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G = Q8×C33order 264 = 23·3·11

Direct product of C33 and Q8

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

Aliases: Q8×C33, C4.C66, C44.3C6, C12.3C22, C132.7C2, C66.24C22, C22.7(C2×C6), C2.2(C2×C66), C6.7(C2×C22), SmallGroup(264,30)

Series: Derived Chief Lower central Upper central

C1C2 — Q8×C33
C1C2C22C66C132 — Q8×C33
C1C2 — Q8×C33
C1C66 — Q8×C33

Generators and relations for Q8×C33
 G = < a,b,c | a33=b4=1, c2=b2, ab=ba, ac=ca, cbc-1=b-1 >


Smallest permutation representation of Q8×C33
Regular action on 264 points
Generators in S264
(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)(199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231)(232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264)
(1 180 162 252)(2 181 163 253)(3 182 164 254)(4 183 165 255)(5 184 133 256)(6 185 134 257)(7 186 135 258)(8 187 136 259)(9 188 137 260)(10 189 138 261)(11 190 139 262)(12 191 140 263)(13 192 141 264)(14 193 142 232)(15 194 143 233)(16 195 144 234)(17 196 145 235)(18 197 146 236)(19 198 147 237)(20 166 148 238)(21 167 149 239)(22 168 150 240)(23 169 151 241)(24 170 152 242)(25 171 153 243)(26 172 154 244)(27 173 155 245)(28 174 156 246)(29 175 157 247)(30 176 158 248)(31 177 159 249)(32 178 160 250)(33 179 161 251)(34 81 229 124)(35 82 230 125)(36 83 231 126)(37 84 199 127)(38 85 200 128)(39 86 201 129)(40 87 202 130)(41 88 203 131)(42 89 204 132)(43 90 205 100)(44 91 206 101)(45 92 207 102)(46 93 208 103)(47 94 209 104)(48 95 210 105)(49 96 211 106)(50 97 212 107)(51 98 213 108)(52 99 214 109)(53 67 215 110)(54 68 216 111)(55 69 217 112)(56 70 218 113)(57 71 219 114)(58 72 220 115)(59 73 221 116)(60 74 222 117)(61 75 223 118)(62 76 224 119)(63 77 225 120)(64 78 226 121)(65 79 227 122)(66 80 228 123)
(1 77 162 120)(2 78 163 121)(3 79 164 122)(4 80 165 123)(5 81 133 124)(6 82 134 125)(7 83 135 126)(8 84 136 127)(9 85 137 128)(10 86 138 129)(11 87 139 130)(12 88 140 131)(13 89 141 132)(14 90 142 100)(15 91 143 101)(16 92 144 102)(17 93 145 103)(18 94 146 104)(19 95 147 105)(20 96 148 106)(21 97 149 107)(22 98 150 108)(23 99 151 109)(24 67 152 110)(25 68 153 111)(26 69 154 112)(27 70 155 113)(28 71 156 114)(29 72 157 115)(30 73 158 116)(31 74 159 117)(32 75 160 118)(33 76 161 119)(34 256 229 184)(35 257 230 185)(36 258 231 186)(37 259 199 187)(38 260 200 188)(39 261 201 189)(40 262 202 190)(41 263 203 191)(42 264 204 192)(43 232 205 193)(44 233 206 194)(45 234 207 195)(46 235 208 196)(47 236 209 197)(48 237 210 198)(49 238 211 166)(50 239 212 167)(51 240 213 168)(52 241 214 169)(53 242 215 170)(54 243 216 171)(55 244 217 172)(56 245 218 173)(57 246 219 174)(58 247 220 175)(59 248 221 176)(60 249 222 177)(61 250 223 178)(62 251 224 179)(63 252 225 180)(64 253 226 181)(65 254 227 182)(66 255 228 183)

