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G = C22×C48order 192 = 26·3

Abelian group of type [2,2,48]

direct product, abelian, monomial, 2-elementary

Aliases: C22×C48, SmallGroup(192,935)

Series: Derived Chief Lower central Upper central

C1 — C22×C48
C1C2C4C8C24C48C2×C48 — C22×C48
C1 — C22×C48
C1 — C22×C48

Generators and relations for C22×C48
 G = < a,b,c | a2=b2=c48=1, ab=ba, ac=ca, bc=cb >

Subgroups: 98, all normal (16 characteristic)
C1, C2, C2 [×6], C3, C4, C4 [×3], C22 [×7], C6, C6 [×6], C8, C8 [×3], C2×C4 [×6], C23, C12, C12 [×3], C2×C6 [×7], C16 [×4], C2×C8 [×6], C22×C4, C24, C24 [×3], C2×C12 [×6], C22×C6, C2×C16 [×6], C22×C8, C48 [×4], C2×C24 [×6], C22×C12, C22×C16, C2×C48 [×6], C22×C24, C22×C48
Quotients: C1, C2 [×7], C3, C4 [×4], C22 [×7], C6 [×7], C8 [×4], C2×C4 [×6], C23, C12 [×4], C2×C6 [×7], C16 [×4], C2×C8 [×6], C22×C4, C24 [×4], C2×C12 [×6], C22×C6, C2×C16 [×6], C22×C8, C48 [×4], C2×C24 [×6], C22×C12, C22×C16, C2×C48 [×6], C22×C24, C22×C48

Smallest permutation representation of C22×C48
Regular action on 192 points
Generators in S192
(1 71)(2 72)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 95)(26 96)(27 49)(28 50)(29 51)(30 52)(31 53)(32 54)(33 55)(34 56)(35 57)(36 58)(37 59)(38 60)(39 61)(40 62)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(97 151)(98 152)(99 153)(100 154)(101 155)(102 156)(103 157)(104 158)(105 159)(106 160)(107 161)(108 162)(109 163)(110 164)(111 165)(112 166)(113 167)(114 168)(115 169)(116 170)(117 171)(118 172)(119 173)(120 174)(121 175)(122 176)(123 177)(124 178)(125 179)(126 180)(127 181)(128 182)(129 183)(130 184)(131 185)(132 186)(133 187)(134 188)(135 189)(136 190)(137 191)(138 192)(139 145)(140 146)(141 147)(142 148)(143 149)(144 150)
(1 180)(2 181)(3 182)(4 183)(5 184)(6 185)(7 186)(8 187)(9 188)(10 189)(11 190)(12 191)(13 192)(14 145)(15 146)(16 147)(17 148)(18 149)(19 150)(20 151)(21 152)(22 153)(23 154)(24 155)(25 156)(26 157)(27 158)(28 159)(29 160)(30 161)(31 162)(32 163)(33 164)(34 165)(35 166)(36 167)(37 168)(38 169)(39 170)(40 171)(41 172)(42 173)(43 174)(44 175)(45 176)(46 177)(47 178)(48 179)(49 104)(50 105)(51 106)(52 107)(53 108)(54 109)(55 110)(56 111)(57 112)(58 113)(59 114)(60 115)(61 116)(62 117)(63 118)(64 119)(65 120)(66 121)(67 122)(68 123)(69 124)(70 125)(71 126)(72 127)(73 128)(74 129)(75 130)(76 131)(77 132)(78 133)(79 134)(80 135)(81 136)(82 137)(83 138)(84 139)(85 140)(86 141)(87 142)(88 143)(89 144)(90 97)(91 98)(92 99)(93 100)(94 101)(95 102)(96 103)
(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)

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

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

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

192 conjugacy classes

class 1 2A···2G3A3B4A···4H6A···6N8A···8P12A···12P16A···16AF24A···24AF48A···48BL
order12···2334···46···68···812···1216···1624···2448···48
size11···1111···11···11···11···11···11···11···1

192 irreducible representations

dim1111111111111111
type+++
imageC1C2C2C3C4C4C6C6C8C8C12C12C16C24C24C48
kernelC22×C48C2×C48C22×C24C22×C16C2×C24C22×C12C2×C16C22×C8C2×C12C22×C6C2×C8C22×C4C2×C6C2×C4C23C22
# reps1612621221241243224864

Matrix representation of C22×C48 in GL3(𝔽97) generated by

100
0960
0096
,
9600
010
001
,
3600
0360
0011
G:=sub<GL(3,GF(97))| [1,0,0,0,96,0,0,0,96],[96,0,0,0,1,0,0,0,1],[36,0,0,0,36,0,0,0,11] >;

C22×C48 in GAP, Magma, Sage, TeX

C_2^2\times C_{48}
% in TeX

G:=Group("C2^2xC48");
// GroupNames label

G:=SmallGroup(192,935);
// by ID

G=gap.SmallGroup(192,935);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,168,102,124]);
// Polycyclic

G:=Group<a,b,c|a^2=b^2=c^48=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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