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## G = C22×C56order 224 = 25·7

### Abelian group of type [2,2,56]

Aliases: C22×C56, SmallGroup(224,164)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22×C56
 Chief series C1 — C2 — C4 — C28 — C56 — C2×C56 — C22×C56
 Lower central C1 — C22×C56
 Upper central 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, C4, C4, C22, C7, C8, C2×C4, C23, C14, C14, C2×C8, C22×C4, C28, C28, C2×C14, C22×C8, C56, C2×C28, C22×C14, C2×C56, C22×C28, C22×C56
Quotients: C1, C2, C4, C22, C7, C8, C2×C4, C23, C14, C2×C8, C22×C4, C28, C2×C14, C22×C8, C56, C2×C28, C22×C14, C2×C56, C22×C28, C22×C56

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

224 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 type + + + image C1 C2 C2 C4 C4 C7 C8 C14 C14 C28 C28 C56 kernel C22×C56 C2×C56 C22×C28 C2×C28 C22×C14 C22×C8 C2×C14 C2×C8 C22×C4 C2×C4 C23 C22 # reps 1 6 1 6 2 6 16 36 6 36 12 96

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

 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
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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