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G = C4×C4.F5order 320 = 26·5

Direct product of C4 and C4.F5

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

Aliases: C4×C4.F5, C204M4(2), C42.10F5, C20.11C42, Dic56M4(2), D10.10C42, C4.4(C4×F5), C51(C4×M4(2)), (C4×C20).11C4, C10.3(C2×C42), (C4×Dic5).34C4, (D5×C42).24C2, C10.2(C2×M4(2)), C2.2(D5⋊M4(2)), C10.C4216C2, C22.26(C22×F5), Dic5.27(C22×C4), (C2×Dic5).313C23, (C4×Dic5).319C22, (C4×C5⋊C8)⋊7C2, C5⋊C81(C2×C4), C2.6(C2×C4×F5), (C2×C4×D5).43C4, C2.1(C2×C4.F5), (C4×D5).64(C2×C4), (C2×C5⋊C8).15C22, (C2×C4).130(C2×F5), (C2×C4.F5).11C2, (C2×C20).120(C2×C4), (C2×C4×D5).389C22, (C2×C10).15(C22×C4), (C2×Dic5).163(C2×C4), (C22×D5).115(C2×C4), SmallGroup(320,1015)

Series: Derived Chief Lower central Upper central

C1C10 — C4×C4.F5
C1C5C10Dic5C2×Dic5C2×C5⋊C8C4×C5⋊C8 — C4×C4.F5
C5C10 — C4×C4.F5
C1C2×C4C42

Generators and relations for C4×C4.F5
 G = < a,b,c,d | a4=b4=c5=1, d4=b2, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c3 >

Subgroups: 426 in 142 conjugacy classes, 74 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×6], C22, C22 [×4], C5, C8 [×8], C2×C4 [×3], C2×C4 [×11], C23, D5 [×2], C10 [×3], C42, C42 [×3], C2×C8 [×4], M4(2) [×8], C22×C4 [×3], Dic5 [×4], Dic5, C20 [×4], C20, D10 [×2], D10 [×2], C2×C10, C4×C8 [×2], C8⋊C4 [×2], C2×C42, C2×M4(2) [×2], C5⋊C8 [×8], C4×D5 [×4], C4×D5 [×4], C2×Dic5 [×3], C2×C20 [×3], C22×D5, C4×M4(2), C4×Dic5 [×3], C4×C20, C4.F5 [×8], C2×C5⋊C8 [×4], C2×C4×D5 [×3], C4×C5⋊C8 [×2], C10.C42 [×2], D5×C42, C2×C4.F5 [×2], C4×C4.F5
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], C23, C42 [×4], M4(2) [×4], C22×C4 [×3], F5, C2×C42, C2×M4(2) [×2], C2×F5 [×3], C4×M4(2), C4.F5 [×2], C4×F5 [×2], C22×F5, C2×C4.F5, D5⋊M4(2), C2×C4×F5, C4×C4.F5

