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G = (C22×C4).F5order 320 = 26·5

1st non-split extension by C22×C4 of F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Dic5)⋊2C8, (C22×C4).1F5, (C22×C20).7C4, C2.1(C23⋊F5), C23.32(C2×F5), C10.6(C22⋊C8), C22.3(D5⋊C8), (C2×C10).5M4(2), C2.7(D10⋊C8), C10.12(C23⋊C4), C22.4(C4.F5), (C2×Dic5).102D4, (C22×Dic5).7C4, C23.2F5.2C2, C10.4(C4.10D4), C22.36(C22⋊F5), C2.2(Dic5.D4), C52(C22.M4(2)), (C22×Dic5).172C22, (C2×C10).8(C2×C8), (C22×C10).43(C2×C4), (C2×C10.D4).1C2, (C2×C10).27(C22⋊C4), SmallGroup(320,252)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C22×C4).F5
C1C5C10C2×C10C2×Dic5C22×Dic5C23.2F5 — (C22×C4).F5
C5C10C2×C10 — (C22×C4).F5
C1C22C23C22×C4

Generators and relations for (C22×C4).F5
 G = < a,b,c,d,e | a2=b2=c4=d5=1, e4=c2, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, cd=dc, ece-1=abc, ede-1=d3 >

Subgroups: 306 in 78 conjugacy classes, 28 normal (22 characteristic)
C1, C2 [×3], C2 [×2], C4 [×5], C22 [×3], C22 [×2], C5, C8 [×2], C2×C4 [×9], C23, C10 [×3], C10 [×2], C4⋊C4 [×2], C2×C8 [×2], C22×C4, C22×C4 [×2], Dic5 [×4], C20, C2×C10 [×3], C2×C10 [×2], C22⋊C8 [×2], C2×C4⋊C4, C5⋊C8 [×2], C2×Dic5 [×4], C2×Dic5 [×3], C2×C20 [×2], C22×C10, C22.M4(2), C10.D4 [×2], C2×C5⋊C8 [×2], C22×Dic5 [×2], C22×C20, C23.2F5 [×2], C2×C10.D4, (C22×C4).F5
Quotients: C1, C2 [×3], C4 [×2], C22, C8 [×2], C2×C4, D4 [×2], C22⋊C4, C2×C8, M4(2), F5, C22⋊C8, C23⋊C4, C4.10D4, C2×F5, C22.M4(2), D5⋊C8, C4.F5, C22⋊F5, D10⋊C8, Dic5.D4, C23⋊F5, (C22×C4).F5

