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G = C22×C56order 224 = 25·7

Abelian group of type [2,2,56]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C56, SmallGroup(224,164)

Series: Derived Chief Lower central Upper central

C1 — C22×C56
C1C2C4C28C56C2×C56 — C22×C56
C1 — C22×C56
C1 — C22×C56

Generators and relations for C22×C56
 G = < a,b,c | a2=b2=c56=1, ab=ba, ac=ca, bc=cb >

Subgroups: 76, all normal (12 characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×7], C7, C8 [×4], C2×C4 [×6], C23, C14, C14 [×6], C2×C8 [×6], C22×C4, C28, C28 [×3], C2×C14 [×7], C22×C8, C56 [×4], C2×C28 [×6], C22×C14, C2×C56 [×6], C22×C28, C22×C56
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C7, C8 [×4], C2×C4 [×6], C23, C14 [×7], C2×C8 [×6], C22×C4, C28 [×4], C2×C14 [×7], C22×C8, C56 [×4], C2×C28 [×6], C22×C14, C2×C56 [×6], C22×C28, C22×C56

Smallest permutation representation of C22×C56
Regular action on 224 points
Generators in S224
(1 121)(2 122)(3 123)(4 124)(5 125)(6 126)(7 127)(8 128)(9 129)(10 130)(11 131)(12 132)(13 133)(14 134)(15 135)(16 136)(17 137)(18 138)(19 139)(20 140)(21 141)(22 142)(23 143)(24 144)(25 145)(26 146)(27 147)(28 148)(29 149)(30 150)(31 151)(32 152)(33 153)(34 154)(35 155)(36 156)(37 157)(38 158)(39 159)(40 160)(41 161)(42 162)(43 163)(44 164)(45 165)(46 166)(47 167)(48 168)(49 113)(50 114)(51 115)(52 116)(53 117)(54 118)(55 119)(56 120)(57 191)(58 192)(59 193)(60 194)(61 195)(62 196)(63 197)(64 198)(65 199)(66 200)(67 201)(68 202)(69 203)(70 204)(71 205)(72 206)(73 207)(74 208)(75 209)(76 210)(77 211)(78 212)(79 213)(80 214)(81 215)(82 216)(83 217)(84 218)(85 219)(86 220)(87 221)(88 222)(89 223)(90 224)(91 169)(92 170)(93 171)(94 172)(95 173)(96 174)(97 175)(98 176)(99 177)(100 178)(101 179)(102 180)(103 181)(104 182)(105 183)(106 184)(107 185)(108 186)(109 187)(110 188)(111 189)(112 190)
(1 101)(2 102)(3 103)(4 104)(5 105)(6 106)(7 107)(8 108)(9 109)(10 110)(11 111)(12 112)(13 57)(14 58)(15 59)(16 60)(17 61)(18 62)(19 63)(20 64)(21 65)(22 66)(23 67)(24 68)(25 69)(26 70)(27 71)(28 72)(29 73)(30 74)(31 75)(32 76)(33 77)(34 78)(35 79)(36 80)(37 81)(38 82)(39 83)(40 84)(41 85)(42 86)(43 87)(44 88)(45 89)(46 90)(47 91)(48 92)(49 93)(50 94)(51 95)(52 96)(53 97)(54 98)(55 99)(56 100)(113 171)(114 172)(115 173)(116 174)(117 175)(118 176)(119 177)(120 178)(121 179)(122 180)(123 181)(124 182)(125 183)(126 184)(127 185)(128 186)(129 187)(130 188)(131 189)(132 190)(133 191)(134 192)(135 193)(136 194)(137 195)(138 196)(139 197)(140 198)(141 199)(142 200)(143 201)(144 202)(145 203)(146 204)(147 205)(148 206)(149 207)(150 208)(151 209)(152 210)(153 211)(154 212)(155 213)(156 214)(157 215)(158 216)(159 217)(160 218)(161 219)(162 220)(163 221)(164 222)(165 223)(166 224)(167 169)(168 170)
(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)

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

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

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

C22×C56 is a maximal subgroup of
C56.91D4  (C2×C56)⋊5C4  C28.9C42  C28.10C42  C28.12C42  Dic7⋊C8⋊C2  C23.22D28  (C22×C8)⋊D7  C5632D4  C23.23D28  C5630D4  C5629D4  C56.82D4

224 conjugacy classes

class 1 2A···2G4A···4H7A···7F8A···8P14A···14AP28A···28AV56A···56CR
order12···24···47···78···814···1428···2856···56
size11···11···11···11···11···11···11···1

224 irreducible representations

dim111111111111
type+++
imageC1C2C2C4C4C7C8C14C14C28C28C56
kernelC22×C56C2×C56C22×C28C2×C28C22×C14C22×C8C2×C14C2×C8C22×C4C2×C4C23C22
# reps16162616366361296

Matrix representation of C22×C56 in GL3(𝔽113) generated by

11200
010
00112
,
100
010
00112
,
10000
0880
0025
G:=sub<GL(3,GF(113))| [112,0,0,0,1,0,0,0,112],[1,0,0,0,1,0,0,0,112],[100,0,0,0,88,0,0,0,25] >;

C22×C56 in GAP, Magma, Sage, TeX

C_2^2\times C_{56}
% in TeX

G:=Group("C2^2xC56");
// GroupNames label

G:=SmallGroup(224,164);
// by ID

G=gap.SmallGroup(224,164);
# by ID

G:=PCGroup([6,-2,-2,-2,-7,-2,-2,336,88]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^56=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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