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G = Q8×C25order 200 = 23·52

Direct product of C25 and Q8

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

Aliases: Q8×C25, C4.C50, C20.4C10, C100.3C2, C50.7C22, C5.(C5×Q8), (C5×Q8)C50, (C5×Q8).C5, C2.2(C2×C50), C10.7(C2×C10), SmallGroup(200,11)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C25
C1C5C10C50C100 — Q8×C25
C1C2 — Q8×C25
C1C50 — Q8×C25

Generators and relations for Q8×C25
 G = < a,b,c | a25=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C25
Regular action on 200 points
Generators in S200
(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)
(1 171 122 33)(2 172 123 34)(3 173 124 35)(4 174 125 36)(5 175 101 37)(6 151 102 38)(7 152 103 39)(8 153 104 40)(9 154 105 41)(10 155 106 42)(11 156 107 43)(12 157 108 44)(13 158 109 45)(14 159 110 46)(15 160 111 47)(16 161 112 48)(17 162 113 49)(18 163 114 50)(19 164 115 26)(20 165 116 27)(21 166 117 28)(22 167 118 29)(23 168 119 30)(24 169 120 31)(25 170 121 32)(51 149 186 88)(52 150 187 89)(53 126 188 90)(54 127 189 91)(55 128 190 92)(56 129 191 93)(57 130 192 94)(58 131 193 95)(59 132 194 96)(60 133 195 97)(61 134 196 98)(62 135 197 99)(63 136 198 100)(64 137 199 76)(65 138 200 77)(66 139 176 78)(67 140 177 79)(68 141 178 80)(69 142 179 81)(70 143 180 82)(71 144 181 83)(72 145 182 84)(73 146 183 85)(74 147 184 86)(75 148 185 87)
(1 74 122 184)(2 75 123 185)(3 51 124 186)(4 52 125 187)(5 53 101 188)(6 54 102 189)(7 55 103 190)(8 56 104 191)(9 57 105 192)(10 58 106 193)(11 59 107 194)(12 60 108 195)(13 61 109 196)(14 62 110 197)(15 63 111 198)(16 64 112 199)(17 65 113 200)(18 66 114 176)(19 67 115 177)(20 68 116 178)(21 69 117 179)(22 70 118 180)(23 71 119 181)(24 72 120 182)(25 73 121 183)(26 140 164 79)(27 141 165 80)(28 142 166 81)(29 143 167 82)(30 144 168 83)(31 145 169 84)(32 146 170 85)(33 147 171 86)(34 148 172 87)(35 149 173 88)(36 150 174 89)(37 126 175 90)(38 127 151 91)(39 128 152 92)(40 129 153 93)(41 130 154 94)(42 131 155 95)(43 132 156 96)(44 133 157 97)(45 134 158 98)(46 135 159 99)(47 136 160 100)(48 137 161 76)(49 138 162 77)(50 139 163 78)

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

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

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

Q8×C25 is a maximal subgroup of   C25⋊Q16  Q8⋊D25  Q82D25

125 conjugacy classes

class 1  2 4A4B4C5A5B5C5D10A10B10C10D20A···20L25A···25T50A···50T100A···100BH
order1244455551010101020···2025···2550···50100···100
size11222111111112···21···11···12···2

125 irreducible representations

dim111111222
type++-
imageC1C2C5C10C25C50Q8C5×Q8Q8×C25
kernelQ8×C25C100C5×Q8C20Q8C4C25C5C1
# reps1341220601420

Matrix representation of Q8×C25 in GL3(𝔽101) generated by

5200
0840
0084
,
100
001
01000
,
10000
086
0693
G:=sub<GL(3,GF(101))| [52,0,0,0,84,0,0,0,84],[1,0,0,0,0,100,0,1,0],[100,0,0,0,8,6,0,6,93] >;

Q8×C25 in GAP, Magma, Sage, TeX

Q_8\times C_{25}
% in TeX

G:=Group("Q8xC25");
// GroupNames label

G:=SmallGroup(200,11);
// by ID

G=gap.SmallGroup(200,11);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,100,221,106,162]);
// Polycyclic

G:=Group<a,b,c|a^25=b^4=1,c^2=b^2,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of Q8×C25 in TeX

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