G:=sub<Sym(264)| (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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231)(232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264), (1,180,162,252)(2,181,163,253)(3,182,164,254)(4,183,165,255)(5,184,133,256)(6,185,134,257)(7,186,135,258)(8,187,136,259)(9,188,137,260)(10,189,138,261)(11,190,139,262)(12,191,140,263)(13,192,141,264)(14,193,142,232)(15,194,143,233)(16,195,144,234)(17,196,145,235)(18,197,146,236)(19,198,147,237)(20,166,148,238)(21,167,149,239)(22,168,150,240)(23,169,151,241)(24,170,152,242)(25,171,153,243)(26,172,154,244)(27,173,155,245)(28,174,156,246)(29,175,157,247)(30,176,158,248)(31,177,159,249)(32,178,160,250)(33,179,161,251)(34,81,229,124)(35,82,230,125)(36,83,231,126)(37,84,199,127)(38,85,200,128)(39,86,201,129)(40,87,202,130)(41,88,203,131)(42,89,204,132)(43,90,205,100)(44,91,206,101)(45,92,207,102)(46,93,208,103)(47,94,209,104)(48,95,210,105)(49,96,211,106)(50,97,212,107)(51,98,213,108)(52,99,214,109)(53,67,215,110)(54,68,216,111)(55,69,217,112)(56,70,218,113)(57,71,219,114)(58,72,220,115)(59,73,221,116)(60,74,222,117)(61,75,223,118)(62,76,224,119)(63,77,225,120)(64,78,226,121)(65,79,227,122)(66,80,228,123), (1,77,162,120)(2,78,163,121)(3,79,164,122)(4,80,165,123)(5,81,133,124)(6,82,134,125)(7,83,135,126)(8,84,136,127)(9,85,137,128)(10,86,138,129)(11,87,139,130)(12,88,140,131)(13,89,141,132)(14,90,142,100)(15,91,143,101)(16,92,144,102)(17,93,145,103)(18,94,146,104)(19,95,147,105)(20,96,148,106)(21,97,149,107)(22,98,150,108)(23,99,151,109)(24,67,152,110)(25,68,153,111)(26,69,154,112)(27,70,155,113)(28,71,156,114)(29,72,157,115)(30,73,158,116)(31,74,159,117)(32,75,160,118)(33,76,161,119)(34,256,229,184)(35,257,230,185)(36,258,231,186)(37,259,199,187)(38,260,200,188)(39,261,201,189)(40,262,202,190)(41,263,203,191)(42,264,204,192)(43,232,205,193)(44,233,206,194)(45,234,207,195)(46,235,208,196)(47,236,209,197)(48,237,210,198)(49,238,211,166)(50,239,212,167)(51,240,213,168)(52,241,214,169)(53,242,215,170)(54,243,216,171)(55,244,217,172)(56,245,218,173)(57,246,219,174)(58,247,220,175)(59,248,221,176)(60,249,222,177)(61,250,223,178)(62,251,224,179)(63,252,225,180)(64,253,226,181)(65,254,227,182)(66,255,228,183)>;

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)(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231)(232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264), (1,180,162,252)(2,181,163,253)(3,182,164,254)(4,183,165,255)(5,184,133,256)(6,185,134,257)(7,186,135,258)(8,187,136,259)(9,188,137,260)(10,189,138,261)(11,190,139,262)(12,191,140,263)(13,192,141,264)(14,193,142,232)(15,194,143,233)(16,195,144,234)(17,196,145,235)(18,197,146,236)(19,198,147,237)(20,166,148,238)(21,167,149,239)(22,168,150,240)(23,169,151,241)(24,170,152,242)(25,171,153,243)(26,172,154,244)(27,173,155,245)(28,174,156,246)(29,175,157,247)(30,176,158,248)(31,177,159,249)(32,178,160,250)(33,179,161,251)(34,81,229,124)(35,82,230,125)(36,83,231,126)(37,84,199,127)(38,85,200,128)(39,86,201,129)(40,87,202,130)(41,88,203,131)(42,89,204,132)(43,90,205,100)(44,91,206,101)(45,92,207,102)(46,93,208,103)(47,94,209,104)(48,95,210,105)(49,96,211,106)(50,97,212,107)(51,98,213,108)(52,99,214,109)(53,67,215,110)(54,68,216,111)(55,69,217,112)(56,70,218,113)(57,71,219,114)(58,72,220,115)(59,73,221,116)(60,74,222,117)(61,75,223,118)(62,76,224,119)(63,77,225,120)(64,78,226,121)(65,79,227,122)(66,80,228,123), (1,77,162,120)(2,78,163,121)(3,79,164,122)(4,80,165,123)(5,81,133,124)(6,82,134,125)(7,83,135,126)(8,84,136,127)(9,85,137,128)(10,86,138,129)(11,87,139,130)(12,88,140,131)(13,89,141,132)(14,90,142,100)(15,91,143,101)(16,92,144,102)(17,93,145,103)(18,94,146,104)(19,95,147,105)(20,96,148,106)(21,97,149,107)(22,98,150,108)(23,99,151,109)(24,67,152,110)(25,68,153,111)(26,69,154,112)(27,70,155,113)(28,71,156,114)(29,72,157,115)(30,73,158,116)(31,74,159,117)(32,75,160,118)(33,76,161,119)(34,256,229,184)(35,257,230,185)(36,258,231,186)(37,259,199,187)(38,260,200,188)(39,261,201,189)(40,262,202,190)(41,263,203,191)(42,264,204,192)(43,232,205,193)(44,233,206,194)(45,234,207,195)(46,235,208,196)(47,236,209,197)(48,237,210,198)(49,238,211,166)(50,239,212,167)(51,240,213,168)(52,241,214,169)(53,242,215,170)(54,243,216,171)(55,244,217,172)(56,245,218,173)(57,246,219,174)(58,247,220,175)(59,248,221,176)(60,249,222,177)(61,250,223,178)(62,251,224,179)(63,252,225,180)(64,253,226,181)(65,254,227,182)(66,255,228,183) );

