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## G = C4○D4×C27order 432 = 24·33

### Direct product of C27 and C4○D4

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C4○D4×C27
 Chief series C1 — C3 — C9 — C18 — C54 — C2×C54 — D4×C27 — C4○D4×C27
 Lower central C1 — C2 — C4○D4×C27
 Upper central C1 — C108 — C4○D4×C27

Generators and relations for C4○D4×C27
G = < a,b,c,d | a27=b4=d2=1, c2=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c >

Subgroups: 92 in 80 conjugacy classes, 68 normal (20 characteristic)
C1, C2, C2 [×3], C3, C4, C4 [×3], C22 [×3], C6, C6 [×3], C2×C4 [×3], D4 [×3], Q8, C9, C12, C12 [×3], C2×C6 [×3], C4○D4, C18, C18 [×3], C2×C12 [×3], C3×D4 [×3], C3×Q8, C27, C36, C36 [×3], C2×C18 [×3], C3×C4○D4, C54, C54 [×3], C2×C36 [×3], D4×C9 [×3], Q8×C9, C108, C108 [×3], C2×C54 [×3], C9×C4○D4, C2×C108 [×3], D4×C27 [×3], Q8×C27, C4○D4×C27
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C9, C2×C6 [×7], C4○D4, C18 [×7], C22×C6, C27, C2×C18 [×7], C3×C4○D4, C54 [×7], C22×C18, C2×C54 [×7], C9×C4○D4, C22×C54, C4○D4×C27

Smallest permutation representation of C4○D4×C27
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)
(1 154 207 39)(2 155 208 40)(3 156 209 41)(4 157 210 42)(5 158 211 43)(6 159 212 44)(7 160 213 45)(8 161 214 46)(9 162 215 47)(10 136 216 48)(11 137 190 49)(12 138 191 50)(13 139 192 51)(14 140 193 52)(15 141 194 53)(16 142 195 54)(17 143 196 28)(18 144 197 29)(19 145 198 30)(20 146 199 31)(21 147 200 32)(22 148 201 33)(23 149 202 34)(24 150 203 35)(25 151 204 36)(26 152 205 37)(27 153 206 38)(55 122 92 178)(56 123 93 179)(57 124 94 180)(58 125 95 181)(59 126 96 182)(60 127 97 183)(61 128 98 184)(62 129 99 185)(63 130 100 186)(64 131 101 187)(65 132 102 188)(66 133 103 189)(67 134 104 163)(68 135 105 164)(69 109 106 165)(70 110 107 166)(71 111 108 167)(72 112 82 168)(73 113 83 169)(74 114 84 170)(75 115 85 171)(76 116 86 172)(77 117 87 173)(78 118 88 174)(79 119 89 175)(80 120 90 176)(81 121 91 177)
(1 105 207 68)(2 106 208 69)(3 107 209 70)(4 108 210 71)(5 82 211 72)(6 83 212 73)(7 84 213 74)(8 85 214 75)(9 86 215 76)(10 87 216 77)(11 88 190 78)(12 89 191 79)(13 90 192 80)(14 91 193 81)(15 92 194 55)(16 93 195 56)(17 94 196 57)(18 95 197 58)(19 96 198 59)(20 97 199 60)(21 98 200 61)(22 99 201 62)(23 100 202 63)(24 101 203 64)(25 102 204 65)(26 103 205 66)(27 104 206 67)(28 124 143 180)(29 125 144 181)(30 126 145 182)(31 127 146 183)(32 128 147 184)(33 129 148 185)(34 130 149 186)(35 131 150 187)(36 132 151 188)(37 133 152 189)(38 134 153 163)(39 135 154 164)(40 109 155 165)(41 110 156 166)(42 111 157 167)(43 112 158 168)(44 113 159 169)(45 114 160 170)(46 115 161 171)(47 116 162 172)(48 117 136 173)(49 118 137 174)(50 119 138 175)(51 120 139 176)(52 121 140 177)(53 122 141 178)(54 123 142 179)
(55 92)(56 93)(57 94)(58 95)(59 96)(60 97)(61 98)(62 99)(63 100)(64 101)(65 102)(66 103)(67 104)(68 105)(69 106)(70 107)(71 108)(72 82)(73 83)(74 84)(75 85)(76 86)(77 87)(78 88)(79 89)(80 90)(81 91)(109 165)(110 166)(111 167)(112 168)(113 169)(114 170)(115 171)(116 172)(117 173)(118 174)(119 175)(120 176)(121 177)(122 178)(123 179)(124 180)(125 181)(126 182)(127 183)(128 184)(129 185)(130 186)(131 187)(132 188)(133 189)(134 163)(135 164)

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

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

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

270 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 4E 6A 6B 6C ··· 6H 9A ··· 9F 12A 12B 12C 12D 12E ··· 12J 18A ··· 18F 18G ··· 18X 27A ··· 27R 36A ··· 36L 36M ··· 36AD 54A ··· 54R 54S ··· 54BT 108A ··· 108AJ 108AK ··· 108CL order 1 2 2 2 2 3 3 4 4 4 4 4 6 6 6 ··· 6 9 ··· 9 12 12 12 12 12 ··· 12 18 ··· 18 18 ··· 18 27 ··· 27 36 ··· 36 36 ··· 36 54 ··· 54 54 ··· 54 108 ··· 108 108 ··· 108 size 1 1 2 2 2 1 1 1 1 2 2 2 1 1 2 ··· 2 1 ··· 1 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 1 ··· 1 2 ··· 2 1 ··· 1 2 ··· 2 1 ··· 1 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 C4○D4 C3×C4○D4 C9×C4○D4 C4○D4×C27 kernel C4○D4×C27 C2×C108 D4×C27 Q8×C27 C9×C4○D4 C2×C36 D4×C9 Q8×C9 C3×C4○D4 C2×C12 C3×D4 C3×Q8 C4○D4 C2×C4 D4 Q8 C27 C9 C3 C1 # reps 1 3 3 1 2 6 6 2 6 18 18 6 18 54 54 18 2 4 12 36

Matrix representation of C4○D4×C27 in GL2(𝔽109) generated by

 15 0 0 15
,
 33 0 0 33
,
 108 65 5 1
,
 1 0 104 108
G:=sub<GL(2,GF(109))| [15,0,0,15],[33,0,0,33],[108,5,65,1],[1,104,0,108] >;

C4○D4×C27 in GAP, Magma, Sage, TeX

C_4\circ D_4\times C_{27}
% in TeX

G:=Group("C4oD4xC27");
// GroupNames label

G:=SmallGroup(432,56);
// by ID

G=gap.SmallGroup(432,56);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,365,142,192,166]);
// Polycyclic

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

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