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G = C721S3order 432 = 24·33

1st semidirect product of C72 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C721S3, C241D9, C91D24, C31D72, C6.9D36, C18.9D12, C36.58D6, C12.58D18, C32.4D24, (C3×C9)⋊6D8, C81(C9⋊S3), (C3×C72)⋊3C2, C36⋊S32C2, C3.(C325D8), C24.2(C3⋊S3), (C3×C24).12S3, (C3×C18).31D4, (C3×C6).56D12, (C3×C12).193D6, C6.3(C12⋊S3), C2.5(C36⋊S3), (C3×C36).60C22, C4.10(C2×C9⋊S3), C12.61(C2×C3⋊S3), SmallGroup(432,172)

Series: Derived Chief Lower central Upper central

C1C3×C36 — C721S3
C1C3C32C3×C9C3×C18C3×C36C36⋊S3 — C721S3
C3×C9C3×C18C3×C36 — C721S3
C1C2C4C8

Generators and relations for C721S3
 G = < a,b,c | a72=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1180 in 110 conjugacy classes, 43 normal (17 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×8], C6, C6 [×3], C8, D4 [×2], C9 [×3], C32, C12, C12 [×3], D6 [×8], D8, D9 [×6], C18 [×3], C3⋊S3 [×2], C3×C6, C24, C24 [×3], D12 [×8], C3×C9, C36 [×3], D18 [×6], C3×C12, C2×C3⋊S3 [×2], D24 [×4], C9⋊S3 [×2], C3×C18, C72 [×3], D36 [×6], C3×C24, C12⋊S3 [×2], C3×C36, C2×C9⋊S3 [×2], D72 [×3], C325D8, C3×C72, C36⋊S3 [×2], C721S3
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D8, D9 [×3], C3⋊S3, D12 [×4], D18 [×3], C2×C3⋊S3, D24 [×4], C9⋊S3, D36 [×3], C12⋊S3, C2×C9⋊S3, D72 [×3], C325D8, C36⋊S3, C721S3

Smallest permutation representation of C721S3
On 216 points
Generators in S216
(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)
(1 191 138)(2 192 139)(3 193 140)(4 194 141)(5 195 142)(6 196 143)(7 197 144)(8 198 73)(9 199 74)(10 200 75)(11 201 76)(12 202 77)(13 203 78)(14 204 79)(15 205 80)(16 206 81)(17 207 82)(18 208 83)(19 209 84)(20 210 85)(21 211 86)(22 212 87)(23 213 88)(24 214 89)(25 215 90)(26 216 91)(27 145 92)(28 146 93)(29 147 94)(30 148 95)(31 149 96)(32 150 97)(33 151 98)(34 152 99)(35 153 100)(36 154 101)(37 155 102)(38 156 103)(39 157 104)(40 158 105)(41 159 106)(42 160 107)(43 161 108)(44 162 109)(45 163 110)(46 164 111)(47 165 112)(48 166 113)(49 167 114)(50 168 115)(51 169 116)(52 170 117)(53 171 118)(54 172 119)(55 173 120)(56 174 121)(57 175 122)(58 176 123)(59 177 124)(60 178 125)(61 179 126)(62 180 127)(63 181 128)(64 182 129)(65 183 130)(66 184 131)(67 185 132)(68 186 133)(69 187 134)(70 188 135)(71 189 136)(72 190 137)
(1 64)(2 63)(3 62)(4 61)(5 60)(6 59)(7 58)(8 57)(9 56)(10 55)(11 54)(12 53)(13 52)(14 51)(15 50)(16 49)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)(25 40)(26 39)(27 38)(28 37)(29 36)(30 35)(31 34)(32 33)(65 72)(66 71)(67 70)(68 69)(73 175)(74 174)(75 173)(76 172)(77 171)(78 170)(79 169)(80 168)(81 167)(82 166)(83 165)(84 164)(85 163)(86 162)(87 161)(88 160)(89 159)(90 158)(91 157)(92 156)(93 155)(94 154)(95 153)(96 152)(97 151)(98 150)(99 149)(100 148)(101 147)(102 146)(103 145)(104 216)(105 215)(106 214)(107 213)(108 212)(109 211)(110 210)(111 209)(112 208)(113 207)(114 206)(115 205)(116 204)(117 203)(118 202)(119 201)(120 200)(121 199)(122 198)(123 197)(124 196)(125 195)(126 194)(127 193)(128 192)(129 191)(130 190)(131 189)(132 188)(133 187)(134 186)(135 185)(136 184)(137 183)(138 182)(139 181)(140 180)(141 179)(142 178)(143 177)(144 176)

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

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

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

111 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A6B6C6D8A8B9A···9I12A···12H18A···18I24A···24P36A···36R72A···72AJ
order1222333346666889···912···1218···1824···2436···3672···72
size11108108222222222222···22···22···22···22···22···2

111 irreducible representations

dim11122222222222222
type+++++++++++++++++
imageC1C2C2S3S3D4D6D6D8D9D12D12D18D24D24D36D72
kernelC721S3C3×C72C36⋊S3C72C3×C24C3×C18C36C3×C12C3×C9C24C18C3×C6C12C9C32C6C3
# reps11231131296291241836

Matrix representation of C721S3 in GL4(𝔽73) generated by

26000
136200
00072
0011
,
07200
17200
007272
0010
,
51800
236800
00720
0011
G:=sub<GL(4,GF(73))| [2,13,0,0,60,62,0,0,0,0,0,1,0,0,72,1],[0,1,0,0,72,72,0,0,0,0,72,1,0,0,72,0],[5,23,0,0,18,68,0,0,0,0,72,1,0,0,0,1] >;

C721S3 in GAP, Magma, Sage, TeX

C_{72}\rtimes_1S_3
% in TeX

G:=Group("C72:1S3");
// GroupNames label

G:=SmallGroup(432,172);
// by ID

G=gap.SmallGroup(432,172);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,92,254,58,6164,662,4037,14118]);
// Polycyclic

G:=Group<a,b,c|a^72=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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