Copied to
clipboard

## G = M4(2)×C52order 400 = 24·52

### Direct product of C52 and M4(2)

direct product, metacyclic, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — M4(2)×C52
 Chief series C1 — C2 — C4 — C20 — C5×C20 — C5×C40 — M4(2)×C52
 Lower central C1 — C2 — M4(2)×C52
 Upper central C1 — C5×C20 — M4(2)×C52

Generators and relations for M4(2)×C52
G = < a,b,c,d | a5=b5=c8=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 88 in 80 conjugacy classes, 72 normal (14 characteristic)
C1, C2, C2, C4, C22, C5, C8, C2×C4, C10, C10, M4(2), C20, C2×C10, C52, C40, C2×C20, C5×C10, C5×C10, C5×M4(2), C5×C20, C102, C5×C40, C10×C20, M4(2)×C52
Quotients: C1, C2, C4, C22, C5, C2×C4, C10, M4(2), C20, C2×C10, C52, C2×C20, C5×C10, C5×M4(2), C5×C20, C102, C10×C20, M4(2)×C52

Smallest permutation representation of M4(2)×C52
On 200 points
Generators in S200
(1 98 95 171 71)(2 99 96 172 72)(3 100 89 173 65)(4 101 90 174 66)(5 102 91 175 67)(6 103 92 176 68)(7 104 93 169 69)(8 97 94 170 70)(9 185 44 128 20)(10 186 45 121 21)(11 187 46 122 22)(12 188 47 123 23)(13 189 48 124 24)(14 190 41 125 17)(15 191 42 126 18)(16 192 43 127 19)(25 113 193 177 77)(26 114 194 178 78)(27 115 195 179 79)(28 116 196 180 80)(29 117 197 181 73)(30 118 198 182 74)(31 119 199 183 75)(32 120 200 184 76)(33 82 166 58 150)(34 83 167 59 151)(35 84 168 60 152)(36 85 161 61 145)(37 86 162 62 146)(38 87 163 63 147)(39 88 164 64 148)(40 81 165 57 149)(49 138 132 109 158)(50 139 133 110 159)(51 140 134 111 160)(52 141 135 112 153)(53 142 136 105 154)(54 143 129 106 155)(55 144 130 107 156)(56 137 131 108 157)
(1 155 115 63 11)(2 156 116 64 12)(3 157 117 57 13)(4 158 118 58 14)(5 159 119 59 15)(6 160 120 60 16)(7 153 113 61 9)(8 154 114 62 10)(17 66 109 30 166)(18 67 110 31 167)(19 68 111 32 168)(20 69 112 25 161)(21 70 105 26 162)(22 71 106 27 163)(23 72 107 28 164)(24 65 108 29 165)(33 41 90 138 182)(34 42 91 139 183)(35 43 92 140 184)(36 44 93 141 177)(37 45 94 142 178)(38 46 95 143 179)(39 47 96 144 180)(40 48 89 137 181)(49 198 150 190 101)(50 199 151 191 102)(51 200 152 192 103)(52 193 145 185 104)(53 194 146 186 97)(54 195 147 187 98)(55 196 148 188 99)(56 197 149 189 100)(73 81 124 173 131)(74 82 125 174 132)(75 83 126 175 133)(76 84 127 176 134)(77 85 128 169 135)(78 86 121 170 136)(79 87 122 171 129)(80 88 123 172 130)
(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)(161 162 163 164 165 166 167 168)(169 170 171 172 173 174 175 176)(177 178 179 180 181 182 183 184)(185 186 187 188 189 190 191 192)(193 194 195 196 197 198 199 200)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(58 62)(60 64)(66 70)(68 72)(74 78)(76 80)(82 86)(84 88)(90 94)(92 96)(97 101)(99 103)(105 109)(107 111)(114 118)(116 120)(121 125)(123 127)(130 134)(132 136)(138 142)(140 144)(146 150)(148 152)(154 158)(156 160)(162 166)(164 168)(170 174)(172 176)(178 182)(180 184)(186 190)(188 192)(194 198)(196 200)

