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## G = C40.Dic3order 480 = 25·3·5

### 2nd non-split extension by C40 of Dic3 acting via Dic3/C3=C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C60 — C40.Dic3
 Chief series C1 — C5 — C15 — C30 — C3×Dic5 — D5×C12 — C12.F5 — C40.Dic3
 Lower central C15 — C30 — C60 — C40.Dic3
 Upper central C1 — C2 — C4 — C8

Generators and relations for C40.Dic3
G = < a,b,c | a40=1, b6=a20, c2=a20b3, bab-1=a9, cac-1=a27, cbc-1=b5 >

Subgroups: 268 in 60 conjugacy classes, 29 normal (27 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C8, C8, C2×C4, D5, C10, C12, C12, C2×C6, C15, C2×C8, M4(2), Dic5, C20, D10, C3⋊C8, C24, C24, C2×C12, C3×D5, C30, C8.C4, C52C8, C40, C5⋊C8, C4×D5, C4.Dic3, C2×C24, C3×Dic5, C60, C6×D5, C8×D5, C4.F5, C24.C4, C3×C52C8, C120, C15⋊C8, D5×C12, C40.C4, D5×C24, C12.F5, C40.Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, F5, Dic6, D12, C2×Dic3, C8.C4, C2×F5, C4⋊Dic3, C3⋊F5, C4⋊F5, C24.C4, C2×C3⋊F5, C40.C4, C60⋊C4, C40.Dic3

Smallest permutation representation of C40.Dic3
On 240 points
Generators in S240
(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)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 213 79 11 223 49 21 233 59 31 203 69)(2 222 80 20 224 58 22 202 60 40 204 78)(3 231 41 29 225 67 23 211 61 9 205 47)(4 240 42 38 226 76 24 220 62 18 206 56)(5 209 43 7 227 45 25 229 63 27 207 65)(6 218 44 16 228 54 26 238 64 36 208 74)(8 236 46 34 230 72 28 216 66 14 210 52)(10 214 48 12 232 50 30 234 68 32 212 70)(13 201 51 39 235 77 33 221 71 19 215 57)(15 219 53 17 237 55 35 239 73 37 217 75)(81 134 165 87 148 171 101 154 185 107 128 191)(82 143 166 96 149 180 102 123 186 116 129 200)(83 152 167 105 150 189 103 132 187 85 130 169)(84 121 168 114 151 198 104 141 188 94 131 178)(86 139 170 92 153 176 106 159 190 112 133 196)(88 157 172 110 155 194 108 137 192 90 135 174)(89 126 173 119 156 163 109 146 193 99 136 183)(91 144 175 97 158 181 111 124 195 117 138 161)(93 122 177 115 160 199 113 142 197 95 140 179)(98 127 182 120 125 164 118 147 162 100 145 184)
(1 136 31 146 21 156 11 126)(2 139 40 133 22 159 20 153)(3 142 9 160 23 122 29 140)(4 145 18 147 24 125 38 127)(5 148 27 134 25 128 7 154)(6 151 36 121 26 131 16 141)(8 157 14 135 28 137 34 155)(10 123 32 149 30 143 12 129)(13 132 19 150 33 152 39 130)(15 138 37 124 35 158 17 144)(41 199 47 177 61 179 67 197)(42 162 56 164 62 182 76 184)(43 165 65 191 63 185 45 171)(44 168 74 178 64 188 54 198)(46 174 52 192 66 194 72 172)(48 180 70 166 68 200 50 186)(49 183 79 193 69 163 59 173)(51 189 57 167 71 169 77 187)(53 195 75 181 73 175 55 161)(58 170 80 196 78 190 60 176)(81 209 107 207 101 229 87 227)(82 212 116 234 102 232 96 214)(83 215 85 221 103 235 105 201)(84 218 94 208 104 238 114 228)(86 224 112 222 106 204 92 202)(88 230 90 236 108 210 110 216)(89 233 99 223 109 213 119 203)(91 239 117 237 111 219 97 217)(93 205 95 211 113 225 115 231)(98 220 100 226 118 240 120 206)

