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## G = C32×C24order 216 = 23·33

### Abelian group of type [3,3,24]

Aliases: C32×C24, SmallGroup(216,85)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32×C24
 Chief series C1 — C2 — C4 — C12 — C3×C12 — C32×C12 — C32×C24
 Lower central C1 — C32×C24
 Upper central C1 — C32×C24

Generators and relations for C32×C24
G = < a,b,c | a3=b3=c24=1, ab=ba, ac=ca, bc=cb >

Subgroups: 112, all normal (8 characteristic)
C1, C2, C3 [×13], C4, C6 [×13], C8, C32 [×13], C12 [×13], C3×C6 [×13], C24 [×13], C33, C3×C12 [×13], C32×C6, C3×C24 [×13], C32×C12, C32×C24
Quotients: C1, C2, C3 [×13], C4, C6 [×13], C8, C32 [×13], C12 [×13], C3×C6 [×13], C24 [×13], C33, C3×C12 [×13], C32×C6, C3×C24 [×13], C32×C12, C32×C24

Smallest permutation representation of C32×C24
Regular action on 216 points
Generators in S216
(1 215 33)(2 216 34)(3 193 35)(4 194 36)(5 195 37)(6 196 38)(7 197 39)(8 198 40)(9 199 41)(10 200 42)(11 201 43)(12 202 44)(13 203 45)(14 204 46)(15 205 47)(16 206 48)(17 207 25)(18 208 26)(19 209 27)(20 210 28)(21 211 29)(22 212 30)(23 213 31)(24 214 32)(49 110 93)(50 111 94)(51 112 95)(52 113 96)(53 114 73)(54 115 74)(55 116 75)(56 117 76)(57 118 77)(58 119 78)(59 120 79)(60 97 80)(61 98 81)(62 99 82)(63 100 83)(64 101 84)(65 102 85)(66 103 86)(67 104 87)(68 105 88)(69 106 89)(70 107 90)(71 108 91)(72 109 92)(121 161 182)(122 162 183)(123 163 184)(124 164 185)(125 165 186)(126 166 187)(127 167 188)(128 168 189)(129 145 190)(130 146 191)(131 147 192)(132 148 169)(133 149 170)(134 150 171)(135 151 172)(136 152 173)(137 153 174)(138 154 175)(139 155 176)(140 156 177)(141 157 178)(142 158 179)(143 159 180)(144 160 181)
(1 172 99)(2 173 100)(3 174 101)(4 175 102)(5 176 103)(6 177 104)(7 178 105)(8 179 106)(9 180 107)(10 181 108)(11 182 109)(12 183 110)(13 184 111)(14 185 112)(15 186 113)(16 187 114)(17 188 115)(18 189 116)(19 190 117)(20 191 118)(21 192 119)(22 169 120)(23 170 97)(24 171 98)(25 167 54)(26 168 55)(27 145 56)(28 146 57)(29 147 58)(30 148 59)(31 149 60)(32 150 61)(33 151 62)(34 152 63)(35 153 64)(36 154 65)(37 155 66)(38 156 67)(39 157 68)(40 158 69)(41 159 70)(42 160 71)(43 161 72)(44 162 49)(45 163 50)(46 164 51)(47 165 52)(48 166 53)(73 206 126)(74 207 127)(75 208 128)(76 209 129)(77 210 130)(78 211 131)(79 212 132)(80 213 133)(81 214 134)(82 215 135)(83 216 136)(84 193 137)(85 194 138)(86 195 139)(87 196 140)(88 197 141)(89 198 142)(90 199 143)(91 200 144)(92 201 121)(93 202 122)(94 203 123)(95 204 124)(96 205 125)
(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)| (1,215,33)(2,216,34)(3,193,35)(4,194,36)(5,195,37)(6,196,38)(7,197,39)(8,198,40)(9,199,41)(10,200,42)(11,201,43)(12,202,44)(13,203,45)(14,204,46)(15,205,47)(16,206,48)(17,207,25)(18,208,26)(19,209,27)(20,210,28)(21,211,29)(22,212,30)(23,213,31)(24,214,32)(49,110,93)(50,111,94)(51,112,95)(52,113,96)(53,114,73)(54,115,74)(55,116,75)(56,117,76)(57,118,77)(58,119,78)(59,120,79)(60,97,80)(61,98,81)(62,99,82)(63,100,83)(64,101,84)(65,102,85)(66,103,86)(67,104,87)(68,105,88)(69,106,89)(70,107,90)(71,108,91)(72,109,92)(121,161,182)(122,162,183)(123,163,184)(124,164,185)(125,165,186)(126,166,187)(127,167,188)(128,168,189)(129,145,190)(130,146,191)(131,147,192)(132,148,169)(133,149,170)(134,150,171)(135,151,172)(136,152,173)(137,153,174)(138,154,175)(139,155,176)(140,156,177)(141,157,178)(142,158,179)(143,159,180)(144,160,181), (1,172,99)(2,173,100)(3,174,101)(4,175,102)(5,176,103)(6,177,104)(7,178,105)(8,179,106)(9,180,107)(10,181,108)(11,182,109)(12,183,110)(13,184,111)(14,185,112)(15,186,113)(16,187,114)(17,188,115)(18,189,116)(19,190,117)(20,191,118)(21,192,119)(22,169,120)(23,170,97)(24,171,98)(25,167,54)(26,168,55)(27,145,56)(28,146,57)(29,147,58)(30,148,59)(31,149,60)(32,150,61)(33,151,62)(34,152,63)(35,153,64)(36,154,65)(37,155,66)(38,156,67)(39,157,68)(40,158,69)(41,159,70)(42,160,71)(43,161,72)(44,162,49)(45,163,50)(46,164,51)(47,165,52)(48