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## G = C7×Q32order 224 = 25·7

### Direct product of C7 and Q32

direct product, metacyclic, nilpotent (class 4), monomial, 2-elementary

Aliases: C7×Q32, C16.C14, Q16.C14, C112.2C2, C14.17D8, C28.38D4, C56.26C22, C4.3(C7×D4), C2.5(C7×D8), C8.4(C2×C14), (C7×Q16).2C2, SmallGroup(224,62)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C8 — C7×Q32
 Chief series C1 — C2 — C4 — C8 — C56 — C7×Q16 — C7×Q32
 Lower central C1 — C2 — C4 — C8 — C7×Q32
 Upper central C1 — C14 — C28 — C56 — C7×Q32

Generators and relations for C7×Q32
G = < a,b,c | a7=b16=1, c2=b8, ab=ba, ac=ca, cbc-1=b-1 >

Smallest permutation representation of C7×Q32
Regular action on 224 points
Generators in S224
(1 178 116 29 168 224 82)(2 179 117 30 169 209 83)(3 180 118 31 170 210 84)(4 181 119 32 171 211 85)(5 182 120 17 172 212 86)(6 183 121 18 173 213 87)(7 184 122 19 174 214 88)(8 185 123 20 175 215 89)(9 186 124 21 176 216 90)(10 187 125 22 161 217 91)(11 188 126 23 162 218 92)(12 189 127 24 163 219 93)(13 190 128 25 164 220 94)(14 191 113 26 165 221 95)(15 192 114 27 166 222 96)(16 177 115 28 167 223 81)(33 53 208 67 151 110 136)(34 54 193 68 152 111 137)(35 55 194 69 153 112 138)(36 56 195 70 154 97 139)(37 57 196 71 155 98 140)(38 58 197 72 156 99 141)(39 59 198 73 157 100 142)(40 60 199 74 158 101 143)(41 61 200 75 159 102 144)(42 62 201 76 160 103 129)(43 63 202 77 145 104 130)(44 64 203 78 146 105 131)(45 49 204 79 147 106 132)(46 50 205 80 148 107 133)(47 51 206 65 149 108 134)(48 52 207 66 150 109 135)
(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)(193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208)(209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224)
(1 205 9 197)(2 204 10 196)(3 203 11 195)(4 202 12 194)(5 201 13 193)(6 200 14 208)(7 199 15 207)(8 198 16 206)(17 103 25 111)(18 102 26 110)(19 101 27 109)(20 100 28 108)(21 99 29 107)(22 98 30 106)(23 97 31 105)(24 112 32 104)(33 213 41 221)(34 212 42 220)(35 211 43 219)(36 210 44 218)(37 209 45 217)(38 224 46 216)(39 223 47 215)(40 222 48 214)(49 91 57 83)(50 90 58 82)(51 89 59 81)(52 88 60 96)(53 87 61 95)(54 86 62 94)(55 85 63 93)(56 84 64 92)(65 185 73 177)(66 184 74 192)(67 183 75 191)(68 182 76 190)(69 181 77 189)(70 180 78 188)(71 179 79 187)(72 178 80 186)(113 151 121 159)(114 150 122 158)(115 149 123 157)(116 148 124 156)(117 147 125 155)(118 146 126 154)(119 145 127 153)(120 160 128 152)(129 164 137 172)(130 163 138 171)(131 162 139 170)(132 161 140 169)(133 176 141 168)(134 175 142 167)(135 174 143 166)(136 173 144 165)

