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G = C557D4order 440 = 23·5·11

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C557D4, C22⋊D55, D1102C2, C2.5D110, Dic551C2, C22.12D10, C10.12D22, C110.12C22, (C2×C22)⋊2D5, (C2×C110)⋊2C2, (C2×C10)⋊2D11, C113(C5⋊D4), C53(C11⋊D4), SmallGroup(440,38)

Series: Derived Chief Lower central Upper central

C1C110 — C557D4
C1C11C55C110D110 — C557D4
C55C110 — C557D4
C1C2C22

Generators and relations for C557D4
 G = < a,b,c | a55=b4=c2=1, bab-1=cac=a-1, cbc=b-1 >

2C2
110C2
55C22
55C4
2C10
22D5
2C22
10D11
55D4
11D10
11Dic5
5D22
5Dic11
2C110
2D55
11C5⋊D4
5C11⋊D4

Smallest permutation representation of C557D4
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)
(1 137 71 167)(2 136 72 166)(3 135 73 220)(4 134 74 219)(5 133 75 218)(6 132 76 217)(7 131 77 216)(8 130 78 215)(9 129 79 214)(10 128 80 213)(11 127 81 212)(12 126 82 211)(13 125 83 210)(14 124 84 209)(15 123 85 208)(16 122 86 207)(17 121 87 206)(18 120 88 205)(19 119 89 204)(20 118 90 203)(21 117 91 202)(22 116 92 201)(23 115 93 200)(24 114 94 199)(25 113 95 198)(26 112 96 197)(27 111 97 196)(28 165 98 195)(29 164 99 194)(30 163 100 193)(31 162 101 192)(32 161 102 191)(33 160 103 190)(34 159 104 189)(35 158 105 188)(36 157 106 187)(37 156 107 186)(38 155 108 185)(39 154 109 184)(40 153 110 183)(41 152 56 182)(42 151 57 181)(43 150 58 180)(44 149 59 179)(45 148 60 178)(46 147 61 177)(47 146 62 176)(48 145 63 175)(49 144 64 174)(50 143 65 173)(51 142 66 172)(52 141 67 171)(53 140 68 170)(54 139 69 169)(55 138 70 168)
(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 48)(10 47)(11 46)(12 45)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 36)(22 35)(23 34)(24 33)(25 32)(26 31)(27 30)(28 29)(56 86)(57 85)(58 84)(59 83)(60 82)(61 81)(62 80)(63 79)(64 78)(65 77)(66 76)(67 75)(68 74)(69 73)(70 72)(87 110)(88 109)(89 108)(90 107)(91 106)(92 105)(93 104)(94 103)(95 102)(96 101)(97 100)(98 99)(111 193)(112 192)(113 191)(114 190)(115 189)(116 188)(117 187)(118 186)(119 185)(120 184)(121 183)(122 182)(123 181)(124 180)(125 179)(126 178)(127 177)(128 176)(129 175)(130 174)(131 173)(132 172)(133 171)(134 170)(135 169)(136 168)(137 167)(138 166)(139 220)(140 219)(141 218)(142 217)(143 216)(144 215)(145 214)(146 213)(147 212)(148 211)(149 210)(150 209)(151 208)(152 207)(153 206)(154 205)(155 204)(156 203)(157 202)(158 201)(159 200)(160 199)(161 198)(162 197)(163 196)(164 195)(165 194)

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

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

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

113 conjugacy classes

class 1 2A2B2C 4 5A5B10A···10F11A···11E22A···22O55A···55T110A···110BH
order122245510···1011···1122···2255···55110···110
size112110110222···22···22···22···22···2

113 irreducible representations

dim11112222222222
type+++++++++++
imageC1C2C2C2D4D5D10D11C5⋊D4D22C11⋊D4D55D110C557D4
kernelC557D4Dic55D110C2×C110C55C2×C22C22C2×C10C11C10C5C22C2C1
# reps111112254510202040

Matrix representation of C557D4 in GL2(𝔽661) generated by

69482
357530
,
614355
30147
,
60519
49656
G:=sub<GL(2,GF(661))| [69,357,482,530],[614,301,355,47],[605,496,19,56] >;

C557D4 in GAP, Magma, Sage, TeX

C_{55}\rtimes_7D_4
% in TeX

G:=Group("C55:7D4");
// GroupNames label

G:=SmallGroup(440,38);
// by ID

G=gap.SmallGroup(440,38);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-11,61,643,10004]);
// Polycyclic

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

Export

Subgroup lattice of C557D4 in TeX

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