metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C7⋊2Q32, Q16.D7, C28.6D4, C8.7D14, C14.11D8, C56.5C22, Dic28.2C2, C7⋊C16.C2, C2.7(D4⋊D7), C4.4(C7⋊D4), (C7×Q16).1C2, SmallGroup(224,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C7⋊Q32
G = < a,b,c | a7=b16=1, c2=b8, bab-1=a-1, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C7⋊Q32
►Regular action on 224 points
Generators in S224
(1 24 88 175 203 56 216)(2 217 57 204 176 89 25)(3 26 90 161 205 58 218)(4 219 59 206 162 91 27)(5 28 92 163 207 60 220)(6 221 61 208 164 93 29)(7 30 94 165 193 62 222)(8 223 63 194 166 95 31)(9 32 96 167 195 64 224)(10 209 49 196 168 81 17)(11 18 82 169 197 50 210)(12 211 51 198 170 83 19)(13 20 84 171 199 52 212)(14 213 53 200 172 85 21)(15 22 86 173 201 54 214)(16 215 55 202 174 87 23)(33 78 111 124 156 139 185)(34 186 140 157 125 112 79)(35 80 97 126 158 141 187)(36 188 142 159 127 98 65)(37 66 99 128 160 143 189)(38 190 144 145 113 100 67)(39 68 101 114 146 129 191)(40 192 130 147 115 102 69)(41 70 103 116 148 131 177)(42 178 132 149 117 104 71)(43 72 105 118 150 133 179)(44 180 134 151 119 106 73)(45 74 107 120 152 135 181)(46 182 136 153 121 108 75)(47 76 109 122 154 137 183)(48 184 138 155 123 110 77)
(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 44 9 36)(2 43 10 35)(3 42 11 34)(4 41 12 33)(5 40 13 48)(6 39 14 47)(7 38 15 46)(8 37 16 45)(17 187 25 179)(18 186 26 178)(19 185 27 177)(20 184 28 192)(21 183 29 191)(22 182 30 190)(23 181 31 189)(24 180 32 188)(49 97 57 105)(50 112 58 104)(51 111 59 103)(52 110 60 102)(53 109 61 101)(54 108 62 100)(55 107 63 99)(56 106 64 98)(65 216 73 224)(66 215 74 223)(67 214 75 222)(68 213 76 221)(69 212 77 220)(70 211 78 219)(71 210 79 218)(72 209 80 217)(81 141 89 133)(82 140 90 132)(83 139 91 131)(84 138 92 130)(85 137 93 129)(86 136 94 144)(87 135 95 143)(88 134 96 142)(113 201 121 193)(114 200 122 208)(115 199 123 207)(116 198 124 206)(117 197 125 205)(118 196 126 204)(119 195 127 203)(120 194 128 202)(145 173 153 165)(146 172 154 164)(147 171 155 163)(148 170 156 162)(149 169 157 161)(150 168 158 176)(151 167 159 175)(152 166 160 174)
G:=sub<Sym(224)| (1,24,88,175,203,56,216)(2,217,57,204,176,89,25)(3,26,90,161,205,58,218)(4,219,59,206,162,91,27)(5,28,92,163,207,60,220)(6,221,61,208,164,93,29)(7,30,94,165,193,62,222)(8,223,63,194,166,95,31)(9,32,96,167,195,64,224)(10,209,49,196,168,81,17)(11,18,82,169,197,50,210)(12,211,51,198,170,83,19)(13,20,84,171,199,52,212)(14,213,53,200,172,85,21)(15,22,86,173,201,54,214)(16,215,55,202,174,87,23)(33,78,111,124,156,139,185)(34,186,140,157,125,112,79)(35,80,97,126,158,141,187)(36,188,142,159,127,98,65)(37,66,99,128,160,143,189)(38,190,144,145,113,100,67)(39,68,101,114,146,129,191)(40,192,130,147,115,102,69)(41,70,103,116,148,131,177)(42,178,132,149,117,104,71)(43,72,105,118,150,133,179)(44,180,134,151,119,106,73)(45,74,107,120,152,135,181)(46,182,136,153,121,108,75)(47,76,109,122,154,137,183)(48,184,138,155,123,110,77), (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), (1