Smallest permutation representation of (C22×C4).F5
On 160 points
Generators in S160
(2 78)(4 80)(6 74)(8 76)(9 133)(11 135)(13 129)(15 131)(17 48)(19 42)(21 44)(23 46)(25 152)(27 146)(29 148)(31 150)(34 101)(36 103)(38 97)(40 99)(49 93)(51 95)(53 89)(55 91)(57 123)(59 125)(61 127)(63 121)(66 160)(68 154)(70 156)(72 158)(81 144)(83 138)(85 140)(87 142)(105 118)(107 120)(109 114)(111 116)
(1 77)(2 78)(3 79)(4 80)(5 73)(6 74)(7 75)(8 76)(9 133)(10 134)(11 135)(12 136)(13 129)(14 130)(15 131)(16 132)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(25 152)(26 145)(27 146)(28 147)(29 148)(30 149)(31 150)(32 151)(33 100)(34 101)(35 102)(36 103)(37 104)(38 97)(39 98)(40 99)(49 93)(50 94)(51 95)(52 96)(53 89)(54 90)(55 91)(56 92)(57 123)(58 124)(59 125)(60 126)(61 127)(62 128)(63 121)(64 122)(65 159)(66 160)(67 153)(68 154)(69 155)(70 156)(71 157)(72 158)(81 144)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(105 118)(106 119)(107 120)(108 113)(109 114)(110 115)(111 116)(112 117)
(1 137 5 141)(2 83 6 87)(3 84 7 88)(4 140 8 144)(9 57 13 61)(10 58 14 62)(11 125 15 121)(12 126 16 122)(17 97 21 101)(18 98 22 102)(19 40 23 36)(20 33 24 37)(25 68 29 72)(26 69 30 65)(27 156 31 160)(28 157 32 153)(34 48 38 44)(35 41 39 45)(42 99 46 103)(43 100 47 104)(49 109 53 105)(50 110 54 106)(51 116 55 120)(52 117 56 113)(59 131 63 135)(60 132 64 136)(66 146 70 150)(67 147 71 151)(73 86 77 82)(74 142 78 138)(75 143 79 139)(76 81 80 85)(89 118 93 114)(90 119 94 115)(91 107 95 111)(92 108 96 112)(123 129 127 133)(124 130 128 134)(145 155 149 159)(148 158 152 154)
(1 100 60 155 54)(2 156 101 55 61)(3 56 157 62 102)(4 63 49 103 158)(5 104 64 159 50)(6 160 97 51 57)(7 52 153 58 98)(8 59 53 99 154)(9 83 31 17 120)(10 18 84 113 32)(11 114 19 25 85)(12 26 115 86 20)(13 87 27 21 116)(14 22 88 117 28)(15 118 23 29 81)(16 30 119 82 24)(33 126 69 90 77)(34 91 127 78 70)(35 79 92 71 128)(36 72 80 121 93)(37 122 65 94 73)(38 95 123 74 66)(39 75 96 67 124)(40 68 76 125 89)(41 139 108 151 134)(42 152 140 135 109)(43 136 145 110 141)(44 111 129 142 146)(45 143 112 147 130)(46 148 144 131 105)(47 132 149 106 137)(48 107 133 138 150)
(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)| (2,78)(4,80)(6,74)(8,76)(9,133)(11,135)(13,129)(15,131)(17,48)(19,42)(21,44)(23,46)(25,152)(27,146)(29,148)(31,150)(34,101)(36,103)(38,97)(40,99)(49,93)(51,95)(53,89)(55,91)(57,123)(59,125)(61,127)(63,121)(66,160)(68,154)(70,156)(72,158)(81,144)(83,138)(85,140)(87,142)(105,118)(107,120)(109,114)(111,116), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,133)(10,134)(11,135)(12,136)(13,129)(14,130)(15,131)(16,132)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,121)(64,122)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,144)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,137,5,141)(2,83,6,87)(3,84,7,88)(4,140,8,144)(9,57,13,61)(10,58,14,62)(11,125,15,121)(12,126,16,122)(17,97,21,101)(18,98,22,102)(19,40,23,36)(20,33,24,37)(25,68,29,72)(26,69,30,65)(27,156,31,160)(28,157,32,153)(34,48,38,44)(35,41,39,45)(42,99,46,103)(43,100,47,104)(49,109,53,105)(50,110,54,106)(51,116,55,120)(52,117,56,113)(59,131,63,135)(60,132,64,136)(66,146,70,150)(67,147,71,151)(73,86,77,82)(74,142,78,138)(75,143,79,139)(76,81,80,85)(89,118,93,114)(90,119,94,115)(91,107,95,111)(92,108,96,112)(123,129,127,133)(124,130,128,134)(145,155,149,159)(148,158,152,154), (1,100,60,155,54)(2,156,101,55,61)(3,56,157,62,102)(4,63,49,103,158)(5,104,64,159,50)(6,160,97,51,57)(7,52,153,58,98)(8,59,53,99,154)(9,83,31,17,120)(10,18,84,113,32)(11,114,19,25,85)(12,26,115,86,20)(13,87,27,21,116)(14,22,88,117,28)(15,118,23,29,81)(16,30,119,82,24)(33,126,69,90,77)(34,91,127,78,70)(35,79,92,71,128)(36,72,80,121,93)(37,122,65,94,73)(38,95,123,74,66)(39,75,96,67,124)(40,68,76,125,89)(41,139,108,151,134)(42,152,140,135,109)(43,136,145,110,141)(44,111,129,142,146)(45,143,112,147,130)(46,148,144,131,105)(47,132,149,106,137)(48,107,133,138,150), (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( (2,78)(4,80)(6,74)(8,76)(9,133)(11,135)(13,129)(15,131)(17,48)(19,42)(21,44)(23,46)(25,152)(27,146)(29,148)(31,150)(34,101)(36,103)(38,97)(40,99)(49,93)(51,95)(53,89)(55,91)(57,123)(59,125)(61,127)(63,121)(66,160)(68,154)(70,156)(72,158)(81,144)(83,138)(85,140)(87,142)(105,118)(107,120)(109,114)(111,116), (1,77)(2,78)(3,79)(