G:=sub<Sym(224)| (1,178,116,29,168,224,82)(2,179,117,30,169,209,83)(3,180,118,31,170,210,84)(4,181,119,32,171,211,85)(5,182,120,17,172,212,86)(6,183,121,18,173,213,87)(7,184,122,19,174,214,88)(8,185,123,20,175,215,89)(9,186,124,21,176,216,90)(10,187,125,22,161,217,91)(11,188,126,23,162,218,92)(12,189,127,24,163,219,93)(13,190,128,25,164,220,94)(14,191,113,26,165,221,95)(15,192,114,27,166,222,96)(16,177,115,28,167,223,81)(33,53,208,67,151,110,136)(34,54,193,68,152,111,137)(35,55,194,69,153,112,138)(36,56,195,70,154,97,139)(37,57,196,71,155,98,140)(38,58,197,72,156,99,141)(39,59,198,73,157,100,142)(40,60,199,74,158,101,143)(41,61,200,75,159,102,144)(42,62,201,76,160,103,129)(43,63,202,77,145,104,130)(44,64,203,78,146,105,131)(45,49,204,79,147,106,132)(46,50,205,80,148,107,133)(47,51,206,65,149,108,134)(48,52,207,66,150,109,135), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,205,9,197)(2,204,10,196)(3,203,11,195)(4,202,12,194)(5,201,13,193)(6,200,14,208)(7,199,15,207)(8,198,16,206)(17,103,25,111)(18,102,26,110)(19,101,27,109)(20,100,28,108)(21,99,29,107)(22,98,30,106)(23,97,31,105)(24,112,32,104)(33,213,41,221)(34,212,42,220)(35,211,43,219)(36,210,44,218)(37,209,45,217)(38,224,46,216)(39,223,47,215)(40,222,48,214)(49,91,57,83)(50,90,58,82)(51,89,59,81)(52,88,60,96)(53,87,61,95)(54,86,62,94)(55,85,63,93)(56,84,64,92)(65,185,73,177)(66,184,74,192)(67,183,75,191)(68,182,76,190)(69,181,77,189)(70,180,78,188)(71,179,79,187)(72,178,80,186)(113,151,121,159)(114,150,122,158)(115,149,123,157)(116,148,124,156)(117,147,125,155)(118,146,126,154)(119,145,127,153)(120,160,128,152)(129,164,137,172)(130,163,138,171)(131,162,139,170)(132,161,140,169)(133,176,141,168)(134,175,142,167)(135,174,143,166)(136,173,144,165)>;

G:=Group( (1,178,116,29,168,224,82)(2,179,117,30,169,209,83)(3,180,118,31,170,210,84)(4,181,119,32,171,211,85)(5,182,120,17,172,212,86)(6,183,121,18,173,213,87)(7,184,122,19,174,214,88)(8,185,123,20,175,215,89)(9,186,124,21,176,216,90)(10,187,125,22,161,217,91)(11,188,126,23,162,218,92)(12,189,127,24,163,219,93)(13,190,128,25,164,220,94)(14,191,113,26,165,221,95)(15,192,114,27,166,222,96)(16,177,115,28,167,223,81)(33,53,208,67,151,110,136)(34,54,193,68,152,111,137)(35,55,194,69,153,112,138)(36,56,195,70,154,97,139)(37,57,196,71,155,98,140)(38,58,197,72,156,99,141)(39,59,198,73,157,100,142)(40,60,199,74,158,101,143)(41,61,200,75,159,102,144)(42,62,201,76,160,103,129)(43,63,202,77,145,104,130)(44,64,203,78,146,105,131)(45,49,204,79,147,106,132)(46,50,205,80,148,107,133)(47,51,206,65,149,108,134)(48,52,207,66,150,109,135), (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)(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208)(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224), (1,205,9,197)(2,204,10,196)(3,203,11,195)(4,202,12,194)(5,201,13,193)(6,200,14,208)(7,199,15,207)(8,198,16,206)(17,103,25,111)(18,102,26,110)(19,101,27,109)(20,100,28,108)(21,99,29,107)(22,98,30,106)(23,97,31,105)(24,112,32,104)(33,213,41,221)(34,212,42,220)(35,211,43,219)(36,210,44,218)(37,209,45,217)(38,224,46,216)(39,223,47,215)(40,222,48,214)(49,91,57,83)(50,90,58,82)(51,89,59,81)(52,88,60,96)(53,87,61,95)(54,86,62,94)(55,85,63,93)(56,84,64,92)(65,185,73,177)(66,184,74,192)(67,183,75,191)(68,182,76,190)(69,181,77,189)(70,180,78,188)(71,179,79,187)(72,178,80,186)(113,151,121,159)(114,150,122,158)(115,149,123,157)(116,148,124,156)(117,147,125,155)(118,146,126,154)(119,145,127,153)(120,160,128,152)(129,164,137,172)(130,163,138,171)(131,162,139,170)(132,161,140,169)(133,176,141,168)(134,175,142,167)(135,174,143,166)(136,173,144,165) );

