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## G = C3×C24.S3order 432 = 24·33

### Direct product of C3 and C24.S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C32 — C3×C24.S3
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C3×C24 — C32×C24 — C3×C24.S3
 Lower central C32 — C3×C24.S3
 Upper central C1 — C24

Generators and relations for C3×C24.S3
G = < a,b,c,d | a3=b24=c3=1, d2=b9, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b17, dcd-1=c-1 >

Subgroups: 156 in 92 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C3, C3, C3, C4, C6, C6, C6, C8, C32, C32, C32, C12, C12, C12, C16, C3×C6, C3×C6, C3×C6, C24, C24, C24, C33, C3×C12, C3×C12, C3×C12, C3⋊C16, C48, C32×C6, C3×C24, C3×C24, C3×C24, C32×C12, C3×C3⋊C16, C24.S3, C32×C24, C3×C24.S3
Quotients: C1, C2, C3, C4, S3, C6, C8, Dic3, C12, C16, C3×S3, C3⋊S3, C3⋊C8, C24, C3×Dic3, C3⋊Dic3, C3⋊C16, C48, C3×C3⋊S3, C3×C3⋊C8, C324C8, C3×C3⋊Dic3, C3×C3⋊C16, C24.S3, C3×C324C8, C3×C24.S3

Smallest permutation representation of C3×C24.S3
On 144 points
Generators in S144
(1 131 82)(2 132 83)(3 133 84)(4 134 85)(5 135 86)(6 136 87)(7 137 88)(8 138 89)(9 139 90)(10 140 91)(11 141 92)(12 142 93)(13 143 94)(14 144 95)(15 121 96)(16 122 73)(17 123 74)(18 124 75)(19 125 76)(20 126 77)(21 127 78)(22 128 79)(23 129 80)(24 130 81)(25 64 109)(26 65 110)(27 66 111)(28 67 112)(29 68 113)(30 69 114)(31 70 115)(32 71 116)(33 72 117)(34 49 118)(35 50 119)(36 51 120)(37 52 97)(38 53 98)(39 54 99)(40 55 100)(41 56 101)(42 57 102)(43 58 103)(44 59 104)(45 60 105)(46 61 106)(47 62 107)(48 63 108)
(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)
(1 131 82)(2 132 83)(3 133 84)(4 134 85)(5 135 86)(6 136 87)(7 137 88)(8 138 89)(9 139 90)(10 140 91)(11 141 92)(12 142 93)(13 143 94)(14 144 95)(15 121 96)(16 122 73)(17 123 74)(18 124 75)(19 125 76)(20 126 77)(21 127 78)(22 128 79)(23 129 80)(24 130 81)(25 109 64)(26 110 65)(27 111 66)(28 112 67)(29 113 68)(30 114 69)(31 115 70)(32 116 71)(33 117 72)(34 118 49)(35 119 50)(36 120 51)(37 97 52)(38 98 53)(39 99 54)(40 100 55)(41 101 56)(42 102 57)(43 103 58)(44 104 59)(45 105 60)(46 106 61)(47 107 62)(48 108 63)
(1 38 10 47 19 32 4 41 13 26 22 35 7 44 16 29)(2 31 11 40 20 25 5 34 14 43 23 28 8 37 17 46)(3 48 12 33 21 42 6 27 15 36 24 45 9 30 18 39)(49 144 58 129 67 138 52 123 61 132 70 141 55 126 64 135)(50 137 59 122 68 131 53 140 62 125 71 134 56 143 65 128)(51 130 60 139 69 124 54 133 63 142 72 127 57 136 66 121)(73 113 82 98 91 107 76 116 85 101 94 110 79 119 88 104)(74 106 83 115 92 100 77 109 86 118 95 103 80 112 89 97)(75 99 84 108 93 117 78 102 87 111 96 120 81 105 90 114)

G:=sub<Sym(144)| (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,64,109)(26,65,110)(27,66,111)(28,67,112)(29,68,113)(30,69,114)(31,70,115)(32,71,116)(33,72,117)(34,49,118)(35,50,119)(36,51,120)(37,52,97)(38,53,98)(39,54,99)(40,55,100)(41,56,101)(42,57,102)(43,58,103)(44,59,104)(45,60,105)(46,61,106)(47,62,107)(48,63,108), (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), (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,109,64)(26,110,65)(27,111,66)(28,112,67)(29,113,68)(30,114,69)(31,115,70)(32,116,71)(33,117,72)(34,118,49)(35,119,50)(36,120,51)(37,97,52)(38,98,53)(39,99,54)(40,100,55)(41,101,56)(42,102,57)(43,103,58)(44,104,59)(45,105,60)(46,106,61)(47,107,62)(48,108,63), (1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29)(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46)(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39)(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135)(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128)(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121)(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104)(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97)(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114)>;

G:=Group( (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,64,109)(26,65,110)(27,66,111)(28,67,112)(29,68,113)(30,69,114)(31,70,115)(32,71,116)(33,72,117)(34,49,118)(35,50,119)(36,51,120)(37,52,97)(38,53,98)(39,54,99)(40,55,100)(41,56,101)(42,57,102)(43,58,103)(44,59,104)(45,60,105)(46,61,106)(47,62,107)(48,63,108), (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), (1,131,82)(2,132,83)(3,133,84)(4,134,85)(5,135,86)(6,136,87)(7,137,88)(8,138,89)(9,139,90)(10,140,91)(11,141,92)(12,142,93)(13,143,94)(14,144,95)(15,121,96)(16,122,73)(17,123,74)(18,124,75)(19,125,76)(20,126,77)(21,127,78)(22,128,79)(23,129,80)(24,130,81)(25,109,64)(26,110,65)(27,111,66)(28,112,67)(29,113,68)(30,114,69)(31,115,70)(32,116,71)(33,117,72)(34,118,49)(35,119,50)(36,120,51)(37,97,52)(38,98,53)(39,99,54)(40,100,55)(41,101,56)(42,102,57)(43,103,58)(44,104,59)(45,105,60)(46,106,61)(47,107,62)(48,108,63), (1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29)(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46)(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39)(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135)(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128)(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121)(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104)(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97)(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114) );

