C27:C8 - GroupNames

(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)

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

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

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

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

C₂₇⋊C₈ is a maximal subgroup of C₈×D₂₇ C₈⋊D₂₇ C₄.Dic₂₇ D₄.D₂₇ D₄⋊D₂₇ C₂₇⋊Q₁₆ Q₈⋊₂D₂₇
C₂₇⋊C₈ is a maximal quotient of C₂₇⋊C₁₆

60 conjugacy classes

class	1	2	3	4A	4B	6	8A	8B	8C	8D	9A	9B	9C	12A	12B	18A	18B	18C	27A	···	27I	36A	···	36F	54A	···	54I	108A	···	108R
order	1	2	3	4	4	6	8	8	8	8	9	9	9	12	12	18	18	18	27	···	27	36	···	36	54	···	54	108	···	108
size	1	1	2	1	1	2	27	27	27	27	2	2	2	2	2	2	2	2	2	···	2	2	···	2	2	···	2	2	···	2

60 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	2	2	2
type	+	+			+	-	+		-	+		-
image	C₁	C₂	C₄	C₈	S₃	Dic₃	D₉	C₃⋊C₈	Dic₉	D₂₇	C₉⋊C₈	Dic₂₇	C₂₇⋊C₈
kernel	C₂₇⋊C₈	C₁₀₈	C₅₄	C₂₇	C₃₆	C₁₈	C₁₂	C₉	C₆	C₄	C₃	C₂	C₁
# reps	1	1	2	4	1	1	3	2	3	9	6	9	18

Matrix representation of C₂₇⋊C₈ ►in GL₂(𝔽₄₃₃) generated by

401	130
303	271

91	298
207	342

G:=sub<GL(2,GF(433))| [401,303,130,271],[91,207,298,342] >;

C₂₇⋊C₈ in GAP, Magma, Sage, TeX

C_{27}\rtimes C_8

% in TeX

G:=Group("C27:C8");

// GroupNames label

G:=SmallGroup(216,1);

// by ID

G=gap.SmallGroup(216,1);

# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,963,381,3604,208,5189]);

// Polycyclic

G:=Group<a,b|a^27=b^8=1,b*a*b^-1=a^-1>;

// generators/relations

Export

Subgroup lattice of C₂₇⋊C₈ in TeX

G = C27⋊C8 order 216 = 23·33

The semidirect product of C27 and C8 acting via C8/C4=C2

G = C₂₇⋊C₈ order 216 = 2³·3³

The semidirect product of C₂₇ and C₈ acting via C₈/C₄=C₂