,44,9,36)(2,43,10,35)(3,42,11,34)(4,41,12,33)(5,40,13,48)(6,39,14,47)(7,38,15,46)(8,37,16,45)(17,187,25,179)(18,186,26,178)(19,185,27,177)(20,184,28,192)(21,183,29,191)(22,182,30,190)(23,181,31,189)(24,180,32,188)(49,97,57,105)(50,112,58,104)(51,111,59,103)(52,110,60,102)(53,109,61,101)(54,108,62,100)(55,107,63,99)(56,106,64,98)(65,216,73,224)(66,215,74,223)(67,214,75,222)(68,213,76,221)(69,212,77,220)(70,211,78,219)(71,210,79,218)(72,209,80,217)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(113,201,121,193)(114,200,122,208)(115,199,123,207)(116,198,124,206)(117,197,125,205)(118,196,126,204)(119,195,127,203)(120,194,128,202)(145,173,153,165)(146,172,154,164)(147,171,155,163)(148,170,156,162)(149,169,157,161)(150,168,158,176)(151,167,159,175)(152,166,160,174)>;
G:=Group( (1,24,88,175,203,56,216)(2,217,57,204,176,89,25)(3,26,90,161,205,58,218)(4,219,59,206,162,91,27)(5,28,92,163,207,60,220)(6,221,61,208,164,93,29)(7,30,94,165,193,62,222)(8,223,63,194,166,95,31)(9,32,96,167,195,64,224)(10,209,49,196,168,81,17)(11,18,82,169,197,50,210)(12,211,51,198,170,83,19)(13,20,84,171,199,52,212)(14,213,53,200,172,85,21)(15,22,86,173,201,54,214)(16,215,55,202,174,87,23)(33,78,111,124,156,139,185)(34,186,140,157,125,112,79)(35,80,97,126,158,141,187)(36,188,142,159,127,98,65)(37,66,99,128,160,143,189)(38,190,144,145,113,100,67)(39,68,101,114,146,129,191)(40,192,130,147,115,102,69)(41,70,103,116,148,131,177)(42,178,132,149,117,104,71)(43,72,105,118,150,133,179)(44,180,134,151,119,106,73)(45,74,107,120,152,135,181)(46,182,136,153,121,108,75)(47,76,109,122,154,137,183)(48,184,138,155,123,110,77), (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), (1,44,9,36)(2,43,10,35)(3,42,11,34)(4,41,12,33)(5,40,13,48)(6,39,14,47)(7,38,15,46)(8,37,16,45)(17,187,25,179)(18,186,26,178)(19,185,27,177)(20,184,28,192)(21,183,29,191)(22,182,30,190)(23,181,31,189)(24,180,32,188)(49,97,57,105)(50,112,58,104)(51,111,59,103)(52,110,60,102)(53,109,61,101)(54,108,62,100)(55,107,63,99)(56,106,64,98)(65,216,73,224)(66,215,74,223)(67,214,75,222)(68,213,76,221)(69,212,77,220)(70,211,78,219)(71,210,79,218)(72,209,80,217)(81,141,89,133)(82,140,90,132)(83,139,91,131)(84,138,92,130)(85,137,93,129)(86,136,94,144)(87,135,95,143)(88,134,96,142)(113,201,121,193)(114,200,122,208)(115,199,123,207)(116,198,124,206)(117,197,125,205)(118,196,126,204)(119,195,127,203)(120,194,128,202)(145,173,153,165)(146,172,154,164)(147,171,155,163)(148,170,156,162)(149,169,157,161)(150,168,158,176)(151,167,159,175)(152,166,160,174) );
G=PermutationGroup([[(1,24,88,175,203,56,216),(2,217,57,204,176,89,25),(3,26,90,161,205,58,218),(4,219,59,206,162,91,27),(5,28,92,163,207,60,220),(6,221,61,208,164,93,29),(7,30,94,165,193,62,222),(8,223,63,194,166,95,31),(9,32,96,167,195,64,224),(10,209,49,196,168,81,17),(11,18,82,169,197,50,210),(12,211,51,198,170,83,19),(13,20,84,171,199,52,212),(14,213,53,200,172,85,21),(15,22,86,173,201,54,214),(16,215,55,202,174,87,23),(33,78,111,124,156,139,185),(34,186,140,157,125,112,79),(35,80,97,126,