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G = C337C8order 216 = 23·33

3rd semidirect product of C33 and C8 acting via C8/C4=C2

metabelian, supersoluble, monomial, A-group

Aliases: C337C8, C326(C3⋊C8), C3⋊(C324C8), C12.6(C3⋊S3), (C3×C12).20S3, C2.(C335C4), (C32×C6).5C4, (C32×C12).6C2, C6.3(C3⋊Dic3), (C3×C6).11Dic3, C4.2(C33⋊C2), SmallGroup(216,84)

Series: Derived Chief Lower central Upper central

C1C33 — C337C8
C1C3C32C33C32×C6C32×C12 — C337C8
C33 — C337C8
C1C4

Generators and relations for C337C8
 G = < a,b,c,d | a3=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, bc=cb, dbd-1=b-1, dcd-1=c-1 >

Subgroups: 268 in 112 conjugacy classes, 85 normal (7 characteristic)
C1, C2, C3 [×13], C4, C6 [×13], C8, C32 [×13], C12 [×13], C3×C6 [×13], C3⋊C8 [×13], C33, C3×C12 [×13], C32×C6, C324C8 [×13], C32×C12, C337C8
Quotients: C1, C2, C4, S3 [×13], C8, Dic3 [×13], C3⋊S3 [×13], C3⋊C8 [×13], C3⋊Dic3 [×13], C33⋊C2, C324C8 [×13], C335C4, C337C8

Smallest permutation representation of C337C8
Regular action on 216 points
Generators in S216
(1 54 148)(2 149 55)(3 56 150)(4 151 49)(5 50 152)(6 145 51)(7 52 146)(8 147 53)(9 159 85)(10 86 160)(11 153 87)(12 88 154)(13 155 81)(14 82 156)(15 157 83)(16 84 158)(17 123 105)(18 106 124)(19 125 107)(20 108 126)(21 127 109)(22 110 128)(23 121 111)(24 112 122)(25 36 184)(26 177 37)(27 38 178)(28 179 39)(29 40 180)(30 181 33)(31 34 182)(32 183 35)(41 66 136)(42 129 67)(43 68 130)(44 131 69)(45 70 132)(46 133 71)(47 72 134)(48 135 65)(57 202 91)(58 92 203)(59 204 93)(60 94 205)(61 206 95)(62 96 207)(63 208 89)(64 90 201)(73 168 176)(74 169 161)(75 162 170)(76 171 163)(77 164 172)(78 173 165)(79 166 174)(80 175 167)(97 189 113)(98 114 190)(99 191 115)(100 116 192)(101 185 117)(102 118 186)(103 187 119)(104 120 188)(137 196 211)(138 212 197)(139 198 213)(140 214 199)(141 200 215)(142 216 193)(143 194 209)(144 210 195)
(1 198 182)(2 183 199)(3 200 184)(4 177 193)(5 194 178)(6 179 195)(7 196 180)(8 181 197)(9 64 65)(10 66 57)(11 58 67)(12 68 59)(13 60 69)(14 70 61)(15 62 71)(16 72 63)(17 76 118)(18 119 77)(19 78 120)(20 113 79)(21 80 114)(22 115 73)(23 74 116)(24 117 75)(25 56 215)(26 216 49)(27 50 209)(28 210 51)(29 52 211)(30 212 53)(31 54 213)(32 214 55)(33 138 147)(34 148 139)(35 140 149)(36 150 141)(37 142 151)(38 152 143)(39 144 145)(40 146 137)(41 91 160)(42 153 92)(43 93 154)(44 155 94)(45 95 156)(46 157 96)(47 89 158)(48 159 90)(81 205 131)(82 132 206)(83 207 133)(84 134 208)(85 201 135)(86 136 202)(87 203 129)(88 130 204)(97 166 108)(98 109 167)(99 168 110)(100 111 161)(101 162 112)(102 105 163)(103 164 106)(104 107 165)(121 169 192)(122 185 170)(123 171 186)(124 187 172)(125 173 188)(126 189 174)(127 175 190)(128 191 176)
(1 108 13)(2 14 109)(3 110 15)(4 16 111)(5 112 9)(6 10 105)(7 106 11)(8 12 107)(17 145 86)(18 87 146)(19 147 88)(20 81 148)(21 149 82)(22 83 150)(23 151 84)(24 85 152)(25 176 46)(26 47 169)(27 170 48)(28 41 171)(29 172 42)(30 43 173)(31 174 44)(32 45 175)(33 130 78)(34 79 131)(35 132 80)(36 73 133)(37 134 74)(38 75 135)(39 136 76)(40 77 129)(49 158 121)(50 122 159)(51 160 123)(52 124 153)(53 154 125)(54 126 155)(55 156 127)(56 128 157)(57 102 195)(58 196 103)(59 104 197)(60 198 97)(61 98 199)(62 200 99)(63 100 193)(64 194 101)(65 178 162)(66 163 179)(67 180 164)(68 165 181)(69 182 166)(70 167 183)(71 184 168)(72 161 177)(89 192 216)(90 209 185)(91 186 210)(92 211 187)(93 188 212)(94 213 189)(95 190 214)(96 215 191)(113 205 139)(114 140 206)(115 207 141)(116 142 208)(117 201 143)(118 144 202)(119 203 137)(120 138 204)
(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)

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

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

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

C337C8 is a maximal subgroup of
S3×C324C8  C3⋊S3×C3⋊C8  C337M4(2)  C338M4(2)  C336D8  C3312SD16  C3313SD16  C336Q16  C8×C33⋊C2  C3315M4(2)  C3318M4(2)  C3315D8  C3324SD16  C3327SD16  C3315Q16
C337C8 is a maximal quotient of
C337C16

60 conjugacy classes

class 1  2 3A···3M4A4B6A···6M8A8B8C8D12A···12Z
order123···3446···6888812···12
size112···2112···2272727272···2

60 irreducible representations

dim1111222
type+++-
imageC1C2C4C8S3Dic3C3⋊C8
kernelC337C8C32×C12C32×C6C33C3×C12C3×C6C32
# reps1124131326

Matrix representation of C337C8 in GL6(𝔽73)

010000
72720000
00727200
001000
0000721
0000720
,
010000
72720000
000100
00727200
0000072
0000172
,
100000
010000
001000
000100
0000721
0000720
,
59540000
68140000
0055900
00546800
00007070
0000673

G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[59,68,0,0,0,0,54,14,0,0,0,0,0,0,5,54,0,0,0,0,59,68,0,0,0,0,0,0,70,67,0,0,0,0,70,3] >;

C337C8 in GAP, Magma, Sage, TeX

C_3^3\rtimes_7C_8
% in TeX

G:=Group("C3^3:7C8");
// GroupNames label

G:=SmallGroup(216,84);
// by ID

G=gap.SmallGroup(216,84);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,12,31,387,1444,5189]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