4,80)(5,73)(6,74)(7,75)(8,76)(9,133)(10,134)(11,135)(12,136)(13,129)(14,130)(15,131)(16,132)(17,48)(18,41)(19,42)(20,43)(21,44)(22,45)(23,46)(24,47)(25,152)(26,145)(27,146)(28,147)(29,148)(30,149)(31,150)(32,151)(33,100)(34,101)(35,102)(36,103)(37,104)(38,97)(39,98)(40,99)(49,93)(50,94)(51,95)(52,96)(53,89)(54,90)(55,91)(56,92)(57,123)(58,124)(59,125)(60,126)(61,127)(62,128)(63,121)(64,122)(65,159)(66,160)(67,153)(68,154)(69,155)(70,156)(71,157)(72,158)(81,144)(82,137)(83,138)(84,139)(85,140)(86,141)(87,142)(88,143)(105,118)(106,119)(107,120)(108,113)(109,114)(110,115)(111,116)(112,117), (1,137,5,141)(2,83,6,87)(3,84,7,88)(4,140,8,144)(9,57,13,61)(10,58,14,62)(11,125,15,121)(12,126,16,122)(17,97,21,101)(18,98,22,102)(19,40,23,36)(20,33,24,37)(25,68,29,72)(26,69,30,65)(27,156,31,160)(28,157,32,153)(34,48,38,44)(35,41,39,45)(42,99,46,103)(43,100,47,104)(49,109,53,105)(50,110,54,106)(51,116,55,120)(52,117,56,113)(59,131,63,135)(60,132,64,136)(66,146,70,150)(67,147,71,151)(73,86,77,82)(74,142,78,138)(75,143,79,139)(76,81,80,85)(89,118,93,114)(90,119,94,115)(91,107,95,111)(92,108,96,112)(123,129,127,133)(124,130,128,134)(145,155,149,159)(148,158,152,154), (1,100,60,155,54)(2,156,101,55,61)(3,56,157,62,102)(4,63,49,103,158)(5,104,64,159,50)(6,160,97,51,57)(7,52,153,58,98)(8,59,53,99,154)(9,83,31,17,120)(10,18,84,113,32)(11,114,19,25,85)(12,26,115,86,20)(13,87,27,21,116)(14,22,88,117,28)(15,118,23,29,81)(16,30,119,82,24)(33,126,69,90,77)(34,91,127,78,70)(35,79,92,71,128)(36,72,80,121,93)(37,122,65,94,73)(38,95,123,74,66)(39,75,96,67,124)(40,68,76,125,89)(41,139,108,151,134)(42,152,140,135,109)(43,136,145,110,141)(44,111,129,142,146)(45,143,112,147,130)(46,148,144,131,105)(47,132,149,106,137)(48,107,133,138,150), (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([(2,78),(4,80),(6,74),(8,76),(9,133),(11,135),(13,129),(15,131),(17,48),(19,42),(21,44),(23,46),(25,152),(27,146),(29,148),(31,150),(34,101),(36,103),(38,97),(40,99),(49,93),(51,95),(53,89),(55,91),(57,123),(59,125),(61,127),(63,121),(66,160),(68,154),(70,156),(72,158),(81,144),(83,138),(85,140),(87,142),(105,118),(107,120),(109,114),(111,116)], [(1,77),(2,78),(3,79),(4,80),(5,73),(6,74),(7,75),(8,76),(9,133),(10,134),(11,135),(12,136),(13,129),(14,130),(15,131),(16,132),(17,48),(18,41),(19,42),(20,43),(21,44),(22,45),(23,46),(24,47),(25,152),(26,145),(27,146),(28,147),(29,148),(30,149),(31,150),(32,151),(33,100),(34,101),(35,102),(36,103),(37,104),(38,97),(39,98),(40,99),(49,93),(50,94),(51,95),(52,96),(53,89),(54,90),(55,91),(56,92),(57,123),(58,124),(59,125),(60,126),(61,127),(62,128),(63,121),(64,122),(65,159),(66,160),(67,153),(68,154),(69,155),(70,156),(71,157),(72,158),(81,144),(82,137),(83,138),(84,139),(85,140),(86,141),(87,142),(88,143),(105,118),(106,119),(107,120),(108,113),(109,114),(110,115),(111,116),(112,117)], [(1,137,5,141),(2,83,6,87),(3,84,7,88),(4,140,8,144),(9,57,13,61),(10,58,14,62),(11,125,15,121),(12,126,16,122),(17,97,21,101),(18,98,22,102),(19,40,23,36),(20,33,24,37),(25,68,29,72),(26,69,30,65),(27,156,31,160),(28,157,32,153),(34,48,38,44),(35,41,39,45),(42,99,46,103),(43,100,47,104),(49,109,53,105),(50,110,54,106),(51,116,55,120),(52,117,56,113),(59,131,63,135),(60,132,64,136),(66,146,70,150),(67,147,71,151),(73,86,77,82),(74,142,78,138),(75,143,79,139),(76,81,80,85),(89,118,93,114),(90,119,94,115),(91,107,95,111),(92,108,96,112),(123,129,127,133),(124,130,128,134),(145,155,149,159),(148,158,152,154)], [(1,100,60,155,54),(2,156,101,55,61),(3,56,157,62,102),(4,63,49,103,158),(5,104,64,159,50),(6,160,97,51,57),(7,52,153,58,98),(8,59,53,99,154),(9,83,31,17,120),(10,18,84,113,32),(11,114,19,25,85),(12,26,115,86,20),(13,87,27,21,116),(14,22,88,117,28),(15,118,23,29,81),(16,30,119,82,24),(33,126,69,90,77),(34,91,127,78,70),(35,79,92,71,128),(36,72,80,121,93),(37,122,65,94,73),(38,95,123,74,66),(39,75,96,67,124),(40,68,76,125,89),(41,139,108,151,134),(42,152,140,135,109),(43,136,145,110,141),(44,111,129,142,146),(45,143,112,147,130),(46,148,144,131,105),(47,132,149,106,137),(48,107,133,138,150)], [(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++++++-++-
imageC1C2C2C4C4C8D4M4(2)F5C23⋊C4C4.10D4C2×F5D5⋊C8C4.F5C22⋊F5Dic5.D4C23⋊F5
kernel(C22×C4).F5C23.2F5C2×C10.D4C22×Dic5C22×C20C2×Dic5C2×Dic5C2×C10C22×C4C10C10C23C22C22C22C2C2
# reps12122822111122244

