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## G = D4×C54order 432 = 24·33

### Direct product of C54 and D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — D4×C54
 Chief series C1 — C3 — C9 — C18 — C54 — C2×C54 — D4×C27 — D4×C54
 Lower central C1 — C2 — D4×C54
 Upper central C1 — C2×C54 — D4×C54

Generators and relations for D4×C54
G = < a,b,c | a54=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 140 in 108 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, D4, C23, C9, C12, C2×C6, C2×C6, C2×C6, C2×D4, C18, C18, C18, C2×C12, C3×D4, C22×C6, C27, C36, C2×C18, C2×C18, C2×C18, C6×D4, C54, C54, C54, C2×C36, D4×C9, C22×C18, C108, C2×C54, C2×C54, C2×C54, D4×C18, C2×C108, D4×C27, C22×C54, D4×C54
Quotients: C1, C2, C3, C22, C6, D4, C23, C9, C2×C6, C2×D4, C18, C3×D4, C22×C6, C27, C2×C18, C6×D4, C54, D4×C9, C22×C18, C2×C54, D4×C18, D4×C27, C22×C54, D4×C54

Smallest permutation representation of D4×C54
On 216 points
Generators in S216
(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)
(1 169 131 75)(2 170 132 76)(3 171 133 77)(4 172 134 78)(5 173 135 79)(6 174 136 80)(7 175 137 81)(8 176 138 82)(9 177 139 83)(10 178 140 84)(11 179 141 85)(12 180 142 86)(13 181 143 87)(14 182 144 88)(15 183 145 89)(16 184 146 90)(17 185 147 91)(18 186 148 92)(19 187 149 93)(20 188 150 94)(21 189 151 95)(22 190 152 96)(23 191 153 97)(24 192 154 98)(25 193 155 99)(26 194 156 100)(27 195 157 101)(28 196 158 102)(29 197 159 103)(30 198 160 104)(31 199 161 105)(32 200 162 106)(33 201 109 107)(34 202 110 108)(35 203 111 55)(36 204 112 56)(37 205 113 57)(38 206 114 58)(39 207 115 59)(40 208 116 60)(41 209 117 61)(42 210 118 62)(43 211 119 63)(44 212 120 64)(45 213 121 65)(46 214 122 66)(47 215 123 67)(48 216 124 68)(49 163 125 69)(50 164 126 70)(51 165 127 71)(52 166 128 72)(53 167 129 73)(54 168 130 74)
(1 158)(2 159)(3 160)(4 161)(5 162)(6 109)(7 110)(8 111)(9 112)(10 113)(11 114)(12 115)(13 116)(14 117)(15 118)(16 119)(17 120)(18 121)(19 122)(20 123)(21 124)(22 125)(23 126)(24 127)(25 128)(26 129)(27 130)(28 131)(29 132)(30 133)(31 134)(32 135)(33 136)(34 137)(35 138)(36 139)(37 140)(38 141)(39 142)(40 143)(41 144)(42 145)(43 146)(44 147)(45 148)(46 149)(47 150)(48 151)(49 152)(50 153)(51 154)(52 155)(53 156)(54 157)(55 82)(56 83)(57 84)(58 85)(59 86)(60 87)(61 88)(62 89)(63 90)(64 91)(65 92)(66 93)(67 94)(68 95)(69 96)(70 97)(71 98)(72 99)(73 100)(74 101)(75 102)(76 103)(77 104)(78 105)(79 106)(80 107)(81 108)(163 190)(164 191)(165 192)(166 193)(167 194)(168 195)(169 196)(170 197)(171 198)(172 199)(173 200)(174 201)(175 202)(176 203)(177 204)(178 205)(179 206)(180 207)(181 208)(182 209)(183 210)(184 211)(185 212)(186 213)(187 214)(188 215)(189 216)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 9A ··· 9F 12A 12B 12C 12D 18A ··· 18R 18S ··· 18AP 27A ··· 27R 36A ··· 36L 54A ··· 54BB 54BC ··· 54DV 108A ··· 108AJ order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 9 ··· 9 12 12 12 12 18 ··· 18 18 ··· 18 27 ··· 27 36 ··· 36 54 ··· 54 54 ··· 54 108 ··· 108 size 1 1 1 1 2 2 2 2 1 1 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 2 2 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2

270 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 type + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 C27 C54 C54 C54 D4 C3×D4 D4×C9 D4×C27 kernel D4×C54 C2×C108 D4×C27 C22×C54 D4×C18 C2×C36 D4×C9 C22×C18 C6×D4 C2×C12 C3×D4 C22×C6 C2×D4 C2×C4 D4 C23 C54 C18 C6 C2 # reps 1 1 4 2 2 2 8 4 6 6 24 12 18 18 72 36 2 4 12 36

Matrix representation of D4×C54 in GL3(𝔽109) generated by

 108 0 0 0 74 0 0 0 74
,
 1 0 0 0 0 108 0 1 0
,
 1 0 0 0 1 0 0 0 108
G:=sub<GL(3,GF(109))| [108,0,0,0,74,0,0,0,74],[1,0,0,0,0,1,0,108,0],[1,0,0,0,1,0,0,0,108] >;

D4×C54 in GAP, Magma, Sage, TeX

D_4\times C_{54}
% in TeX

G:=Group("D4xC54");
// GroupNames label

G:=SmallGroup(432,54);
// by ID

G=gap.SmallGroup(432,54);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,192,166]);
// Polycyclic

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

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