,120,103)(54,104,113,157)(55,154,114,101)(56,102,115,155), (1,159,36,87,65)(2,88,160,66,37)(3,67,81,38,153)(4,39,68,154,82)(5,155,40,83,69)(6,84,156,70,33)(7,71,85,34,157)(8,35,72,158,86)(9,135,80,142,103)(10,143,136,104,73)(11,97,144,74,129)(12,75,98,130,137)(13,131,76,138,99)(14,139,132,100,77)(15,101,140,78,133)(16,79,102,134,141)(17,47,94,149,51)(18,150,48,52,95)(19,53,151,96,41)(20,89,54,42,152)(21,43,90,145,55)(22,146,44,56,91)(23,49,147,92,45)(24,93,50,46,148)(25,108,119,60,128)(26,61,109,121,120)(27,122,62,113,110)(28,114,123,111,63)(29,112,115,64,124)(30,57,105,125,116)(31,126,58,117,106)(32,118,127,107,59), (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([(1,75),(2,80),(3,77),(4,74),(5,79),(6,76),(7,73),(8,78),(9,66),(10,71),(11,68),(12,65),(13,70),(14,67),(15,72),(16,69),(17,123),(18,128),(19,125),(20,122),(21,127),(22,124),(23,121),(24,126),(25,150),(26,147),(27,152),(28,149),(29,146),(30,151),(31,148),(32,145),(33,131),(34,136),(35,133),(36,130),(37,135),(38,132),(39,129),(40,134),(41,105),(42,110),(43,107),(44,112),(45,109),(46,106),(47,111),(48,108),(49,120),(50,117),(51,114),(52,119),(53,116),(54,113),(55,118),(56,115),(57,96),(58,93),(59,90),(60,95),(61,92),(62,89),(63,94),(64,91),(81,139),(82,144),(83,141),(84,138),(85,143),(86,140),(87,137),(88,142),(97,154),(98,159),(99,156),(100,153),(101,158),(102,155),(103,160),(104,157)], [(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)], [(1,48,75,108),(2,105,76,45),(3,46,77,106),(4,111,78,43),(5,44,79,112),(6,109,80,41),(7,42,73,110),(8,107,74,47),(9,151,70,26),(10,27,71,152),(11,149,72,32),(12,25,65,150),(13,147,66,30),(14,31,67,148),(15,145,68,28),(16,29,69,146),(17,86,127,144),(18,137,128,87),(19,84,121,142),(20,143,122,85),(21,82,123,140),(22,141,124,83),(23,88,125,138),(24,139,126,81),(33,61,135,96),(34,89,136,62),(35,59,129,94),(36,95,130,60),(37,57,131,92),(38,93,132,58),(39,63,133,90),(40,91,134,64),(49,160,116,99),(50,100,117,153),(51,158,118,97),(52,98,119,159),(53,156,120,103),(54,104,113,157),(55,154,114,101),(56,102,115,155)], [(1,159,36,87,65),(2,88,160,66,37),(3,67,81,38,153),(4,39,68,154,82),(5,155,40,83,69),(6,84,156,70,33),(7,71,85,34,157),(8,35,72,158,86),(9,135,80,142,103),(10,143,136,104,73),(11,97,144,74,129),(12,75,98,130,137),(13,131,76,138,99),(14,139,132,100,77),(15,101,140,78,133),(16,79,102,134,141),(17,47,94,149,51),(18,150,48,52,95),(19,53,151,96,41),(20,89,54,42,152),(21,43,90,145,55),(22,146,44,56,91),(23,49,147,92,45),(24,93,50,46,148),(25,108,119,60,128),(26,61,109,121,120),(27,122,62,113,110),(28,114,123,111,63),(29,112,115,64,124),(30,57,105,125,116),(31,126,58,117,106),(32,118,127,107,59)], [(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)])

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H 5 8A···8H10A···10G20A···20H
order1222224444444458···810···1020···20
size11112244101010102020420···204···44···4

38 irreducible representations

dim11111122444444444
type++++-++-++-
imageC1C2C2C4C4C4D4Q8F5C4.D4C4.10D4C2×F5C4×F5C4⋊F5C22⋊F5Dic5.D4C23.F5
kernelC22.F5⋊C4C2×C10.D4C2×C22.F5C22.F5C22×Dic5C22×C20C2×Dic5C2×Dic5C22×C4C10C10C23C22C22C22C2C2
# reps11282231111122244

Matrix representation of C22.F5⋊C4 in GL6(𝔽41)

4000000
0400000
0040000
0004000
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
25180000
29160000
00383200
0033300
000039
0000838
,
100000
010000
0031100
0010300
00002834
00001219
,
13130000
9280000
0000400
0000040
0014900
00102700

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[25,29,0,0,0,0,18,16,0,0,0,0,0,0,38,33,0,0,0,0,32,3,0,0,0,0,0,0,3,8,0,0,0,0,9,38],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,10,0,0,0,0,1,3,0,0,0,0,0,0,28,12,0,0,0,0,34,19],[13,9,0,0,0,0,13,28,0,0,0,0,0,0,0,0,14,10,0,0,0,0,9,27,0,0,40,0,0,0,0,0,0,40,0,0] >;

C22.F5⋊C4 in GAP, Magma, Sage, TeX

C_2^2.F_5\rtimes C_4
% in TeX

G:=Group("C2^2.F5:C4");
// GroupNames label

G:=SmallGroup(320,257);
// by ID

G=gap.SmallGroup(320,257);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,253,64,387,184,1684,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=b,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
// generators/relations

׿
×
𝔽