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G = C7⋊M6(2)  order 448 = 26·7

The semidirect product of C7 and M6(2) acting via M6(2)/C2×C16=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C56.6C8, C72M6(2), C112.4C4, C28.1C16, C16.22D14, C16.2Dic7, C112.27C22, C7⋊C325C2, C4.(C7⋊C16), C8.3(C7⋊C8), (C2×C28).9C8, C22.(C7⋊C16), (C2×C16).7D7, C28.43(C2×C8), C14.9(C2×C16), (C2×C56).25C4, C56.76(C2×C4), (C2×C14).3C16, (C2×C112).13C2, C8.22(C2×Dic7), (C2×C8).14Dic7, C4.15(C2×C7⋊C8), C2.4(C2×C7⋊C16), (C2×C4).5(C7⋊C8), SmallGroup(448,56)

Series: Derived Chief Lower central Upper central

C1C14 — C7⋊M6(2)
C1C7C14C28C56C112C7⋊C32 — C7⋊M6(2)
C7C14 — C7⋊M6(2)
C1C16C2×C16

Generators and relations for C7⋊M6(2)
 G = < a,b,c | a7=b32=c2=1, bab-1=a-1, ac=ca, cbc=b17 >

2C2
2C14
7C32
7C32
7M6(2)

Smallest permutation representation of C7⋊M6(2)
On 224 points
Generators in S224
(1 185 197 70 111 129 37)(2 38 130 112 71 198 186)(3 187 199 72 113 131 39)(4 40 132 114 73 200 188)(5 189 201 74 115 133 41)(6 42 134 116 75 202 190)(7 191 203 76 117 135 43)(8 44 136 118 77 204 192)(9 161 205 78 119 137 45)(10 46 138 120 79 206 162)(11 163 207 80 121 139 47)(12 48 140 122 81 208 164)(13 165 209 82 123 141 49)(14 50 142 124 83 210 166)(15 167 211 84 125 143 51)(16 52 144 126 85 212 168)(17 169 213 86 127 145 53)(18 54 146 128 87 214 170)(19 171 215 88 97 147 55)(20 56 148 98 89 216 172)(21 173 217 90 99 149 57)(22 58 150 100 91 218 174)(23 175 219 92 101 151 59)(24 60 152 102 93 220 176)(25 177 221 94 103 153 61)(26 62 154 104 95 222 178)(27 179 223 96 105 155 63)(28 64 156 106 65 224 180)(29 181 193 66 107 157 33)(30 34 158 108 67 194 182)(31 183 195 68 109 159 35)(32 36 160 110 69 196 184)
(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)
(2 18)(4 20)(6 22)(8 24)(10 26)(12 28)(14 30)(16 32)(34 50)(36 52)(38 54)(40 56)(42 58)(44 60)(46 62)(48 64)(65 81)(67 83)(69 85)(71 87)(73 89)(75 91)(77 93)(79 95)(98 114)(100 116)(102 118)(104 120)(106 122)(108 124)(110 126)(112 128)(130 146)(132 148)(134 150)(136 152)(138 154)(140 156)(142 158)(144 160)(162 178)(164 180)(166 182)(168 184)(170 186)(172 188)(174 190)(176 192)(194 210)(196 212)(198 214)(200 216)(202 218)(204 220)(206 222)(208 224)

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

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

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

136 conjugacy classes

class 1 2A2B4A4B4C7A7B7C8A8B8C8D8E8F14A···14I16A···16H16I16J16K16L28A···28L32A···32P56A···56X112A···112AV
order12244477788888814···1416···161616161628···2832···3256···56112···112
size1121122221111222···21···122222···214···142···22···2

136 irreducible representations

dim1111111112222222222
type++++-+-
imageC1C2C2C4C4C8C8C16C16D7Dic7D14Dic7C7⋊C8C7⋊C8M6(2)C7⋊C16C7⋊C16C7⋊M6(2)
kernelC7⋊M6(2)C7⋊C32C2×C112C112C2×C56C56C2×C28C28C2×C14C2×C16C16C16C2×C8C8C2×C4C7C4C22C1
# reps1212244883333668121248

Matrix representation of C7⋊M6(2) in GL2(𝔽449) generated by

3240
0176
,
01
350
,
10
0448
G:=sub<GL(2,GF(449))| [324,0,0,176],[0,35,1,0],[1,0,0,448] >;

C7⋊M6(2) in GAP, Magma, Sage, TeX

C_7\rtimes M_6(2)
% in TeX

G:=Group("C7:M6(2)");
// GroupNames label

G:=SmallGroup(448,56);
// by ID

G=gap.SmallGroup(448,56);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,28,477,58,80,102,18822]);
// Polycyclic

G:=Group<a,b,c|a^7=b^32=c^2=1,b*a*b^-1=a^-1,a*c=c*a,c*b*c=b^17>;
// generators/relations

Export

Subgroup lattice of C7⋊M6(2) in TeX

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