,166,53)(73,206,126)(74,207,127)(75,208,128)(76,209,129)(77,210,130)(78,211,131)(79,212,132)(80,213,133)(81,214,134)(82,215,135)(83,216,136)(84,193,137)(85,194,138)(86,195,139)(87,196,140)(88,197,141)(89,198,142)(90,199,143)(91,200,144)(92,201,121)(93,202,122)(94,203,123)(95,204,124)(96,205,125), (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( (1,215,33)(2,216,34)(3,193,35)(4,194,36)(5,195,37)(6,196,38)(7,197,39)(8,198,40)(9,199,41)(10,200,42)(11,201,43)(12,202,44)(13,203,45)(14,204,46)(15,205,47)(16,206,48)(17,207,25)(18,208,26)(19,209,27)(20,210,28)(21,211,29)(22,212,30)(23,213,31)(24,214,32)(49,110,93)(50,111,94)(51,112,95)(52,113,96)(53,114,73)(54,115,74)(55,116,75)(56,117,76)(57,118,77)(58,119,78)(59,120,79)(60,97,80)(61,98,81)(62,99,82)(63,100,83)(64,101,84)(65,102,85)(66,103,86)(67,104,87)(68,105,88)(69,106,89)(70,107,90)(71,108,91)(72,109,92)(121,161,182)(122,162,183)(123,163,184)(124,164,185)(125,165,186)(126,166,187)(127,167,188)(128,168,189)(129,145,190)(130,146,191)(131,147,192)(132,148,169)(133,149,170)(134,150,171)(135,151,172)(136,152,173)(137,153,174)(138,154,175)(139,155,176)(140,156,177)(141,157,178)(142,158,179)(143,159,180)(144,160,181), (1,172,99)(2,173,100)(3,174,101)(4,175,102)(5,176,103)(6,177,104)(7,178,105)(8,179,106)(9,180,107)(10,181,108)(11,182,109)(12,183,110)(13,184,111)(14,185,112)(15,186,113)(16,187,114)(17,188,115)(18,189,116)(19,190,117)(20,191,118)(21,192,119)(22,169,120)(23,170,97)(24,171,98)(25,167,54)(26,168,55)(27,145,56)(28,146,57)(29,147,58)(30,148,59)(31,149,60)(32,150,61)(33,151,62)(34,152,63)(35,153,64)(36,154,65)(37,155,66)(38,156,67)(39,157,68)(40,158,69)(41,159,70)(42,160,71)(43,161,72)(44,162,49)(45,163,50)(46,164,51)(47,165,52)(48,166,53)(73,206,126)(74,207,127)(75,208,128)(76,209,129)(77,210,130)(78,211,131)(79,212,132)(80,213,133)(81,214,134)(82,215,135)(83,216,136)(84,193,137)(85,194,138)(86,195,139)(87,196,140)(88,197,141)(89,198,142)(90,199,143)(91,200,144)(92,201,121)(93,202,122)(94,203,123)(95,204,124)(96,205,125), (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([(1,215,33),(2,216,34),(3,193,35),(4,194,36),(5,195,37),(6,196,38),(7,197,39),(8,198,40),(9,199,41),(10,200,42),(11,201,43),(12,202,44),(13,203,45),(14,204,46),(15,205,47),(16,206,48),(17,207,25),(18,208,26),(19,209,27),(20,210,28),(21,211,29),(22,212,30),(23,213,31),(24,214,32),(49,110,93),(50,111,94),(51,112,95),(52,113,96),(53,114,73),(54,115,74),(55,116,75),(56,117,76),(57,118,77),(58,119,78),(59,120,79),(60,97,80),(61,98,81),(62,99,82),(63,100,83),(64,101,84),(65,102,85),(66,103,86),(67,104,87),(68,105,88),(69,106,89),(70,107,90),(71,108,91),(72,109,92),(121,161,182),(122,162,183),(123,163,184),(124,164,185),(125,165,186),(126,166,187),(127,167,188),(128,168,189),(129,145,190),(130,146,191),(131,147,192),(132,148,169),(133,149,170),(134,150,171),(135,151,172),(136,152,173),(137,153,174),(138,154,175),(139,155,176),(140,156,177),(141,157,178),(142,158,179),(143,159,180),(144,160,181)], [(1,172,99),(2,173,100),(3,174,101),(4,175,102),(5,176,103),(6,177,104),(7,178,105),(8,179,106),(9,180,107),(10,181,108),(11,182,109),(12,183,110),(13,184,111),(14,185,112),(15,186,113),(16,187,114),(17,188,115),(18,189,116),(19,190,117),(20,191,118),(21,192,119),(22,169,120),(23,170,97),(24,171,98),(25,167,54),(26,168,55),(27,145,56),(28,146,57),(29,147,58),(30,148,59),(31,149,60),(32,150,61),(33,151,62),(34,152,63),(35,153,64),(36,154,65),(37,155,66),(38,156,67),(39,157,68),(40,158,69),(41,159,70),(42,160,71),(43,161,72),(44,162,49),(45,163,50),(46,164,51),(47,165,52),(48,166,53),(73,206,126),(74,207,127),(75,208,128),(76,209,129),(77,210,130),(78,211,131),(79,212,132),(80,213,133),(81,214,134),(82,215,135),(83,216,136),(84,193,137),(85,194,138),(86,195,139),(87,196,140),(88,197,141),(89,198,142),(90,199,143),(91,200,144),(92,201,121),(93,202,122),(94,203,123),(95,204,124),(96,205,125)], [(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)])