Matrix representation of (C22×C4).F5 in GL10(𝔽41)

40000000000
04000000000
0010000000
0001000000
00004000000
00000400000
0000001000
0000000100
0000000010
0000000001
,
1000000000
0100000000
00400000000
00040000000
00004000000
00000400000
0000001000
0000000100
0000000010
0000000001
,
0900000000
9000000000
00320000000
002038000000
00003200000
000020380000
00000040000
00000004000
00000000400
00000000040
,
1000000000
0100000000
0010000000
0001000000
0000100000
0000010000
00000000040
00000010040
00000001040
00000000140
,
03800000000
3000000000
0000100000
0000010000
0001000000
00400000000
000000110424
000000534625
0000007351629
00000017394031

G:=sub<GL(10,GF(41))| [40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,3,20,0,0,0,0,0,0,0,0,20,38,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40],[0,3,0,0,0,0,0,0,0,0,38,0,0,0,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,5,7,17,0,0,0,0,0,0,10,34,35,39,0,0,0,0,0,0,4,6,16,40,0,0,0,0,0,0,24,25,29,31] >;

(C22×C4).F5 in GAP, Magma, Sage, TeX

(C_2^2\times C_4).F_5
% in TeX

G:=Group("(C2^2xC4).F5");
// GroupNames label

G:=SmallGroup(320,252);
// by ID

G=gap.SmallGroup(320,252);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,120,387,100,1123,6278,3156]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=1,e^4=c^2,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

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