G:=sub<Sym(200)| (1,98,95,171,71)(2,99,96,172,72)(3,100,89,173,65)(4,101,90,174,66)(5,102,91,175,67)(6,103,92,176,68)(7,104,93,169,69)(8,97,94,170,70)(9,185,44,128,20)(10,186,45,121,21)(11,187,46,122,22)(12,188,47,123,23)(13,189,48,124,24)(14,190,41,125,17)(15,191,42,126,18)(16,192,43,127,19)(25,113,193,177,77)(26,114,194,178,78)(27,115,195,179,79)(28,116,196,180,80)(29,117,197,181,73)(30,118,198,182,74)(31,119,199,183,75)(32,120,200,184,76)(33,82,166,58,150)(34,83,167,59,151)(35,84,168,60,152)(36,85,161,61,145)(37,86,162,62,146)(38,87,163,63,147)(39,88,164,64,148)(40,81,165,57,149)(49,138,132,109,158)(50,139,133,110,159)(51,140,134,111,160)(52,141,135,112,153)(53,142,136,105,154)(54,143,129,106,155)(55,144,130,107,156)(56,137,131,108,157), (1,155,115,63,11)(2,156,116,64,12)(3,157,117,57,13)(4,158,118,58,14)(5,159,119,59,15)(6,160,120,60,16)(7,153,113,61,9)(8,154,114,62,10)(17,66,109,30,166)(18,67,110,31,167)(19,68,111,32,168)(20,69,112,25,161)(21,70,105,26,162)(22,71,106,27,163)(23,72,107,28,164)(24,65,108,29,165)(33,41,90,138,182)(34,42,91,139,183)(35,43,92,140,184)(36,44,93,141,177)(37,45,94,142,178)(38,46,95,143,179)(39,47,96,144,180)(40,48,89,137,181)(49,198,150,190,101)(50,199,151,191,102)(51,200,152,192,103)(52,193,145,185,104)(53,194,146,186,97)(54,195,147,187,98)(55,196,148,188,99)(56,197,149,189,100)(73,81,124,173,131)(74,82,125,174,132)(75,83,126,175,133)(76,84,127,176,134)(77,85,128,169,135)(78,86,121,170,136)(79,87,122,171,129)(80,88,123,172,130), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(97,101)(99,103)(105,109)(107,111)(114,118)(116,120)(121,125)(123,127)(130,134)(132,136)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(162,166)(164,168)(170,174)(172,176)(178,182)(180,184)(186,190)(188,192)(194,198)(196,200)>;

G:=Group( (1,98,95,171,71)(2,99,96,172,72)(3,100,89,173,65)(4,101,90,174,66)(5,102,91,175,67)(6,103,92,176,68)(7,104,93,169,69)(8,97,94,170,70)(9,185,44,128,20)(10,186,45,121,21)(11,187,46,122,22)(12,188,47,123,23)(13,189,48,124,24)(14,190,41,125,17)(15,191,42,126,18)(16,192,43,127,19)(25,113,193,177,77)(26,114,194,178,78)(27,115,195,179,79)(28,116,196,180,80)(29,117,197,181,73)(30,118,198,182,74)(31,119,199,183,75)(32,120,200,184,76)(33,82,166,58,150)(34,83,167,59,151)(35,84,168,60,152)(36,85,161,61,145)(37,86,162,62,146)(38,87,163,63,147)(39,88,164,64,148)(40,81,165,57,149)(49,138,132,109,158)(50,139,133,110,159)(51,140,134,111,160)(52,141,135,112,153)(53,142,136,105,154)(54,143,129,106,155)(55,144,130,107,156)(56,137,131,108,157), (1,155,115,63,11)(2,156,116,64,12)(3,157,117,57,13)(4,158,118,58,14)(5,159,119,59,15)(6,160,120,60,16)(7,153,113,61,9)(8,154,114,62,10)(17,66,109,30,166)(18,67,110,31,167)(19,68,111,32,168)(20,69,112,25,161)(21,70,105,26,162)(22,71,106,27,163)(23,72,107,28,164)(24,65,108,29,165)(33,41,90,138,182)(34,42,91,139,183)(35,43,92,140,184)(36,44,93,141,177)(37,45,94,142,178)(38,46,95,143,179)(39,47,96,144,180)(40,48,89,137,181)(49,198,150,190,101)(50,199,151,191,102)(51,200,152,192,103)(52,193,145,185,104)(53,194,146,186,97)(54,195,147,187,98)(55,196,148,188,99)(56,197,149,189,100)(73,81,124,173,131)(74,82,125,174,132)(75,83,126,175,133)(76,84,127,176,134)(77,85,128,169,135)(78,86,121,170,136)(79,87,122,171,129)(80,88,123,172,130), (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)(161,162,163,164,165,166,167,168)(169,170,171,172,173,174,175,176)(177,178,179,180,181,182,183,184)(185,186,187,188,189,190,191,192)(193,194,195,196,197,198,199,200), (2,6)(4,8)(10,14)(12,16)(17,21)(19,23)(26,30)(28,32)(33,37)(35,39)(41,45)(43,47)(49,53)(51,55)(58,62)(60,64)(66,70)(68,72)(74,78)(76,80)(82,86)(84,88)(90,94)(92,96)(97,101)(99,103)(105,109)(107,111)(114,118)(116,120)(121,125)(123,127)(130,134)(132,136)(138,142)(140,144)(146,150)(148,152)(154,158)(156,160)(162,166)(164,168)(170,174)(172,176)(178,182)(180,184)(186,190)(188,192)(194,198)(196,200) );

