Copied to
clipboard

G = C55⋊D4order 440 = 23·5·11

1st semidirect product of C55 and D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C551D4, D221D5, D101D11, C10.4D22, C22.4D10, Dic554C2, C110.4C22, (D5×C22)⋊1C2, C52(C11⋊D4), C112(C5⋊D4), C2.4(D5×D11), (C10×D11)⋊1C2, SmallGroup(440,20)

Series: Derived Chief Lower central Upper central

C1C110 — C55⋊D4
C1C11C55C110C10×D11 — C55⋊D4
C55C110 — C55⋊D4
C1C2

Generators and relations for C55⋊D4
 G = < a,b,c | a55=b4=c2=1, bab-1=a-1, cac=a21, cbc=b-1 >

10C2
22C2
5C22
11C22
55C4
2D5
22C10
2D11
10C22
55D4
11C2×C10
11Dic5
5Dic11
5C2×C22
2D5×C11
2C5×D11
11C5⋊D4
5C11⋊D4

Smallest permutation representation of C55⋊D4
On 220 points
Generators in S220
(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)
(1 117 57 208)(2 116 58 207)(3 115 59 206)(4 114 60 205)(5 113 61 204)(6 112 62 203)(7 111 63 202)(8 165 64 201)(9 164 65 200)(10 163 66 199)(11 162 67 198)(12 161 68 197)(13 160 69 196)(14 159 70 195)(15 158 71 194)(16 157 72 193)(17 156 73 192)(18 155 74 191)(19 154 75 190)(20 153 76 189)(21 152 77 188)(22 151 78 187)(23 150 79 186)(24 149 80 185)(25 148 81 184)(26 147 82 183)(27 146 83 182)(28 145 84 181)(29 144 85 180)(30 143 86 179)(31 142 87 178)(32 141 88 177)(33 140 89 176)(34 139 90 175)(35 138 91 174)(36 137 92 173)(37 136 93 172)(38 135 94 171)(39 134 95 170)(40 133 96 169)(41 132 97 168)(42 131 98 167)(43 130 99 166)(44 129 100 220)(45 128 101 219)(46 127 102 218)(47 126 103 217)(48 125 104 216)(49 124 105 215)(50 123 106 214)(51 122 107 213)(52 121 108 212)(53 120 109 211)(54 119 110 210)(55 118 56 209)
(2 22)(3 43)(4 9)(5 30)(6 51)(7 17)(8 38)(10 25)(11 46)(13 33)(14 54)(15 20)(16 41)(18 28)(19 49)(21 36)(24 44)(26 31)(27 52)(29 39)(32 47)(35 55)(37 42)(40 50)(48 53)(56 91)(58 78)(59 99)(60 65)(61 86)(62 107)(63 73)(64 94)(66 81)(67 102)(69 89)(70 110)(71 76)(72 97)(74 84)(75 105)(77 92)(80 100)(82 87)(83 108)(85 95)(88 103)(93 98)(96 106)(104 109)(111 192)(112 213)(113 179)(114 200)(115 166)(116 187)(117 208)(118 174)(119 195)(120 216)(121 182)(122 203)(123 169)(124 190)(125 211)(126 177)(127 198)(128 219)(129 185)(130 206)(131 172)(132 193)(133 214)(134 180)(135 201)(136 167)(137 188)(138 209)(139 175)(140 196)(141 217)(142 183)(143 204)(144 170)(145 191)(146 212)(147 178)(148 199)(149 220)(150 186)(151 207)(152 173)(153 194)(154 215)(155 181)(156 202)(157 168)(158 189)(159 210)(160 176)(161 197)(162 218)(163 184)(164 205)(165 171)

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

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

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

53 conjugacy classes

class 1 2A2B2C 4 5A5B10A10B10C10D10E10F11A···11E22A···22E22F···22O55A···55J110A···110J
order122245510101010101011···1122···2222···2255···55110···110
size1110221102222222222222···22···210···104···44···4

53 irreducible representations

dim1111222222244
type++++++++++-
imageC1C2C2C2D4D5D10D11C5⋊D4D22C11⋊D4D5×D11C55⋊D4
kernelC55⋊D4Dic55C10×D11D5×C22C55D22C22D10C11C10C5C2C1
# reps1111122545101010

Matrix representation of C55⋊D4 in GL4(𝔽661) generated by

66044400
21715700
00356177
00325247
,
49727200
16616400
0040131
00570260
,
121700
066000
0010
0001
G:=sub<GL(4,GF(661))| [660,217,0,0,444,157,0,0,0,0,356,325,0,0,177,247],[497,166,0,0,272,164,0,0,0,0,401,570,0,0,31,260],[1,0,0,0,217,660,0,0,0,0,1,0,0,0,0,1] >;

C55⋊D4 in GAP, Magma, Sage, TeX

C_{55}\rtimes D_4
% in TeX

G:=Group("C55:D4");
// GroupNames label

G:=SmallGroup(440,20);
// by ID

G=gap.SmallGroup(440,20);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,61,328,10004]);
// Polycyclic

G:=Group<a,b,c|a^55=b^4=c^2=1,b*a*b^-1=a^-1,c*a*c=a^21,c*b*c=b^-1>;
// generators/relations

Export

Subgroup lattice of C55⋊D4 in TeX

׿
×
𝔽