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## G = C4×C4.F5order 320 = 26·5

### Direct product of C4 and C4.F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C4.F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C2×C5⋊C8 — C4×C5⋊C8 — C4×C4.F5
 Lower central C5 — C10 — C4×C4.F5
 Upper central C1 — C2×C4 — C42

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, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, D5, C10, C42, C42, C2×C8, M4(2), C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C5⋊C8, C4×D5, C4×D5, C2×Dic5, C2×C20, C22×D5, C4×M4(2), C4×Dic5, C4×C20, C4.F5, C2×C5⋊C8, C2×C4×D5, C4×C5⋊C8, C10.C42, D5×C42, C2×C4.F5, C4×C4.F5
Quotients: C1, C2, C4, C22, C2×C4, C23, C42, M4(2), C22×C4, F5, C2×C42, C2×M4(2), C2×F5, C4×M4(2), C4.F5, C4×F5, 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 64 28 21)(2 57 29 22)(3 58 30 23)(4 59 31 24)(5 60 32 17)(6 61 25 18)(7 62 26 19)(8 63 27 20)(9 111 96 150)(10 112 89 151)(11 105 90 152)(12 106 91 145)(13 107 92 146)(14 108 93 147)(15 109 94 148)(16 110 95 149)(33 141 42 81)(34 142 43 82)(35 143 44 83)(36 144 45 84)(37 137 46 85)(38 138 47 86)(39 139 48 87)(40 140 41 88)(49 121 115 97)(50 122 116 98)(51 123 117 99)(52 124 118 100)(53 125 119 101)(54 126 120 102)(55 127 113 103)(56 128 114 104)(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 58 21 62)(18 63 22 59)(19 60 23 64)(20 57 24 61)(33 123 37 127)(34 128 38 124)(35 125 39 121)(36 122 40 126)(41 102 45 98)(42 99 46 103)(43 104 47 100)(44 101 48 97)(49 83 53 87)(50 88 54 84)(51 85 55 81)(52 82 56 86)(73 150 77 146)(74 147 78 151)(75 152 79 148)(76 149 80 145)(89 136 93 132)(90 133 94 129)(91 130 95 134)(92 135 96 131)(105 159 109 155)(106 156 110 160)(107 153 111 157)(108 158 112 154)(113 141 117 137)(114 138 118 142)(115 143 119 139)(116 140 120 144)
(1 155 48 123 146)(2 124 156 147 41)(3 148 125 42 157)(4 43 149 158 126)(5 159 44 127 150)(6 128 160 151 45)(7 152 121 46 153)(8 47 145 154 122)(9 60 133 83 113)(10 84 61 114 134)(11 115 85 135 62)(12 136 116 63 86)(13 64 129 87 117)(14 88 57 118 130)(15 119 81 131 58)(16 132 120 59 82)(17 72 143 55 96)(18 56 65 89 144)(19 90 49 137 66)(20 138 91 67 50)(21 68 139 51 92)(22 52 69 93 140)(23 94 53 141 70)(24 142 95 71 54)(25 104 80 112 36)(26 105 97 37 73)(27 38 106 74 98)(28 75 39 99 107)(29 100 76 108 40)(30 109 101 33 77)(31 34 110 78 102)(32 79 35 103 111)
(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)| (1,64,28,21)(2,57,29,22)(3,58,30,23)(4,59,31,24)(5,60,32,17)(6,61,25,18)(7,62,26,19)(8,63,27,20)(9,111,96,150)(10,112,89,151)(11,105,90,152)(12,106,91,145)(13,107,92,146)(14,108,93,147)(15,109,94,148)(16,110,95,149)(33,141,42,81)(34,142,43,82)(35,143,44,83)(36,144,45,84)(37,137,46,85)(38,138,47,86)(39,139,48,87)(40,140,41,88)(49,121,115,97)(50,122,116,98)(51,123,117,99)(52,124,118,100)(53,125,119,101)(54,126,120,102)(55,127,113,103)(56,128,114,104)(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)(17,58,21,62)(18,63,22,59)(19,60,23,64)(20,57,24,61)(33,123,37,127)(34,128,38,124)(35,125,39,121)(36,122,40,126)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(49,83,53,87)(50,88,54,84)(51,85,55,81)(52,82,56,86)(73,150,77,146)(74,147,78,151)(75,152,79,148)(76,149,80,145)(89,136,93,132)(90,133,94,129)(91,130,95,134)(92,135,96,131)(105,159,109,155)(106,156,110,160)(107,153,111,157)(108,158,112,154)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,48,123,146)(2,124,156,147,41)(3,148,125,42,157)(4,43,149,158,126)(5,159,44,127,150)(6,128,160,151,45)(7,152,121,46,153)(8,47,145,154,122)(9,60,133,83,113)(10,84,61,114,134)(11,115,85,135,62)(12,136,116,63,86)(13,64,129,87,117)(14,88,57,118,130)(15,119,81,131,58)(16,132,120,59,82)(17,72,143,55,96)(18,56,65,89,144)(19,90,49,137,66)(20,138,91,67,50)(21,68,139,51,92)(22,52,69,93,140)(23,94,53,141,70)(24,142,95,71,54)(25,104,80,112,36)(26,105,97,37,73)(27,38,106,74,98)(28,75,39,99,107)(29,100,76,108,40)(30,109,101,33,77)(31,34,110,78,102)(32,79,35,103,111), (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( (1,64,28,21)(2,57,29,22)(3,58,30,23)(4,59,31,24)(5,60,32,17)(6,61,25,18)(7,62,26,19)(8,63,27,20)(9,111,96,150)(10,112,89,151)(11,105,90,152)(12,106,91,145)(13,107,92,146)(14,108,93,147)(15,109