G=PermutationGroup([[(1,131,82),(2,132,83),(3,133,84),(4,134,85),(5,135,86),(6,136,87),(7,137,88),(8,138,89),(9,139,90),(10,140,91),(11,141,92),(12,142,93),(13,143,94),(14,144,95),(15,121,96),(16,122,73),(17,123,74),(18,124,75),(19,125,76),(20,126,77),(21,127,78),(22,128,79),(23,129,80),(24,130,81),(25,64,109),(26,65,110),(27,66,111),(28,67,112),(29,68,113),(30,69,114),(31,70,115),(32,71,116),(33,72,117),(34,49,118),(35,50,119),(36,51,120),(37,52,97),(38,53,98),(39,54,99),(40,55,100),(41,56,101),(42,57,102),(43,58,103),(44,59,104),(45,60,105),(46,61,106),(47,62,107),(48,63,108)], [(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)], [(1,131,82),(2,132,83),(3,133,84),(4,134,85),(5,135,86),(6,136,87),(7,137,88),(8,138,89),(9,139,90),(10,140,91),(11,141,92),(12,142,93),(13,143,94),(14,144,95),(15,121,96),(16,122,73),(17,123,74),(18,124,75),(19,125,76),(20,126,77),(21,127,78),(22,128,79),(23,129,80),(24,130,81),(25,109,64),(26,110,65),(27,111,66),(28,112,67),(29,113,68),(30,114,69),(31,115,70),(32,116,71),(33,117,72),(34,118,49),(35,119,50),(36,120,51),(37,97,52),(38,98,53),(39,99,54),(40,100,55),(41,101,56),(42,102,57),(43,103,58),(44,104,59),(45,105,60),(46,106,61),(47,107,62),(48,108,63)], [(1,38,10,47,19,32,4,41,13,26,22,35,7,44,16,29),(2,31,11,40,20,25,5,34,14,43,23,28,8,37,17,46),(3,48,12,33,21,42,6,27,15,36,24,45,9,30,18,39),(49,144,58,129,67,138,52,123,61,132,70,141,55,126,64,135),(50,137,59,122,68,131,53,140,62,125,71,134,56,143,65,128),(51,130,60,139,69,124,54,133,63,142,72,127,57,136,66,121),(73,113,82,98,91,107,76,116,85,101,94,110,79,119,88,104),(74,106,83,115,92,100,77,109,86,118,95,103,80,112,89,97),(75,99,84,108,93,117,78,102,87,111,96,120,81,105,90,114)]])

144 conjugacy classes

 class 1 2 3A 3B 3C ··· 3N 4A 4B 6A 6B 6C ··· 6N 8A 8B 8C 8D 12A 12B 12C 12D 12E ··· 12AB 16A ··· 16H 24A ··· 24H 24I ··· 24BD 48A ··· 48P order 1 2 3 3 3 ··· 3 4 4 6 6 6 ··· 6 8 8 8 8 12 12 12 12 12 ··· 12 16 ··· 16 24 ··· 24 24 ··· 24 48 ··· 48 size 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 1 1 1 1 1 1 1 1 2 ··· 2 9 ··· 9 1 ··· 1 2 ··· 2 9 ··· 9

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + - image C1 C2 C3 C4 C6 C8 C12 C16 C24 C48 S3 Dic3 C3×S3 C3⋊C8 C3×Dic3 C3⋊C16 C3×C3⋊C8 C3×C3⋊C16 kernel C3×C24.S3 C32×C24 C24.S3 C32×C12 C3×C24 C32×C6 C3×C12 C33 C3×C6 C32 C3×C24 C3×C12 C24 C3×C6 C12 C32 C6 C3 # reps 1 1 2 2 2 4 4 8 8 16 4 4 8 8 8 16 16 32

Matrix representation of C3×C24.S3 in GL4(𝔽97) generated by

 1 0 0 0 0 1 0 0 0 0 61 0 0 0 0 61
,
 66 57 0 0 55 64 0 0 0 0 73 0 0 0 30 88
,
 1 0 0 0 0 1 0 0 0 0 61 0 0 0 45 35
,
 17 96 0 0 31 80 0 0 0 0 68 34 0 0 39 29
G:=sub<GL(4,GF(97))| [1,0,0,0,0,1,0,0,0,0,61,0,0,0,0,61],[66,55,0,0,57,64,0,0,0,0,73,30,0,0,0,88],[1,0,0,0,0,1,0,0,0,0,61,45,0,0,0,35],[17,31,0,0,96,80,0,0,0,0,68,39,0,0,34,29] >;

C3×C24.S3 in GAP, Magma, Sage, TeX

C_3\times C_{24}.S_3
% in TeX

G:=Group("C3xC24.S3");
// GroupNames label

G:=SmallGroup(432,230);
// by ID

G=gap.SmallGroup(432,230);
# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,-3,-3,42,58,80,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^24=c^3=1,d^2=b^9,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^17,d*c*d^-1=c^-1>;
// generators/relations

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