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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

 Derived series C1 — C2 — Q8×C25
 Chief series C1 — C5 — C10 — C50 — C100 — Q8×C25
 Lower central C1 — C2 — Q8×C25
 Upper central C1 — C50 — 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 72 137 42)(2 73 138 43)(3 74 139 44)(4 75 140 45)(5 51 141 46)(6 52 142 47)(7 53 143 48)(8 54 144 49)(9 55 145 50)(10 56 146 26)(11 57 147 27)(12 58 148 28)(13 59 149 29)(14 60 150 30)(15 61 126 31)(16 62 127 32)(17 63 128 33)(18 64 129 34)(19 65 130 35)(20 66 131 36)(21 67 132 37)(22 68 133 38)(23 69 134 39)(24 70 135 40)(25 71 136 41)(76 116 166 178)(77 117 167 179)(78 118 168 180)(79 119 169 181)(80 120 170 182)(81 121 171 183)(82 122 172 184)(83 123 173 185)(84 124 174 186)(85 125 175 187)(86 101 151 188)(87 102 152 189)(88 103 153 190)(89 104 154 191)(90 105 155 192)(91 106 156 193)(92 107 157 194)(93 108 158 195)(94 109 159 196)(95 110 160 197)(96 111 161 198)(97 112 162 199)(98 113 163 200)(99 114 164 176)(100 115 165 177)
(1 175 137 85)(2 151 138 86)(3 152 139 87)(4 153 140 88)(5 154 141 89)(6 155 142 90)(7 156 143 91)(8 157 144 92)(9 158 145 93)(10 159 146 94)(11 160 147 95)(12 161 148 96)(13 162 149 97)(14 163 150 98)(15 164 126 99)(16 165 127 100)(17 166 128 76)(18 167 129 77)(19 168 130 78)(20 169 131 79)(21 170 132 80)(22 171 133 81)(23 172 134 82)(24 173 135 83)(25 174 136 84)(26 196 56 109)(27 197 57 110)(28 198 58 111)(29 199 59 112)(30 200 60 113)(31 176 61 114)(32 177 62 115)(33 178 63 116)(34 179 64 117)(35 180 65 118)(36 181 66 119)(37 182 67 120)(38 183 68 121)(39 184 69 122)(40 185 70 123)(41 186 71 124)(42 187 72 125)(43 188 73 101)(44 189 74 102)(45 190 75 103)(46 191 51 104)(47 192 52 105)(48 193 53 106)(49 194 54 107)(50 195 55 108)

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

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

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

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

125 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 5C 5D 10A 10B 10C 10D 20A ··· 20L 25A ··· 25T 50A ··· 50T 100A ··· 100BH order 1 2 4 4 4 5 5 5 5 10 10 10 10 20 ··· 20 25 ··· 25 50 ··· 50 100 ··· 100 size 1 1 2 2 2 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2

125 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + - image C1 C2 C5 C10 C25 C50 Q8 C5×Q8 Q8×C25 kernel Q8×C25 C100 C5×Q8 C20 Q8 C4 C25 C5 C1 # reps 1 3 4 12 20 60 1 4 20

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

 52 0 0 0 84 0 0 0 84
,
 1 0 0 0 0 1 0 100 0
,
 100 0 0 0 8 6 0 6 93
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

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