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## G = C5×C4.C42order 320 = 26·5

### Direct product of C5 and C4.C42

direct product, metabelian, nilpotent (class 3), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C4 — C5×C4.C42
 Chief series C1 — C2 — C4 — C2×C4 — C22×C4 — C22×C20 — C10×M4(2) — C5×C4.C42
 Lower central C1 — C2 — C4 — C5×C4.C42
 Upper central C1 — C2×C20 — C22×C20 — C5×C4.C42

Generators and relations for C5×C4.C42
G = < a,b,c,d | a5=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c >

Subgroups: 122 in 90 conjugacy classes, 58 normal (22 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C8, C2×C4, C2×C4, C23, C10, C10, C10, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C20, C20, C2×C10, C2×C10, C2×C10, C22×C8, C2×M4(2), C40, C2×C20, C2×C20, C22×C10, C4.C42, C2×C40, C2×C40, C5×M4(2), C5×M4(2), C22×C20, C22×C40, C10×M4(2), C5×C4.C42
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, Q8, C10, C42, C22⋊C4, C4⋊C4, C20, C2×C10, C2.C42, C8.C4, C2×C20, C5×D4, C5×Q8, C4.C42, C4×C20, C5×C22⋊C4, C5×C4⋊C4, C5×C2.C42, C5×C8.C4, C5×C4.C42

Smallest permutation representation of C5×C4.C42
On 160 points
Generators in S160
(1 105 25 97 17)(2 106 26 98 18)(3 107 27 99 19)(4 108 28 100 20)(5 109 29 101 21)(6 110 30 102 22)(7 111 31 103 23)(8 112 32 104 24)(9 46 118 38 90)(10 47 119 39 91)(11 48 120 40 92)(12 41 113 33 93)(13 42 114 34 94)(14 43 115 35 95)(15 44 116 36 96)(16 45 117 37 89)(49 121 137 57 129)(50 122 138 58 130)(51 123 139 59 131)(52 124 140 60 132)(53 125 141 61 133)(54 126 142 62 134)(55 127 143 63 135)(56 128 144 64 136)(65 87 153 73 145)(66 88 154 74 146)(67 81 155 75 147)(68 82 156 76 148)(69 83 157 77 149)(70 84 158 78 150)(71 85 159 79 151)(72 86 160 80 152)
(1 7 5 3)(2 4 6 8)(9 11 13 15)(10 16 14 12)(17 23 21 19)(18 20 22 24)(25 31 29 27)(26 28 30 32)(33 39 37 35)(34 36 38 40)(41 47 45 43)(42 44 46 48)(49 55 53 51)(50 52 54 56)(57 63 61 59)(58 60 62 64)(65 71 69 67)(66 68 70 72)(73 79 77 75)(74 76 78 80)(81 87 85 83)(82 84 86 88)(89 95 93 91)(90 92 94 96)(97 103 101 99)(98 100 102 104)(105 111 109 107)(106 108 110 112)(113 119 117 115)(114 116 118 120)(121 127 125 123)(122 124 126 128)(129 135 133 131)(130 132 134 136)(137 143 141 139)(138 140 142 144)(145 151 149 147)(146 148 150 152)(153 159 157 155)(154 156 158 160)
(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 67 123 37 5 71 127 33)(2 66 128 40 6 70 124 36)(3 69 125 39 7 65 121 35)(4 68 122 34 8 72 126 38)(9 28 156 58 13 32 160 62)(10 31 153 57 14 27 157 61)(11 30 158 60 15 26 154 64)(12 25 155 59 16 29 159 63)(17 147 51 117 21 151 55 113)(18 146 56 120 22 150 52 116)(19 149 53 119 23 145 49 115)(20 148 50 114 24 152 54 118)(41 97 75 131 45 101 79 135)(42 104 80 134 46 100 76 130)(43 99 77 133 47 103 73 129)(44 98 74 136 48 102 78 132)(81 139 89 109 85 143 93 105)(82 138 94 112 86 142 90 108)(83 141 91 111 87 137 95 107)(84 140 96 106 88 144 92 110)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 8A ··· 8H 8I ··· 8P 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20X 40A ··· 40AF 40AG ··· 40BL order 1 2 2 2 2 2 4 4 4 4 4 4 5 5 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 40 ··· 40 size 1 1 1 1 2 2 1 1 1 1 2 2 1 1 1 1 2 ··· 2 4 ··· 4 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + - image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D4 Q8 C8.C4 C5×D4 C5×Q8 C5×C8.C4 kernel C5×C4.C42 C22×C40 C10×M4(2) C2×C40 C5×M4(2) C4.C42 C22×C8 C2×M4(2) C2×C8 M4(2) C2×C20 C22×C10 C10 C2×C4 C23 C2 # reps 1 1 2 4 8 4 4 8 16 32 3 1 8 12 4 32

Matrix representation of C5×C4.C42 in GL3(𝔽41) generated by

 1 0 0 0 37 0 0 0 37
,
 1 0 0 0 9 0 0 24 32
,
 32 0 0 0 30 39 0 24 11
,
 32 0 0 0 14 0 0 1 3
G:=sub<GL(3,GF(41))| [1,0,0,0,37,0,0,0,37],[1,0,0,0,9,24,0,0,32],[32,0,0,0,30,24,0,39,11],[32,0,0,0,14,1,0,0,3] >;

C5×C4.C42 in GAP, Magma, Sage, TeX

C_5\times C_4.C_4^2
% in TeX

G:=Group("C5xC4.C4^2");
// GroupNames label

G:=SmallGroup(320,146);
// by ID

G=gap.SmallGroup(320,146);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,568,5043,248,172,10085,124]);
// Polycyclic

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

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