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## G = M4(2)×C20order 320 = 26·5

### Direct product of C20 and M4(2)

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C20
 Chief series C1 — C2 — C22 — C2×C4 — C2×C20 — C2×C40 — C4×C40 — M4(2)×C20
 Lower central C1 — C2 — M4(2)×C20
 Upper central C1 — C4×C20 — M4(2)×C20

Generators and relations for M4(2)×C20
G = < a,b,c | a20=b8=c2=1, ab=ba, ac=ca, cbc=b5 >

Subgroups: 162 in 142 conjugacy classes, 122 normal (22 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), C40, C2×C20, C2×C20, C2×C20, C22×C10, C4×M4(2), C4×C20, C4×C20, C2×C40, C5×M4(2), C22×C20, C22×C20, C4×C40, C5×C8⋊C4, C2×C4×C20, C10×M4(2), M4(2)×C20
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, C42, M4(2), C22×C4, C20, C2×C10, C2×C42, C2×M4(2), C2×C20, C22×C10, C4×M4(2), C4×C20, C5×M4(2), C22×C20, C2×C4×C20, C10×M4(2), M4(2)×C20

Smallest permutation representation of M4(2)×C20
On 160 points
Generators in S160
(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 52 112 91 157 136 37 62)(2 53 113 92 158 137 38 63)(3 54 114 93 159 138 39 64)(4 55 115 94 160 139 40 65)(5 56 116 95 141 140 21 66)(6 57 117 96 142 121 22 67)(7 58 118 97 143 122 23 68)(8 59 119 98 144 123 24 69)(9 60 120 99 145 124 25 70)(10 41 101 100 146 125 26 71)(11 42 102 81 147 126 27 72)(12 43 103 82 148 127 28 73)(13 44 104 83 149 128 29 74)(14 45 105 84 150 129 30 75)(15 46 106 85 151 130 31 76)(16 47 107 86 152 131 32 77)(17 48 108 87 153 132 33 78)(18 49 109 88 154 133 34 79)(19 50 110 89 155 134 35 80)(20 51 111 90 156 135 36 61)
(41 125)(42 126)(43 127)(44 128)(45 129)(46 130)(47 131)(48 132)(49 133)(50 134)(51 135)(52 136)(53 137)(54 138)(55 139)(56 140)(57 121)(58 122)(59 123)(60 124)(61 90)(62 91)(63 92)(64 93)(65 94)(66 95)(67 96)(68 97)(69 98)(70 99)(71 100)(72 81)(73 82)(74 83)(75 84)(76 85)(77 86)(78 87)(79 88)(80 89)

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

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

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

200 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A ··· 4L 4M ··· 4R 5A 5B 5C 5D 8A ··· 8P 10A ··· 10L 10M ··· 10T 20A ··· 20AV 20AW ··· 20BT 40A ··· 40BL order 1 2 2 2 2 2 4 ··· 4 4 ··· 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 1 1 2 2 1 ··· 1 2 ··· 2 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

200 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C2 C4 C4 C4 C5 C10 C10 C10 C10 C20 C20 C20 M4(2) C5×M4(2) kernel M4(2)×C20 C4×C40 C5×C8⋊C4 C2×C4×C20 C10×M4(2) C4×C20 C5×M4(2) C22×C20 C4×M4(2) C4×C8 C8⋊C4 C2×C42 C2×M4(2) C42 M4(2) C22×C4 C20 C4 # reps 1 2 2 1 2 4 16 4 4 8 8 4 8 16 64 16 8 32

Matrix representation of M4(2)×C20 in GL4(𝔽41) generated by

 9 0 0 0 0 10 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 32 2 0 0 5 9
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 9 40
G:=sub<GL(4,GF(41))| [9,0,0,0,0,10,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,32,5,0,0,2,9],[40,0,0,0,0,40,0,0,0,0,1,9,0,0,0,40] >;

M4(2)×C20 in GAP, Magma, Sage, TeX

M_4(2)\times C_{20}
% in TeX

G:=Group("M4(2)xC20");
// GroupNames label

G:=SmallGroup(320,905);
// by ID

G=gap.SmallGroup(320,905);
# by ID

G:=PCGroup([7,-2,-2,-2,-5,-2,-2,-2,280,568,3446,172]);
// Polycyclic

G:=Group<a,b,c|a^20=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations

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