Smallest permutation representation of C4×C4.F5
On 160 points
Generators in S160
(1 81 28 21)(2 82 29 22)(3 83 30 23)(4 84 31 24)(5 85 32 17)(6 86 25 18)(7 87 26 19)(8 88 27 20)(9 44 53 128)(10 45 54 121)(11 46 55 122)(12 47 56 123)(13 48 49 124)(14 41 50 125)(15 42 51 126)(16 43 52 127)(33 141 109 94)(34 142 110 95)(35 143 111 96)(36 144 112 89)(37 137 105 90)(38 138 106 91)(39 139 107 92)(40 140 108 93)(57 149 118 100)(58 150 119 101)(59 151 120 102)(60 152 113 103)(61 145 114 104)(62 146 115 97)(63 147 116 98)(64 148 117 99)(65 160 134 80)(66 153 135 73)(67 154 136 74)(68 155 129 75)(69 156 130 76)(70 157 131 77)(71 158 132 78)(72 159 133 79)
(1 26 5 30)(2 31 6 27)(3 28 7 32)(4 25 8 29)(9 70 13 66)(10 67 14 71)(11 72 15 68)(12 69 16 65)(17 83 21 87)(18 88 22 84)(19 85 23 81)(20 82 24 86)(33 148 37 152)(34 145 38 149)(35 150 39 146)(36 147 40 151)(41 158 45 154)(42 155 46 159)(43 160 47 156)(44 157 48 153)(49 135 53 131)(50 132 54 136)(51 129 55 133)(52 134 56 130)(57 95 61 91)(58 92 62 96)(59 89 63 93)(60 94 64 90)(73 128 77 124)(74 125 78 121)(75 122 79 126)(76 127 80 123)(97 111 101 107)(98 108 102 112)(99 105 103 109)(100 110 104 106)(113 141 117 137)(114 138 118 142)(115 143 119 139)(116 140 120 144)
(1 155 107 148 124)(2 149 156 125 108)(3 126 150 109 157)(4 110 127 158 151)(5 159 111 152 128)(6 145 160 121 112)(7 122 146 105 153)(8 106 123 154 147)(9 85 133 96 113)(10 89 86 114 134)(11 115 90 135 87)(12 136 116 88 91)(13 81 129 92 117)(14 93 82 118 130)(15 119 94 131 83)(16 132 120 84 95)(17 72 143 60 53)(18 61 65 54 144)(19 55 62 137 66)(20 138 56 67 63)(21 68 139 64 49)(22 57 69 50 140)(23 51 58 141 70)(24 142 52 71 59)(25 104 80 45 36)(26 46 97 37 73)(27 38 47 74 98)(28 75 39 99 48)(29 100 76 41 40)(30 42 101 33 77)(31 34 43 78 102)(32 79 35 103 44)
(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)| (1,81,28,21)(2,82,29,22)(3,83,30,23)(4,84,31,24)(5,85,32,17)(6,86,25,18)(7,87,26,19)(8,88,27,20)(9,44,53,128)(10,45,54,121)(11,46,55,122)(12,47,56,123)(13,48,49,124)(14,41,50,125)(15,42,51,126)(16,43,52,127)(33,141,109,94)(34,142,110,95)(35,143,111,96)(36,144,112,89)(37,137,105,90)(38,138,106,91)(39,139,107,92)(40,140,108,93)(57,149,118,100)(58,150,119,101)(59,151,120,102)(60,152,113,103)(61,145,114,104)(62,146,115,97)(63,147,116,98)(64,148,117,99)(65,160,134,80)(66,153,135,73)(67,154,136,74)(68,155,129,75)(69,156,130,76)(70,157,131,77)(71,158,132,78)(72,159,133,79), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,70,13,66)(10,67,14,71)(11,72,15,68)(12,69,16,65)(17,83,21,87)(18,88,22,84)(19,85,23,81)(20,82,24,86)(33,148,37,152)(34,145,38,149)(35,150,39,146)(36,147,40,151)(41,158,45,154)(42,155,46,159)(43,160,47,156)(44,157,48,153)(49,135,53,131)(50,132,54,136)(51,129,55,133)(52,134,56,130)(57,95,61,91)(58,92,62,96)(59,89,63,93)(60,94,64,90)(73,128,77,124)(74,125,78,121)(75,122,79,126)(76,127,80,123)(97,111,101,107)(98,108,102,112)(99,105,103,109)(100,110,104,106)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,107,148,124)(2,149,156,125,108)(3,126,150,109,157)(4,110,127,158,151)(5,159,111,152,128)(6,145,160,121,112)(7,122,146,105,153)(8,106,123,154,147)(9,85,133,96,113)(10,89,86,114,134)(11,115,90,135,87)(12,136,116,88,91)(13,81,129,92,117)(14,93,82,118,130)(15,119,94,131,83)(16,132,120,84,95)(17,72,143,60,53)(18,61,65,54,144)(19,55,62,137,66)(20,138,56,67,63)(21,68,139,64,49)(22,57,69,50,140)(23,51,58,141,70)(24,142,52,71,59)(25,104,80,45,36)(26,46,97,37,73)(27,38,47,74,98)(28,75,39,99,48)(29,100,76,41,40)(30,42,101,33,77)(31,34,43,78,102)(32,79,35,103,44), (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( (1,81,28,21)(2,82,29,22)(3,83,30,23)(4,84,31,24)(5,85,32,17)(6,86,25,18)(7,87,26,19)(8,88,27,20)(9,44,53,128)(10,45,54,121)(11,46,55,122)(12,47,56,123)(13,48,49,124)(14,41,50,125)(15,42,51,126)(16,43,52,127)(33,141,109,94)(34,142,110,95)(35,143,111,96)(36,144,112,89)(37,137,105,90)(38,138,106,91)(39,139,107,92)(40,140,108,93)(57,149,118,100)(58,150,119,101)(59,151,120,102)(60,152,113,103)(61,145,114,104)(62,146,115,97)(63,147,116,98)(64,148,117,99)(65,160,134,80)(66,153,135,73)(67,154,136,74)(68,155,129,75)(69,156,130,76)(70,157,131,77)(71,158,132,78)(72,159,133,79), (1,26,5,30)(2,31,6,27)(3,28,7,32)(4,25,8,29)(9,70,13,66)(10,67,14,71)(11,72,15,68)(12,69,16,65)(17,83,21,87)(18,88,22,84)(19,85,23,81)(20,82,24,86)(33,148,37,152)(34,145,38,149)(35,150,39,146)(36,147,40,151)(41,158,45,154)(42,155,46,159)(43,160,47,156)(44,157,48,153)(49,135,53,131)(50,132,54,136)(51,129,55,133)(52,134,56,130)(57,95,61,91)(58,92,62,96)(59,89,63,93)(60,94,64,90)(73,128,77,124)(74,125,78,121)(75,122,79,126)(76,127,80,123)(97,111,101,107)(98,108,102,112)(99,105,103,109)(100,110,104,106)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,107,148,124)(2,149,156,125,108)(3,126,150,109,157)(4,110,127,158,151)(5,159,111,152,128)(6,145,160,121,112)(7,122,146,105,153)(8,106,123,154,147)(9,85,133,96,113)(10,89,86,114,134)(11,115,90,135,87)(12,136,116,88,91)(13,81,129,92,117)(14,93,82,118,130)(15,119,94,131,83)(16,132,120,84,95)(17,72,143,60,53)(18,61,65,54,144)(19,55,62,137,66)(20,138,56,67,63)(21,68,139,64,49)(22,57,69,50,140)(23,51,58,141,70)(24,142,52,71,59)(25,104,80,45,36)(26,46,97,37,73)(27,38,47,74,98)(28,75,39,99,48)(29,100,76,41,40)(30,42,101,33,77)(31,34,43,78,102)(32,79,35,103,44), (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([(1,81,28,21),(2,82,29,22),(3,83,30,23),(4,84,31,24),(5,85,32,17),(6,86,25,18),(7,87,26,19),(8,88,27,20),(9,44,53,128),(10,45,54,121),(11,46,55,122),(12,47,56,123),(13,48,49,124),(14,41,50,125),(15,42,51,126),(16,43,52,127),(33,141,109,94),(34,142,110,95),(35,143,111,96),(36,144,112,89),(37,137,105,90),(38,138,106,91),(39,139,107,92),(40,140,108,93),(57,149,118,100),(58,150,119,101),(59,151,120,102),(60,152,113,103),(61,145,114,104),(62,146,115,97),(63,147,116,98),(64,148,117,99),(65,160,134,80),(66,153,135,73),(67,154,136,74),(68,155,129,75),(69,156,130,76),(70,157,131,77),(71,158,132,78),(72,159,133,79)], [(1,26,5,30),(2,31,6,27),(3,28,7,32),(4,25,8,29),(9,70,13,66),(10,67,14,71),(11,72,15,68),(12,69,16,65),(17,83,21,87),(18,88,22,84),(19,85,23,81),(20,82,24,86),(33,148,37,152),(34,145,38,149),(35,150,39,146),(36,147,40,151),(41,158,45,154),(42,155,46,159),(43,160,47,156),(44,157,48,153),(49,135,53,131),(50,132,54,136),(51,129,55,133),(52,134,56,130),(57,95,61,91),(58,92,62,96),(59,89,63,93),(60,94,64,90),(73,128,77,124),(74,125,78,121),(75,122,79,126),(76,127,80,123),(97,111,101,107),(98,108,102,112),(99,105,103,109),(100,110,104,106),(113,141,117,137),(114,138,118,142),(115,143,119,139),(116,140,120,144)], [(1,155,107,148,124),(2,149,156,125,108),(3,126,150,109,157),(4,110,127,158,151),(5,159,111,152,128),(6,145,160,121,112),(7,122,146,105,153),(8,106,123,154,147),(9,85,133,96,113),(10,89,86,114,134),(11,115,90,135,87),(12,136,116,88,91),(13,81,129,92,117),(14,93,82,118,130),(15,119,94,131,83),(16,132,120,84,95),(17,72,143,60,53),(18,61,65,54,144),(19,55,62,137,66),(20,138,56,67,63),(21,68,139,64,49),(22,57,69,50,140),(23,51,58,141,70),(24,142,52,71,59),(25,104,80,45,36),(26,46,97,37,73),(27,38,47,74,98),(28,75,39,99,48),(29,100,76,41,40),(30,42,101,33,77),(31,34,43,78,102),(32,79,35,103,44)], [(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 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4P4Q4R 5 8A···8P10A10B10C20A···20L
order122222444444444···44458···810101020···20
size11111010111122225···51010410···104444···4