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),(199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231),(232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,247,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264)], [(1,180,162,252),(2,181,163,253),(3,182,164,254),(4,183,165,255),(5,184,133,256),(6,185,134,257),(7,186,135,258),(8,187,136,259),(9,188,137,260),(10,189,138,261),(11,190,139,262),(12,191,140,263),(13,192,141,264),(14,193,142,232),(15,194,143,233),(16,195,144,234),(17,196,145,235),(18,197,146,236),(19,198,147,237),(20,166,148,238),(21,167,149,239),(22,168,150,240),(23,169,151,241),(24,170,152,242),(25,171,153,243),(26,172,154,244),(27,173,155,245),(28,174,156,246),(29,175,157,247),(30,176,158,248),(31,177,159,249),(32,178,160,250),(33,179,161,251),(34,81,229,124),(35,82,230,125),(36,83,231,126),(37,84,199,127),(38,85,200,128),(39,86,201,129),(40,87,202,130),(41,88,203,131),(42,89,204,132),(43,90,205,100),(44,91,206,101),(45,92,207,102),(46,93,208,103),(47,94,209,104),(48,95,210,105),(49,96,211,106),(50,97,212,107),(51,98,213,108),(52,99,214,109),(53,67,215,110),(54,68,216,111),(55,69,217,112),(56,70,218,113),(57,71,219,114),(58,72,220,115),(59,73,221,116),(60,74,222,117),(61,75,223,118),(62,76,224,119),(63,77,225,120),(64,78,226,121),(65,79,227,122),(66,80,228,123)], [(1,77,162,120),(2,78,163,121),(3,79,164,122),(4,80,165,123),(5,81,133,124),(6,82,134,125),(7,83,135,126),(8,84,136,127),(9,85,137,128),(10,86,138,129),(11,87,139,130),(12,88,140,131),(13,89,141,132),(14,90,142,100),(15,91,143,101),(16,92,144,102),(17,93,145,103),(18,94,146,104),(19,95,147,105),(20,96,148,106),(21,97,149,107),(22,98,150,108),(23,99,151,109),(24,67,152,110),(25,68,153,111),(26,69,154,112),(27,70,155,113),(28,71,156,114),(29,72,157,115),(30,73,158,116),(31,74,159,117),(32,75,160,118),(33,76,161,119),(34,256,229,184),(35,257,230,185),(36,258,231,186),(37,259,199,187),(38,260,200,188),(39,261,201,189),(40,262,202,190),(41,263,203,191),(42,264,204,192),(43,232,205,193),(44,233,206,194),(45,234,207,195),(46,235,208,196),(47,236,209,197),(48,237,210,198),(49,238,211,166),(50,239,212,167),(51,240,213,168),(52,241,214,169),(53,242,215,170),(54,243,216,171),(55,244,217,172),(56,245,218,173),(57,246,219,174),(58,247,220,175),(59,248,221,176),(60,249,222,177),(61,250,223,178),(62,251,224,179),(63,252,225,180),(64,253,226,181),(65,254,227,182),(66,255,228,183)])

165 conjugacy classes

class 1  2 3A3B4A4B4C6A6B11A···11J12A···12F22A···22J33A···33T44A···44AD66A···66T132A···132BH
order12334446611···1112···1222···2233···3344···4466···66132···132
size1111222111···12···21···11···12···21···12···2

165 irreducible representations

dim111111112222
type++-
imageC1C2C3C6C11C22C33C66Q8C3×Q8Q8×C11Q8×C33
kernelQ8×C33C132Q8×C11C44C3×Q8C12Q8C4C33C11C3C1
# reps132610302060121020

Matrix representation of Q8×C33 in GL2(𝔽397) generated by

1200
0120
,
1395
1396
,
26210
131371
G:=sub<GL(2,GF(397))| [120,0,0,120],[1,1,395,396],[26,131,210,371] >;

Q8×C33 in GAP, Magma, Sage, TeX

Q_8\times C_{33}
% in TeX

G:=Group("Q8xC33");
// GroupNames label

G:=SmallGroup(264,30);
// by ID

G=gap.SmallGroup(264,30);
# by ID

G:=PCGroup([5,-2,-2,-3,-11,-2,660,1341,666]);
// Polycyclic

G:=Group<a,b,c|a^33=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×C33 in TeX

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