158,141,187),(36,188,142,159,127,98,65),(37,66,99,128,160,143,189),(38,190,144,145,113,100,67),(39,68,101,114,146,129,191),(40,192,130,147,115,102,69),(41,70,103,116,148,131,177),(42,178,132,149,117,104,71),(43,72,105,118,150,133,179),(44,180,134,151,119,106,73),(45,74,107,120,152,135,181),(46,182,136,153,121,108,75),(47,76,109,122,154,137,183),(48,184,138,155,123,110,77)], [(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)], [(1,44,9,36),(2,43,10,35),(3,42,11,34),(4,41,12,33),(5,40,13,48),(6,39,14,47),(7,38,15,46),(8,37,16,45),(17,187,25,179),(18,186,26,178),(19,185,27,177),(20,184,28,192),(21,183,29,191),(22,182,30,190),(23,181,31,189),(24,180,32,188),(49,97,57,105),(50,112,58,104),(51,111,59,103),(52,110,60,102),(53,109,61,101),(54,108,62,100),(55,107,63,99),(56,106,64,98),(65,216,73,224),(66,215,74,223),(67,214,75,222),(68,213,76,221),(69,212,77,220),(70,211,78,219),(71,210,79,218),(72,209,80,217),(81,141,89,133),(82,140,90,132),(83,139,91,131),(84,138,92,130),(85,137,93,129),(86,136,94,144),(87,135,95,143),(88,134,96,142),(113,201,121,193),(114,200,122,208),(115,199,123,207),(116,198,124,206),(117,197,125,205),(118,196,126,204),(119,195,127,203),(120,194,128,202),(145,173,153,165),(146,172,154,164),(147,171,155,163),(148,170,156,162),(149,169,157,161),(150,168,158,176),(151,167,159,175),(152,166,160,174)]])
C7⋊Q32 is a maximal subgroup of
SD32⋊D7 SD32⋊3D7 D7×Q32 Q32⋊D7 Q16.D14 C56.30C23 C56.31C23
C7⋊Q32 is a maximal quotient of C8.4Dic14 C56.6D4 C14.Q32
32 conjugacy classes
| class | 1 | 2 | 4A | 4B | 4C | 7A | 7B | 7C | 8A | 8B | 14A | 14B | 14C | 16A | 16B | 16C | 16D | 28A | 28B | 28C | 28D | ··· | 28I | 56A | ··· | 56F |
| order | 1 | 2 | 4 | 4 | 4 | 7 | 7 | 7 | 8 | 8 | 14 | 14 | 14 | 16 | 16 | 16 | 16 | 28 | 28 | 28 | 28 | ··· | 28 | 56 | ··· | 56 |
| size | 1 | 1 | 2 | 8 | 56 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 14 | 14 | 14 | 14 | 4 | 4 | 4 | 8 | ··· | 8 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | D4 | D7 | D8 | D14 | Q32 | C7⋊D4 | D4⋊D7 | C7⋊Q32 |
| kernel | C7⋊Q32 | C7⋊C16 | Dic28 | C7×Q16 | C28 | Q16 | C14 | C8 | C7 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 4 | 6 | 3 | 6 |
Matrix representation of C7⋊Q32 ►in GL4(𝔽113) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 112 | 1 |
| 0 | 0 | 102 | 10 |
| 4 | 18 | 0 | 0 |
| 95 | 4 | 0 | 0 |
| 0 | 0 | 88 | 53 |
| 0 | 0 | 82 | 25 |
| 32 | 52 | 0 | 0 |
| 52 | 81 | 0 | 0 |
| 0 | 0 | 84 | 108 |
| 0 | 0 | 55 | 29 |
G:=sub<GL(4,GF(113))| [1,0,0,0,0,1,0,0,0,0,112,102,0,0,1,10],[4,95,0,0,18,4,0,0,0,0,88,82,0,0,53,25],[32,52,0,0,52,81,0,0,0,0,84,55,0,0,108,29] >;
C7⋊Q32 in GAP, Magma, Sage, TeX
C_7\rtimes Q_{32} % in TeX
G:=Group("C7:Q32"); // GroupNames label
G:=SmallGroup(224,35);
// by ID
G=gap.SmallGroup(224,35);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-7,96,73,103,218,116,122,579,297,69,6917]);
// Polycyclic
G:=Group<a,b,c|a^7=b^16=1,c^2=b^8,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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