56 irreducible representations

dim1111111112244444
type+++++++
imageC1C2C2C2C2C4C4C4C4M4(2)M4(2)F5C2×F5C4.F5C4×F5D5⋊M4(2)
kernelC4×C4.F5C4×C5⋊C8C10.C42D5×C42C2×C4.F5C4×Dic5C4×C20C4.F5C2×C4×D5Dic5C20C42C2×C4C4C4C2
# reps12212221644413444

Matrix representation of C4×C4.F5 in GL6(𝔽41)

100000
010000
0032000
0003200
0000320
0000032
,
32320000
090000
00727014
000342714
001427340
00140277
,
100000
010000
0040100
0040010
0040001
0040000
,
940000
23320000
0039393320
0031182518
0023162310
0021822

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32,0,0,0,0,0,0,32],[32,0,0,0,0,0,32,9,0,0,0,0,0,0,7,0,14,14,0,0,27,34,27,0,0,0,0,27,34,27,0,0,14,14,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[9,23,0,0,0,0,4,32,0,0,0,0,0,0,39,31,23,21,0,0,39,18,16,8,0,0,33,25,23,2,0,0,20,18,10,2] >;

C4×C4.F5 in GAP, Magma, Sage, TeX

C_4\times C_4.F_5
% in TeX

G:=Group("C4xC4.F5");
// GroupNames label

G:=SmallGroup(320,1015);
// by ID

G=gap.SmallGroup(320,1015);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,120,758,184,136,6278,1595]);
// Polycyclic

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

׿
×
𝔽