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## G = Q8×C28order 224 = 25·7

### Direct product of C28 and Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8×C28
 Chief series C1 — C2 — C22 — C2×C14 — C2×C28 — C7×C4⋊C4 — Q8×C28
 Lower central C1 — C2 — Q8×C28
 Upper central C1 — C2×C28 — Q8×C28

Generators and relations for Q8×C28
G = < a,b,c | a28=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 76 in 70 conjugacy classes, 64 normal (16 characteristic)
C1, C2, C4, C4, C22, C7, C2×C4, C2×C4, Q8, C14, C42, C4⋊C4, C2×Q8, C28, C28, C2×C14, C4×Q8, C2×C28, C2×C28, C7×Q8, C4×C28, C7×C4⋊C4, Q8×C14, Q8×C28
Quotients: C1, C2, C4, C22, C7, C2×C4, Q8, C23, C14, C22×C4, C2×Q8, C4○D4, C28, C2×C14, C4×Q8, C2×C28, C7×Q8, C22×C14, C22×C28, Q8×C14, C7×C4○D4, Q8×C28

Smallest permutation representation of Q8×C28
Regular action on 224 points
Generators in S224
(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)
(1 159 201 65)(2 160 202 66)(3 161 203 67)(4 162 204 68)(5 163 205 69)(6 164 206 70)(7 165 207 71)(8 166 208 72)(9 167 209 73)(10 168 210 74)(11 141 211 75)(12 142 212 76)(13 143 213 77)(14 144 214 78)(15 145 215 79)(16 146 216 80)(17 147 217 81)(18 148 218 82)(19 149 219 83)(20 150 220 84)(21 151 221 57)(22 152 222 58)(23 153 223 59)(24 154 224 60)(25 155 197 61)(26 156 198 62)(27 157 199 63)(28 158 200 64)(29 101 175 115)(30 102 176 116)(31 103 177 117)(32 104 178 118)(33 105 179 119)(34 106 180 120)(35 107 181 121)(36 108 182 122)(37 109 183 123)(38 110 184 124)(39 111 185 125)(40 112 186 126)(41 85 187 127)(42 86 188 128)(43 87 189 129)(44 88 190 130)(45 89 191 131)(46 90 192 132)(47 91 193 133)(48 92 194 134)(49 93 195 135)(50 94 196 136)(51 95 169 137)(52 96 170 138)(53 97 171 139)(54 98 172 140)(55 99 173 113)(56 100 174 114)
(1 87 201 129)(2 88 202 130)(3 89 203 131)(4 90 204 132)(5 91 205 133)(6 92 206 134)(7 93 207 135)(8 94 208 136)(9 95 209 137)(10 96 210 138)(11 97 211 139)(12 98 212 140)(13 99 213 113)(14 100 214 114)(15 101 215 115)(16 102 216 116)(17 103 217 117)(18 104 218 118)(19 105 219 119)(20 106 220 120)(21 107 221 121)(22 108 222 122)(23 109 223 123)(24 110 224 124)(25 111 197 125)(26 112 198 126)(27 85 199 127)(28 86 200 128)(29 79 175 145)(30 80 176 146)(31 81 177 147)(32 82 178 148)(33 83 179 149)(34 84 180 150)(35 57 181 151)(36 58 182 152)(37 59 183 153)(38 60 184 154)(39 61 185 155)(40 62 186 156)(41 63 187 157)(42 64 188 158)(43 65 189 159)(44 66 190 160)(45 67 191 161)(46 68 192 162)(47 69 193 163)(48 70 194 164)(49 71 195 165)(50 72 196 166)(51 73 169 167)(52 74 170 168)(53 75 171 141)(54 76 172 142)(55 77 173 143)(56 78 174 144)

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

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

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

140 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E ··· 4P 7A ··· 7F 14A ··· 14R 28A ··· 28X 28Y ··· 28CR order 1 2 2 2 4 4 4 4 4 ··· 4 7 ··· 7 14 ··· 14 28 ··· 28 28 ··· 28 size 1 1 1 1 1 1 1 1 2 ··· 2 1 ··· 1 1 ··· 1 1 ··· 1 2 ··· 2

140 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + - image C1 C2 C2 C2 C4 C7 C14 C14 C14 C28 Q8 C4○D4 C7×Q8 C7×C4○D4 kernel Q8×C28 C4×C28 C7×C4⋊C4 Q8×C14 C7×Q8 C4×Q8 C42 C4⋊C4 C2×Q8 Q8 C28 C14 C4 C2 # reps 1 3 3 1 8 6 18 18 6 48 2 2 12 12

Matrix representation of Q8×C28 in GL3(𝔽29) generated by

 17 0 0 0 23 0 0 0 23
,
 1 0 0 0 0 28 0 1 0
,
 1 0 0 0 15 21 0 21 14
G:=sub<GL(3,GF(29))| [17,0,0,0,23,0,0,0,23],[1,0,0,0,0,1,0,28,0],[1,0,0,0,15,21,0,21,14] >;

Q8×C28 in GAP, Magma, Sage, TeX

Q_8\times C_{28}
% in TeX

G:=Group("Q8xC28");
// GroupNames label

G:=SmallGroup(224,154);
// by ID

G=gap.SmallGroup(224,154);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,672,697,343,770]);
// Polycyclic

G:=Group<a,b,c|a^28=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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