C32×C24 is a maximal subgroup of   C337C16  C3315M4(2)  C3321SD16  C3312D8  C3312Q16

216 conjugacy classes

 class 1 2 3A ··· 3Z 4A 4B 6A ··· 6Z 8A 8B 8C 8D 12A ··· 12AZ 24A ··· 24CZ order 1 2 3 ··· 3 4 4 6 ··· 6 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 ··· 1 1 1 1 ··· 1 1 1 1 1 1 ··· 1 1 ··· 1

216 irreducible representations

 dim 1 1 1 1 1 1 1 1 type + + image C1 C2 C3 C4 C6 C8 C12 C24 kernel C32×C24 C32×C12 C3×C24 C32×C6 C3×C12 C33 C3×C6 C32 # reps 1 1 26 2 26 4 52 104

Matrix representation of C32×C24 in GL3(𝔽73) generated by

 1 0 0 0 8 0 0 0 64
,
 64 0 0 0 8 0 0 0 1
,
 24 0 0 0 24 0 0 0 7
G:=sub<GL(3,GF(73))| [1,0,0,0,8,0,0,0,64],[64,0,0,0,8,0,0,0,1],[24,0,0,0,24,0,0,0,7] >;

C32×C24 in GAP, Magma, Sage, TeX

C_3^2\times C_{24}
% in TeX

G:=Group("C3^2xC24");
// GroupNames label

G:=SmallGroup(216,85);
// by ID

G=gap.SmallGroup(216,85);
# by ID

G:=PCGroup([6,-2,-3,-3,-3,-2,-2,324,88]);
// Polycyclic

G:=Group<a,b,c|a^3=b^3=c^24=1,a*b=b*a,a*c=c*a,b*c=c*b>;
// generators/relations

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