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## G = M4(2)×C2×C10order 320 = 26·5

### Direct product of C2×C10 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)×C2×C10
 Chief series C1 — C2 — C4 — C20 — C40 — C5×M4(2) — C10×M4(2) — M4(2)×C2×C10
 Lower central C1 — C2 — M4(2)×C2×C10
 Upper central C1 — C22×C20 — M4(2)×C2×C10

Generators and relations for M4(2)×C2×C10
G = < a,b,c,d | a2=b10=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 338 in 298 conjugacy classes, 258 normal (18 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C23, C23, C23, C10, C10, C10, C2×C8, M4(2), C22×C4, C22×C4, C24, C20, C20, C2×C10, C2×C10, C22×C8, C2×M4(2), C23×C4, C40, C2×C20, C22×C10, C22×C10, C22×C10, C22×M4(2), C2×C40, C5×M4(2), C22×C20, C22×C20, C23×C10, C22×C40, C10×M4(2), C23×C20, M4(2)×C2×C10
Quotients: C1, C2, C4, C22, C5, C2×C4, C23, C10, M4(2), C22×C4, C24, C20, C2×C10, C2×M4(2), C23×C4, C2×C20, C22×C10, C22×M4(2), C5×M4(2), C22×C20, C23×C10, C10×M4(2), C23×C20, M4(2)×C2×C10

Smallest permutation representation of M4(2)×C2×C10
On 160 points
Generators in S160
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 109)(9 110)(10 101)(11 133)(12 134)(13 135)(14 136)(15 137)(16 138)(17 139)(18 140)(19 131)(20 132)(21 114)(22 115)(23 116)(24 117)(25 118)(26 119)(27 120)(28 111)(29 112)(30 113)(31 149)(32 150)(33 141)(34 142)(35 143)(36 144)(37 145)(38 146)(39 147)(40 148)(41 75)(42 76)(43 77)(44 78)(45 79)(46 80)(47 71)(48 72)(49 73)(50 74)(51 84)(52 85)(53 86)(54 87)(55 88)(56 89)(57 90)(58 81)(59 82)(60 83)(61 92)(62 93)(63 94)(64 95)(65 96)(66 97)(67 98)(68 99)(69 100)(70 91)(121 155)(122 156)(123 157)(124 158)(125 159)(126 160)(127 151)(128 152)(129 153)(130 154)
(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 53 16 67 22 41 34)(2 151 54 17 68 23 42 35)(3 152 55 18 69 24 43 36)(4 153 56 19 70 25 44 37)(5 154 57 20 61 26 45 38)(6 155 58 11 62 27 46 39)(7 156 59 12 63 28 47 40)(8 157 60 13 64 29 48 31)(9 158 51 14 65 30 49 32)(10 159 52 15 66 21 50 33)(71 148 108 122 82 134 94 111)(72 149 109 123 83 135 95 112)(73 150 110 124 84 136 96 113)(74 141 101 125 85 137 97 114)(75 142 102 126 86 138 98 115)(76 143 103 127 87 139 99 116)(77 144 104 128 88 140 100 117)(78 145 105 129 89 131 91 118)(79 146 106 130 90 132 92 119)(80 147 107 121 81 133 93 120)
(11 39)(12 40)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)(21 159)(22 160)(23 151)(24 152)(25 153)(26 154)(27 155)(28 156)(29 157)(30 158)(111 122)(112 123)(113 124)(114 125)(115 126)(116 127)(117 128)(118 129)(119 130)(120 121)(131 145)(132 146)(133 147)(134 148)(135 149)(136 150)(137 141)(138 142)(139 143)(140 144)

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

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

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

200 conjugacy classes

 class 1 2A ··· 2G 2H 2I 2J 2K 4A ··· 4H 4I 4J 4K 4L 5A 5B 5C 5D 8A ··· 8P 10A ··· 10AB 10AC ··· 10AR 20A ··· 20AF 20AG ··· 20AV 40A ··· 40BL order 1 2 ··· 2 2 2 2 2 4 ··· 4 4 4 4 4 5 5 5 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 ··· 1 2 2 2 2 1 ··· 1 2 2 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 2 2 type + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 M4(2) C5×M4(2) kernel M4(2)×C2×C10 C22×C40 C10×M4(2) C23×C20 C22×C20 C23×C10 C22×M4(2) C22×C8 C2×M4(2) C23×C4 C22×C4 C24 C2×C10 C22 # reps 1 2 12 1 14 2 4 8 48 4 56 8 8 32

Matrix representation of M4(2)×C2×C10 in GL4(𝔽41) generated by

 40 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 31 0 0 0 0 31
,
 40 0 0 0 0 1 0 0 0 0 39 2 0 0 14 2
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 2 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,31,0,0,0,0,31],[40,0,0,0,0,1,0,0,0,0,39,14,0,0,2,2],[40,0,0,0,0,40,0,0,0,0,1,2,0,0,0,40] >;

M4(2)×C2×C10 in GAP, Magma, Sage, TeX

M_4(2)\times C_2\times C_{10}
% in TeX

G:=Group("M4(2)xC2xC10");
// GroupNames label

G:=SmallGroup(320,1568);
// by ID

G=gap.SmallGroup(320,1568);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,560,2269,124]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^10=c^8=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations

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