G=PermutationGroup([[(1,98,95,171,71),(2,99,96,172,72),(3,100,89,173,65),(4,101,90,174,66),(5,102,91,175,67),(6,103,92,176,68),(7,104,93,169,69),(8,97,94,170,70),(9,185,44,128,20),(10,186,45,121,21),(11,187,46,122,22),(12,188,47,123,23),(13,189,48,124,24),(14,190,41,125,17),(15,191,42,126,18),(16,192,43,127,19),(25,113,193,177,77),(26,114,194,178,78),(27,115,195,179,79),(28,116,196,180,80),(29,117,197,181,73),(30,118,198,182,74),(31,119,199,183,75),(32,120,200,184,76),(33,82,166,58,150),(34,83,167,59,151),(35,84,168,60,152),(36,85,161,61,145),(37,86,162,62,146),(38,87,163,63,147),(39,88,164,64,148),(40,81,165,57,149),(49,138,132,109,158),(50,139,133,110,159),(51,140,134,111,160),(52,141,135,112,153),(53,142,136,105,154),(54,143,129,106,155),(55,144,130,107,156),(56,137,131,108,157)], [(1,155,115,63,11),(2,156,116,64,12),(3,157,117,57,13),(4,158,118,58,14),(5,159,119,59,15),(6,160,120,60,16),(7,153,113,61,9),(8,154,114,62,10),(17,66,109,30,166),(18,67,110,31,167),(19,68,111,32,168),(20,69,112,25,161),(21,70,105,26,162),(22,71,106,27,163),(23,72,107,28,164),(24,65,108,29,165),(33,41,90,138,182),(34,42,91,139,183),(35,43,92,140,184),(36,44,93,141,177),(37,45,94,142,178),(38,46,95,143,179),(39,47,96,144,180),(40,48,89,137,181),(49,198,150,190,101),(50,199,151,191,102),(51,200,152,192,103),(52,193,145,185,104),(53,194,146,186,97),(54,195,147,187,98),(55,196,148,188,99),(56,197,149,189,100),(73,81,124,173,131),(74,82,125,174,132),(75,83,126,175,133),(76,84,127,176,134),(77,85,128,169,135),(78,86,121,170,136),(79,87,122,171,129),(80,88,123,172,130)], [(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),(161,162,163,164,165,166,167,168),(169,170,171,172,173,174,175,176),(177,178,179,180,181,182,183,184),(185,186,187,188,189,190,191,192),(193,194,195,196,197,198,199,200)], [(2,6),(4,8),(10,14),(12,16),(17,21),(19,23),(26,30),(28,32),(33,37),(35,39),(41,45),(43,47),(49,53),(51,55),(58,62),(60,64),(66,70),(68,72),(74,78),(76,80),(82,86),(84,88),(90,94),(92,96),(97,101),(99,103),(105,109),(107,111),(114,118),(116,120),(121,125),(123,127),(130,134),(132,136),(138,142),(140,144),(146,150),(148,152),(154,158),(156,160),(162,166),(164,168),(170,174),(172,176),(178,182),(180,184),(186,190),(188,192),(194,198),(196,200)]])

250 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A ··· 5X 8A 8B 8C 8D 10A ··· 10X 10Y ··· 10AV 20A ··· 20AV 20AW ··· 20BT 40A ··· 40CR order 1 2 2 4 4 4 5 ··· 5 8 8 8 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 1 1 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

250 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 type + + + image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 M4(2) C5×M4(2) kernel M4(2)×C52 C5×C40 C10×C20 C5×C20 C102 C5×M4(2) C40 C2×C20 C20 C2×C10 C52 C5 # reps 1 2 1 2 2 24 48 24 48 48 2 48

Matrix representation of M4(2)×C52 in GL3(𝔽41) generated by

 37 0 0 0 10 0 0 0 10
,
 10 0 0 0 16 0 0 0 16
,
 40 0 0 0 32 36 0 18 9
,
 40 0 0 0 1 1 0 0 40
G:=sub<GL(3,GF(41))| [37,0,0,0,10,0,0,0,10],[10,0,0,0,16,0,0,0,16],[40,0,0,0,32,18,0,36,9],[40,0,0,0,1,0,0,1,40] >;

M4(2)×C52 in GAP, Magma, Sage, TeX

M_4(2)\times C_5^2
% in TeX

G:=Group("M4(2)xC5^2");
// GroupNames label

G:=SmallGroup(400,112);
// by ID

G=gap.SmallGroup(400,112);
# by ID

G:=PCGroup([6,-2,-2,-5,-5,-2,-2,600,2425,88]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=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

׿
×
𝔽