,94,148)(16,110,95,149)(33,141,42,81)(34,142,43,82)(35,143,44,83)(36,144,45,84)(37,137,46,85)(38,138,47,86)(39,139,48,87)(40,140,41,88)(49,121,115,97)(50,122,116,98)(51,123,117,99)(52,124,118,100)(53,125,119,101)(54,126,120,102)(55,127,113,103)(56,128,114,104)(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)(17,58,21,62)(18,63,22,59)(19,60,23,64)(20,57,24,61)(33,123,37,127)(34,128,38,124)(35,125,39,121)(36,122,40,126)(41,102,45,98)(42,99,46,103)(43,104,47,100)(44,101,48,97)(49,83,53,87)(50,88,54,84)(51,85,55,81)(52,82,56,86)(73,150,77,146)(74,147,78,151)(75,152,79,148)(76,149,80,145)(89,136,93,132)(90,133,94,129)(91,130,95,134)(92,135,96,131)(105,159,109,155)(106,156,110,160)(107,153,111,157)(108,158,112,154)(113,141,117,137)(114,138,118,142)(115,143,119,139)(116,140,120,144), (1,155,48,123,146)(2,124,156,147,41)(3,148,125,42,157)(4,43,149,158,126)(5,159,44,127,150)(6,128,160,151,45)(7,152,121,46,153)(8,47,145,154,122)(9,60,133,83,113)(10,84,61,114,134)(11,115,85,135,62)(12,136,116,63,86)(13,64,129,87,117)(14,88,57,118,130)(15,119,81,131,58)(16,132,120,59,82)(17,72,143,55,96)(18,56,65,89,144)(19,90,49,137,66)(20,138,91,67,50)(21,68,139,51,92)(22,52,69,93,140)(23,94,53,141,70)(24,142,95,71,54)(25,104,80,112,36)(26,105,97,37,73)(27,38,106,74,98)(28,75,39,99,107)(29,100,76,108,40)(30,109,101,33,77)(31,34,110,78,102)(32,79,35,103,111), (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([[(1,64,28,21),(2,57,29,22),(3,58,30,23),(4,59,31,24),(5,60,32,17),(6,61,25,18),(7,62,26,19),(8,63,27,20),(9,111,96,150),(10,112,89,151),(11,105,90,152),(12,106,91,145),(13,107,92,146),(14,108,93,147),(15,109,94,148),(16,110,95,149),(33,141,42,81),(34,142,43,82),(35,143,44,83),(36,144,45,84),(37,137,46,85),(38,138,47,86),(39,139,48,87),(40,140,41,88),(49,121,115,97),(50,122,116,98),(51,123,117,99),(52,124,118,100),(53,125,119,101),(54,126,120,102),(55,127,113,103),(56,128,114,104),(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),(17,58,21,62),(18,63,22,59),(19,60,23,64),(20,57,24,61),(33,123,37,127),(34,128,38,124),(35,125,39,121),(36,122,40,126),(41,102,45,98),(42,99,46,103),(43,104,47,100),(44,101,48,97),(49,83,53,87),(50,88,54,84),(51,85,55,81),(52,82,56,86),(73,150,77,146),(74,147,78,151),(75,152,79,148),(76,149,80,145),(89,136,93,132),(90,133,94,129),(91,130,95,134),(92,135,96,131),(105,159,109,155),(106,156,110,160),(107,153,111,157),(108,158,112,154),(113,141,117,137),(114,138,118,142),(115,143,119,139),(116,140,120,144)], [(1,155,48,123,146),(2,124,156,147,41),(3,148,125,42,157),(4,43,149,158,126),(5,159,44,127,150),(6,128,160,151,45),(7,152,121,46,153),(8,47,145,154,122),(9,60,133,83,113),(10,84,61,114,134),(11,115,85,135,62),(12,136,116,63,86),(13,64,129,87,117),(14,88,57,118,130),(15,119,81,131,58),(16,132,120,59,82),(17,72,143,55,96),(18,56,65,89,144),(19,90,49,137,66),(20,138,91,67,50),(21,68,139,51,92),(22,52,69,93,140),(23,94,53,141,70),(24,142,95,71,54),(25,104,80,112,36),(26,105,97,37,73),(27,38,106,74,98),(28,75,39,99,107),(29,100,76,108,40),(30,109,101,33,77),(31,34,110,78,102),(32,79,35,103,111)], [(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 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 4Q 4R 5 8A ··· 8P 10A 10B 10C 20A ··· 20L order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 4 ··· 4 4 4 5 8 ··· 8 10 10 10 20 ··· 20 size 1 1 1 1 10 10 1 1 1 1 2 2 2 2 5 ··· 5 10 10 4 10 ··· 10 4 4 4 4 ··· 4

56 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 4 4 4 4 4 type + + + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C4 M4(2) M4(2) F5 C2×F5 C4.F5 C4×F5 D5⋊M4(2) kernel C4×C4.F5 C4×C5⋊C8 C10.C42 D5×C42 C2×C4.F5 C4×Dic5 C4×C20 C4.F5 C2×C4×D5 Dic5 C20 C42 C2×C4 C4 C4 C2 # reps 1 2 2 1 2 2 2 16 4 4 4 1 3 4 4 4

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

 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 32 0 0 0 0 0 9 0 0 0 0 0 0 7 27 0 14 0 0 0 34 27 14 0 0 14 27 34 0 0 0 14 0 27 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 1 0 0 0 0 40 0 1 0 0 0 40 0 0 1 0 0 40 0 0 0
,
 9 4 0 0 0 0 23 32 0 0 0 0 0 0 39 39 33 20 0 0 31 18 25 18 0 0 23 16 23 10 0 0 21 8 2 2

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

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