G:=sub<Sym(240)| (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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,213,79,11,223,49,21,233,59,31,203,69)(2,222,80,20,224,58,22,202,60,40,204,78)(3,231,41,29,225,67,23,211,61,9,205,47)(4,240,42,38,226,76,24,220,62,18,206,56)(5,209,43,7,227,45,25,229,63,27,207,65)(6,218,44,16,228,54,26,238,64,36,208,74)(8,236,46,34,230,72,28,216,66,14,210,52)(10,214,48,12,232,50,30,234,68,32,212,70)(13,201,51,39,235,77,33,221,71,19,215,57)(15,219,53,17,237,55,35,239,73,37,217,75)(81,134,165,87,148,171,101,154,185,107,128,191)(82,143,166,96,149,180,102,123,186,116,129,200)(83,152,167,105,150,189,103,132,187,85,130,169)(84,121,168,114,151,198,104,141,188,94,131,178)(86,139,170,92,153,176,106,159,190,112,133,196)(88,157,172,110,155,194,108,137,192,90,135,174)(89,126,173,119,156,163,109,146,193,99,136,183)(91,144,175,97,158,181,111,124,195,117,138,161)(93,122,177,115,160,199,113,142,197,95,140,179)(98,127,182,120,125,164,118,147,162,100,145,184), (1,136,31,146,21,156,11,126)(2,139,40,133,22,159,20,153)(3,142,9,160,23,122,29,140)(4,145,18,147,24,125,38,127)(5,148,27,134,25,128,7,154)(6,151,36,121,26,131,16,141)(8,157,14,135,28,137,34,155)(10,123,32,149,30,143,12,129)(13,132,19,150,33,152,39,130)(15,138,37,124,35,158,17,144)(41,199,47,177,61,179,67,197)(42,162,56,164,62,182,76,184)(43,165,65,191,63,185,45,171)(44,168,74,178,64,188,54,198)(46,174,52,192,66,194,72,172)(48,180,70,166,68,200,50,186)(49,183,79,193,69,163,59,173)(51,189,57,167,71,169,77,187)(53,195,75,181,73,175,55,161)(58,170,80,196,78,190,60,176)(81,209,107,207,101,229,87,227)(82,212,116,234,102,232,96,214)(83,215,85,221,103,235,105,201)(84,218,94,208,104,238,114,228)(86,224,112,222,106,204,92,202)(88,230,90,236,108,210,110,216)(89,233,99,223,109,213,119,203)(91,239,117,237,111,219,97,217)(93,205,95,211,113,225,115,231)(98,220,100,226,118,240,120,206)>;

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)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,213,79,11,223,49,21,233,59,31,203,69)(2,222,80,20,224,58,22,202,60,40,204,78)(3,231,41,29,225,67,23,211,61,9,205,47)(4,240,42,38,226,76,24,220,62,18,206,56)(5,209,43,7,227,45,25,229,63,27,207,65)(6,218,44,16,228,54,26,238,64,36,208,74)(8,236,46,34,230,72,28,216,66,14,210,52)(10,214,48,12,232,50,30,234,68,32,212,70)(13,201,51,39,235,77,33,221,71,19,215,57)(15,219,53,17,237,55,35,239,73,37,217,75)(81,134,165,87,148,171,101,154,185,107,128,191)(82,143,166,96,149,180,102,123,186,116,129,200)(83,152,167,105,150,189,103,132,187,85,130,169)(84,121,168,114,151,198,104,141,188,94,131,178)(86,139,170,92,153,176,106,159,190,112,133,196)(88,157,172,110,155,194,108,137,192,90,135,174)(89,126,173,119,156,163,109,146,193,99,136,183)(91,144,175,97,158,181,111,124,195,117,138,161)(93,122,177,115,160,199,113,142,197,95,140,179)(98,127,182,120,125,164,118,147,162,100,145,184), (1,136,31,146,21,156,11,126)(2,139,40,133,22,159,20,153)(3,142,9,160,23,122,29,140)(4,145,18,147,24,125,38,127)(5,148,27,134,25,128,7,154)(6,151,36,121,26,131,16,141)(8,157,14,135,28,137,34,155)(10,123,32,149,30,143,12,129)(13,132,19,150,33,152,39,130)(15,138,37,124,35,158,17,144)(41,199,47,177,61,179,67,197)(42,162,56,164,62,182,76,184)(43,165,65,191,63,185,45,171)(44,168,74,178,64,188,54,198)(46,174,52,192,66,194,72,172)(48,180,70,166,68,200,50,186)(49,183,79,193,69,163,59,173)(51,189,57,167,71,169,77,187)(53,195,75,181,73,175,55,161)(58,170,80,196,78,190,60,176)(81,209,107,207,101,229,87,227)(82,212,116,234,102,232,96,214)(83,215,85,221,103,235,105,201)(84,218,94,208,104,238,114,228)(86,224,112,222,106,204,92,202)(88,230,90,236,108,210,110,216)(89,233,99,223,109,213,119,203)(91,239,117,237,111,219,97,217)(93,205,95,211,113,225,115,231)(98,220,100,226,118,240,120,206) );

