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G = C4×C4.Dic5order 320 = 26·5

Direct product of C4 and C4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C4×C4.Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C2×C4.Dic5 — C4×C4.Dic5
 Lower central C5 — C10 — C4×C4.Dic5
 Upper central C1 — C42 — C2×C42

Generators and relations for C4×C4.Dic5
G = < a,b,c,d | a4=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >

Subgroups: 270 in 142 conjugacy classes, 95 normal (21 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, C23, C10, C10, C10, C42, C42, C2×C8, M4(2), C22×C4, C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C4×C8, C8⋊C4, C2×C42, C2×M4(2), C52C8, C2×C20, C2×C20, C2×C20, C22×C10, C4×M4(2), C2×C52C8, C4.Dic5, C4×C20, C4×C20, C22×C20, C22×C20, C4×C52C8, C42.D5, C2×C4.Dic5, C2×C4×C20, C4×C4.Dic5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C42, M4(2), C22×C4, Dic5, D10, C2×C42, C2×M4(2), C4×D5, C2×Dic5, C22×D5, C4×M4(2), C4.Dic5, C4×Dic5, C2×C4×D5, C22×Dic5, C2×C4.Dic5, C2×C4×Dic5, C4×C4.Dic5

Smallest permutation representation of C4×C4.Dic5
On 160 points
Generators in S160
(1 61 137 29)(2 62 138 30)(3 63 139 31)(4 64 140 32)(5 65 121 33)(6 66 122 34)(7 67 123 35)(8 68 124 36)(9 69 125 37)(10 70 126 38)(11 71 127 39)(12 72 128 40)(13 73 129 21)(14 74 130 22)(15 75 131 23)(16 76 132 24)(17 77 133 25)(18 78 134 26)(19 79 135 27)(20 80 136 28)(41 91 151 112)(42 92 152 113)(43 93 153 114)(44 94 154 115)(45 95 155 116)(46 96 156 117)(47 97 157 118)(48 98 158 119)(49 99 159 120)(50 100 160 101)(51 81 141 102)(52 82 142 103)(53 83 143 104)(54 84 144 105)(55 85 145 106)(56 86 146 107)(57 87 147 108)(58 88 148 109)(59 89 149 110)(60 90 150 111)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 35 40)(41 56 51 46)(42 57 52 47)(43 58 53 48)(44 59 54 49)(45 60 55 50)(61 66 71 76)(62 67 72 77)(63 68 73 78)(64 69 74 79)(65 70 75 80)(81 96 91 86)(82 97 92 87)(83 98 93 88)(84 99 94 89)(85 100 95 90)(101 116 111 106)(102 117 112 107)(103 118 113 108)(104 119 114 109)(105 120 115 110)(121 126 131 136)(122 127 132 137)(123 128 133 138)(124 129 134 139)(125 130 135 140)(141 156 151 146)(142 157 152 147)(143 158 153 148)(144 159 154 149)(145 160 155 150)
(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)
(1 160 6 145 11 150 16 155)(2 149 7 154 12 159 17 144)(3 158 8 143 13 148 18 153)(4 147 9 152 14 157 19 142)(5 156 10 141 15 146 20 151)(21 88 26 93 31 98 36 83)(22 97 27 82 32 87 37 92)(23 86 28 91 33 96 38 81)(24 95 29 100 34 85 39 90)(25 84 30 89 35 94 40 99)(41 121 46 126 51 131 56 136)(42 130 47 135 52 140 57 125)(43 139 48 124 53 129 58 134)(44 128 49 133 54 138 59 123)(45 137 50 122 55 127 60 132)(61 101 66 106 71 111 76 116)(62 110 67 115 72 120 77 105)(63 119 68 104 73 109 78 114)(64 108 69 113 74 118 79 103)(65 117 70 102 75 107 80 112)

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

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

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

104 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 5A 5B 8A ··· 8P 10A ··· 10N 20A ··· 20AV order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 8 ··· 8 10 ··· 10 20 ··· 20 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 2 2 10 ··· 10 2 ··· 2 2 ··· 2

104 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + - + - + image C1 C2 C2 C2 C2 C4 C4 C4 D5 M4(2) Dic5 D10 Dic5 D10 C4×D5 C4.Dic5 kernel C4×C4.Dic5 C4×C5⋊2C8 C42.D5 C2×C4.Dic5 C2×C4×C20 C4.Dic5 C4×C20 C22×C20 C2×C42 C20 C42 C42 C22×C4 C22×C4 C2×C4 C4 # reps 1 2 2 2 1 16 4 4 2 8 4 4 4 2 16 32

Matrix representation of C4×C4.Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 9 0 0 0 0 9
,
 32 9 0 0 0 9 0 0 0 0 40 0 0 0 0 40
,
 32 0 0 0 0 32 0 0 0 0 34 1 0 0 33 1
,
 9 36 0 0 18 32 0 0 0 0 32 9 0 0 0 9
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,9,0,0,0,0,9],[32,0,0,0,9,9,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,0,32,0,0,0,0,34,33,0,0,1,1],[9,18,0,0,36,32,0,0,0,0,32,0,0,0,9,9] >;

C4×C4.Dic5 in GAP, Magma, Sage, TeX

C_4\times C_4.{\rm Dic}_5
% in TeX

G:=Group("C4xC4.Dic5");
// GroupNames label

G:=SmallGroup(320,549);
// by ID

G=gap.SmallGroup(320,549);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,100,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=c^5,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^9>;
// generators/relations

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