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G = C4.Dic27order 432 = 24·33

The non-split extension by C4 of Dic27 acting via Dic27/C54=C2

metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C4.Dic27, C108.1C4, C36.63D6, C4.16D54, C272M4(2), C12.63D18, C36.1Dic3, C12.1Dic9, C22.Dic27, C108.16C22, C27⋊C85C2, (C2×C54).3C4, C54.7(C2×C4), (C2×C4).2D27, (C2×C36).10S3, (C2×C108).5C2, (C2×C12).10D9, (C2×C6).4Dic9, C6.7(C2×Dic9), C9.(C4.Dic3), C3.(C4.Dic9), C18.7(C2×Dic3), C2.3(C2×Dic27), (C2×C18).4Dic3, SmallGroup(432,10)

Series: Derived Chief Lower central Upper central

C1C54 — C4.Dic27
C1C3C9C27C54C108C27⋊C8 — C4.Dic27
C27C54 — C4.Dic27
C1C4C2×C4

Generators and relations for C4.Dic27
 G = < a,b,c | a4=1, b54=a2, c2=a2b27, ab=ba, cac-1=a-1, cbc-1=b53 >

2C2
2C6
27C8
27C8
2C18
27M4(2)
9C3⋊C8
9C3⋊C8
2C54
9C4.Dic3
3C9⋊C8
3C9⋊C8
3C4.Dic9

Smallest permutation representation of C4.Dic27
On 216 points
Generators in S216
(1 82 55 28)(2 83 56 29)(3 84 57 30)(4 85 58 31)(5 86 59 32)(6 87 60 33)(7 88 61 34)(8 89 62 35)(9 90 63 36)(10 91 64 37)(11 92 65 38)(12 93 66 39)(13 94 67 40)(14 95 68 41)(15 96 69 42)(16 97 70 43)(17 98 71 44)(18 99 72 45)(19 100 73 46)(20 101 74 47)(21 102 75 48)(22 103 76 49)(23 104 77 50)(24 105 78 51)(25 106 79 52)(26 107 80 53)(27 108 81 54)(109 136 163 190)(110 137 164 191)(111 138 165 192)(112 139 166 193)(113 140 167 194)(114 141 168 195)(115 142 169 196)(116 143 170 197)(117 144 171 198)(118 145 172 199)(119 146 173 200)(120 147 174 201)(121 148 175 202)(122 149 176 203)(123 150 177 204)(124 151 178 205)(125 152 179 206)(126 153 180 207)(127 154 181 208)(128 155 182 209)(129 156 183 210)(130 157 184 211)(131 158 185 212)(132 159 186 213)(133 160 187 214)(134 161 188 215)(135 162 189 216)
(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)
(1 136 82 109 55 190 28 163)(2 189 83 162 56 135 29 216)(3 134 84 215 57 188 30 161)(4 187 85 160 58 133 31 214)(5 132 86 213 59 186 32 159)(6 185 87 158 60 131 33 212)(7 130 88 211 61 184 34 157)(8 183 89 156 62 129 35 210)(9 128 90 209 63 182 36 155)(10 181 91 154 64 127 37 208)(11 126 92 207 65 180 38 153)(12 179 93 152 66 125 39 206)(13 124 94 205 67 178 40 151)(14 177 95 150 68 123 41 204)(15 122 96 203 69 176 42 149)(16 175 97 148 70 121 43 202)(17 120 98 201 71 174 44 147)(18 173 99 146 72 119 45 200)(19 118 100 199 73 172 46 145)(20 171 101 144 74 117 47 198)(21 116 102 197 75 170 48 143)(22 169 103 142 76 115 49 196)(23 114 104 195 77 168 50 141)(24 167 105 140 78 113 51 194)(25 112 106 193 79 166 52 139)(26 165 107 138 80 111 53 192)(27 110 108 191 81 164 54 137)

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

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

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

114 conjugacy classes

class 1 2A2B 3 4A4B4C6A6B6C8A8B8C8D9A9B9C12A12B12C12D18A···18I27A···27I36A···36L54A···54AA108A···108AJ
order122344466688889991212121218···1827···2736···3654···54108···108
size11221122225454545422222222···22···22···22···22···2

114 irreducible representations

dim111112222222222222222
type++++-+-+-+-+-+-
imageC1C2C2C4C4S3Dic3D6Dic3M4(2)D9Dic9D18Dic9C4.Dic3D27Dic27D54Dic27C4.Dic9C4.Dic27
kernelC4.Dic27C27⋊C8C2×C108C108C2×C54C2×C36C36C36C2×C18C27C2×C12C12C12C2×C6C9C2×C4C4C4C22C3C1
# reps12122111123333499991236

Matrix representation of C4.Dic27 in GL2(𝔽433) generated by

1790
63254
,
1230
31788
,
347310
11586
G:=sub<GL(2,GF(433))| [179,63,0,254],[123,317,0,88],[347,115,310,86] >;

C4.Dic27 in GAP, Magma, Sage, TeX

C_4.{\rm Dic}_{27}
% in TeX

G:=Group("C4.Dic27");
// GroupNames label

G:=SmallGroup(432,10);
// by ID

G=gap.SmallGroup(432,10);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,58,2804,557,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^4=1,b^54=a^2,c^2=a^2*b^27,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=b^53>;
// generators/relations

Export

Subgroup lattice of C4.Dic27 in TeX

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