G=PermutationGroup([(1,178,116,29,168,224,82),(2,179,117,30,169,209,83),(3,180,118,31,170,210,84),(4,181,119,32,171,211,85),(5,182,120,17,172,212,86),(6,183,121,18,173,213,87),(7,184,122,19,174,214,88),(8,185,123,20,175,215,89),(9,186,124,21,176,216,90),(10,187,125,22,161,217,91),(11,188,126,23,162,218,92),(12,189,127,24,163,219,93),(13,190,128,25,164,220,94),(14,191,113,26,165,221,95),(15,192,114,27,166,222,96),(16,177,115,28,167,223,81),(33,53,208,67,151,110,136),(34,54,193,68,152,111,137),(35,55,194,69,153,112,138),(36,56,195,70,154,97,139),(37,57,196,71,155,98,140),(38,58,197,72,156,99,141),(39,59,198,73,157,100,142),(40,60,199,74,158,101,143),(41,61,200,75,159,102,144),(42,62,201,76,160,103,129),(43,63,202,77,145,104,130),(44,64,203,78,146,105,131),(45,49,204,79,147,106,132),(46,50,205,80,148,107,133),(47,51,206,65,149,108,134),(48,52,207,66,150,109,135)], [(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),(193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208),(209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224)], [(1,205,9,197),(2,204,10,196),(3,203,11,195),(4,202,12,194),(5,201,13,193),(6,200,14,208),(7,199,15,207),(8,198,16,206),(17,103,25,111),(18,102,26,110),(19,101,27,109),(20,100,28,108),(21,99,29,107),(22,98,30,106),(23,97,31,105),(24,112,32,104),(33,213,41,221),(34,212,42,220),(35,211,43,219),(36,210,44,218),(37,209,45,217),(38,224,46,216),(39,223,47,215),(40,222,48,214),(49,91,57,83),(50,90,58,82),(51,89,59,81),(52,88,60,96),(53,87,61,95),(54,86,62,94),(55,85,63,93),(56,84,64,92),(65,185,73,177),(66,184,74,192),(67,183,75,191),(68,182,76,190),(69,181,77,189),(70,180,78,188),(71,179,79,187),(72,178,80,186),(113,151,121,159),(114,150,122,158),(115,149,123,157),(116,148,124,156),(117,147,125,155),(118,146,126,154),(119,145,127,153),(120,160,128,152),(129,164,137,172),(130,163,138,171),(131,162,139,170),(132,161,140,169),(133,176,141,168),(134,175,142,167),(135,174,143,166),(136,173,144,165)])

C7×Q32 is a maximal subgroup of   C7⋊SD64  C7⋊Q64  Q32⋊D7  Q323D7

77 conjugacy classes

 class 1 2 4A 4B 4C 7A ··· 7F 8A 8B 14A ··· 14F 16A 16B 16C 16D 28A ··· 28F 28G ··· 28R 56A ··· 56L 112A ··· 112X order 1 2 4 4 4 7 ··· 7 8 8 14 ··· 14 16 16 16 16 28 ··· 28 28 ··· 28 56 ··· 56 112 ··· 112 size 1 1 2 8 8 1 ··· 1 2 2 1 ··· 1 2 2 2 2 2 ··· 2 8 ··· 8 2 ··· 2 2 ··· 2

77 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + - image C1 C2 C2 C7 C14 C14 D4 D8 Q32 C7×D4 C7×D8 C7×Q32 kernel C7×Q32 C112 C7×Q16 Q32 C16 Q16 C28 C14 C7 C4 C2 C1 # reps 1 1 2 6 6 12 1 2 4 6 12 24

Matrix representation of C7×Q32 in GL2(𝔽113) generated by

 49 0 0 49
,
 109 95 18 109
,
 102 45 45 11
G:=sub<GL(2,GF(113))| [49,0,0,49],[109,18,95,109],[102,45,45,11] >;

C7×Q32 in GAP, Magma, Sage, TeX

C_7\times Q_{32}
% in TeX

G:=Group("C7xQ32");
// GroupNames label

G:=SmallGroup(224,62);
// by ID

G=gap.SmallGroup(224,62);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,-2,-2,672,361,679,2019,1017,165,5044,2530,88]);
// Polycyclic

G:=Group<a,b,c|a^7=b^16=1,c^2=b^8,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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