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),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,213,79,11,223,49,21,233,59,31,203,69),(2,222,80,20,224,58,22,202,60,40,204,78),(3,231,41,29,225,67,23,211,61,9,205,47),(4,240,42,38,226,76,24,220,62,18,206,56),(5,209,43,7,227,45,25,229,63,27,207,65),(6,218,44,16,228,54,26,238,64,36,208,74),(8,236,46,34,230,72,28,216,66,14,210,52),(10,214,48,12,232,50,30,234,68,32,212,70),(13,201,51,39,235,77,33,221,71,19,215,57),(15,219,53,17,237,55,35,239,73,37,217,75),(81,134,165,87,148,171,101,154,185,107,128,191),(82,143,166,96,149,180,102,123,186,116,129,200),(83,152,167,105,150,189,103,132,187,85,130,169),(84,121,168,114,151,198,104,141,188,94,131,178),(86,139,170,92,153,176,106,159,190,112,133,196),(88,157,172,110,155,194,108,137,192,90,135,174),(89,126,173,119,156,163,109,146,193,99,136,183),(91,144,175,97,158,181,111,124,195,117,138,161),(93,122,177,115,160,199,113,142,197,95,140,179),(98,127,182,120,125,164,118,147,162,100,145,184)], [(1,136,31,146,21,156,11,126),(2,139,40,133,22,159,20,153),(3,142,9,160,23,122,29,140),(4,145,18,147,24,125,38,127),(5,148,27,134,25,128,7,154),(6,151,36,121,26,131,16,141),(8,157,14,135,28,137,34,155),(10,123,32,149,30,143,12,129),(13,132,19,150,33,152,39,130),(15,138,37,124,35,158,17,144),(41,199,47,177,61,179,67,197),(42,162,56,164,62,182,76,184),(43,165,65,191,63,185,45,171),(44,168,74,178,64,188,54,198),(46,174,52,192,66,194,72,172),(48,180,70,166,68,200,50,186),(49,183,79,193,69,163,59,173),(51,189,57,167,71,169,77,187),(53,195,75,181,73,175,55,161),(58,170,80,196,78,190,60,176),(81,209,107,207,101,229,87,227),(82,212,116,234,102,232,96,214),(83,215,85,221,103,235,105,201),(84,218,94,208,104,238,114,228),(86,224,112,222,106,204,92,202),(88,230,90,236,108,210,110,216),(89,233,99,223,109,213,119,203),(91,239,117,237,111,219,97,217),(93,205,95,211,113,225,115,231),(98,220,100,226,118,240,120,206)]])

54 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 5 6A 6B 6C 8A 8B 8C 8D 8E 8F 8G 8H 10 12A 12B 12C 12D 15A 15B 20A 20B 24A 24B 24C 24D 24E 24F 24G 24H 30A 30B 40A 40B 40C 40D 60A 60B 60C 60D 120A ··· 120H order 1 2 2 3 4 4 4 5 6 6 6 8 8 8 8 8 8 8 8 10 12 12 12 12 15 15 20 20 24 24 24 24 24 24 24 24 30 30 40 40 40 40 60 60 60 60 120 ··· 120 size 1 1 10 2 2 5 5 4 2 10 10 2 2 10 10 60 60 60 60 4 2 2 10 10 4 4 4 4 2 2 2 2 10 10 10 10 4 4 4 4 4 4 4 4 4 4 4 ··· 4

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + - - - + + - + + image C1 C2 C2 C4 C4 S3 D4 Q8 Dic3 Dic3 D6 D12 Dic6 C8.C4 C24.C4 F5 C2×F5 C3⋊F5 C4⋊F5 C2×C3⋊F5 C40.C4 C60⋊C4 C40.Dic3 kernel C40.Dic3 D5×C24 C12.F5 C3×C5⋊2C8 C120 C8×D5 C3×Dic5 C6×D5 C5⋊2C8 C40 C4×D5 Dic5 D10 C15 C5 C24 C12 C8 C6 C4 C3 C2 C1 # reps 1 1 2 2 2 1 1 1 1 1 1 2 2 4 8 1 1 2 2 2 4 4 8

Matrix representation of C40.Dic3 in GL6(𝔽241)

 211 84 0 0 0 0 0 233 0 0 0 0 0 0 0 240 1 0 0 0 0 240 0 1 0 0 0 240 0 0 0 0 1 240 0 0
,
 237 71 0 0 0 0 0 181 0 0 0 0 0 0 126 115 12 0 0 0 0 127 12 229 0 0 12 127 0 114 0 0 12 115 126 229
,
 11 32 0 0 0 0 66 230 0 0 0 0 0 0 239 3 238 2 0 0 0 3 236 5 0 0 3 0 238 5 0 0 3 239 0 2

G:=sub<GL(6,GF(241))| [211,0,0,0,0,0,84,233,0,0,0,0,0,0,0,0,0,1,0,0,240,240,240,240,0,0,1,0,0,0,0,0,0,1,0,0],[237,0,0,0,0,0,71,181,0,0,0,0,0,0,126,0,12,12,0,0,115,127,127,115,0,0,12,12,0,126,0,0,0,229,114,229],[11,66,0,0,0,0,32,230,0,0,0,0,0,0,239,0,3,3,0,0,3,3,0,239,0,0,238,236,238,0,0,0,2,5,5,2] >;

C40.Dic3 in GAP, Magma, Sage, TeX

C_{40}.{\rm Dic}_3
% in TeX

G:=Group("C40.Dic3");
// GroupNames label

G:=SmallGroup(480,300);
// by ID

G=gap.SmallGroup(480,300);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-5,28,141,64,100,675,80,2693,14118,4724]);
// Polycyclic

G:=Group<a,b,c|a^40=1,b^6=a^20,c^2=a^20*b^3,b*a*b^-1=a^9,c*a*c^-1=a^27,c*b*c^-1=b^5>;
// generators/relations

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