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## G = S3×Q8×C9order 432 = 24·33

### Direct product of C9, S3 and Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C6 — S3×Q8×C9
 Chief series C1 — C3 — C32 — C3×C6 — C3×C18 — S3×C18 — S3×C36 — S3×Q8×C9
 Lower central C3 — C6 — S3×Q8×C9
 Upper central C1 — C18 — Q8×C9

Generators and relations for S3×Q8×C9
G = < a,b,c,d,e | a9=b3=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 216 in 126 conjugacy classes, 75 normal (24 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C2×Q8, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, S3×C9, C3×C18, C2×C36, Q8×C9, Q8×C9, C3×Dic6, S3×C12, Q8×C32, C9×Dic3, C3×C36, S3×C18, Q8×C18, C3×S3×Q8, C9×Dic6, S3×C36, Q8×C3×C9, S3×Q8×C9
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, C9, D6, C2×C6, C2×Q8, C18, C3×S3, C3×Q8, C22×S3, C22×C6, C2×C18, S3×C6, S3×Q8, C6×Q8, S3×C9, Q8×C9, C22×C18, S3×C2×C6, S3×C18, Q8×C18, C3×S3×Q8, S3×C2×C18, S3×Q8×C9

Smallest permutation representation of S3×Q8×C9
On 144 points
Generators in S144
(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 4 7)(2 5 8)(3 6 9)(10 16 13)(11 17 14)(12 18 15)(19 22 25)(20 23 26)(21 24 27)(28 31 34)(29 32 35)(30 33 36)(37 40 43)(38 41 44)(39 42 45)(46 49 52)(47 50 53)(48 51 54)(55 58 61)(56 59 62)(57 60 63)(64 67 70)(65 68 71)(66 69 72)(73 76 79)(74 77 80)(75 78 81)(82 88 85)(83 89 86)(84 90 87)(91 97 94)(92 98 95)(93 99 96)(100 106 103)(101 107 104)(102 108 105)(109 115 112)(110 116 113)(111 117 114)(118 124 121)(119 125 122)(120 126 123)(127 133 130)(128 134 131)(129 135 132)(136 142 139)(137 143 140)(138 144 141)
(1 92)(2 93)(3 94)(4 95)(5 96)(6 97)(7 98)(8 99)(9 91)(10 67)(11 68)(12 69)(13 70)(14 71)(15 72)(16 64)(17 65)(18 66)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 82)(27 83)(28 116)(29 117)(30 109)(31 110)(32 111)(33 112)(34 113)(35 114)(36 115)(37 102)(38 103)(39 104)(40 105)(41 106)(42 107)(43 108)(44 100)(45 101)(46 134)(47 135)(48 127)(49 128)(50 129)(51 130)(52 131)(53 132)(54 133)(55 120)(56 121)(57 122)(58 123)(59 124)(60 125)(61 126)(62 118)(63 119)(73 138)(74 139)(75 140)(76 141)(77 142)(78 143)(79 144)(80 136)(81 137)
(1 38 20 31)(2 39 21 32)(3 40 22 33)(4 41 23 34)(5 42 24 35)(6 43 25 36)(7 44 26 28)(8 45 27 29)(9 37 19 30)(10 121 139 128)(11 122 140 129)(12 123 141 130)(13 124 142 131)(14 125 143 132)(15 126 144 133)(16 118 136 134)(17 119 137 135)(18 120 138 127)(46 64 62 80)(47 65 63 81)(48 66 55 73)(49 67 56 74)(50 68 57 75)(51 69 58 76)(52 70 59 77)(53 71 60 78)(54 72 61 79)(82 116 98 100)(83 117 99 101)(84 109 91 102)(85 110 92 103)(86 111 93 104)(87 112 94 105)(88 113 95 106)(89 114 96 107)(90 115 97 108)
(1 56 20 49)(2 57 21 50)(3 58 22 51)(4 59 23 52)(5 60 24 53)(6 61 25 54)(7 62 26 46)(8 63 27 47)(9 55 19 48)(10 110 139 103)(11 111 140 104)(12 112 141 105)(13 113 142 106)(14 114 143 107)(15 115 144 108)(16 116 136 100)(17 117 137 101)(18 109 138 102)(28 80 44 64)(29 81 45 65)(30 73 37 66)(31 74 38 67)(32 75 39 68)(33 76 40 69)(34 77 41 70)(35 78 42 71)(36 79 43 72)(82 134 98 118)(83 135 99 119)(84 127 91 120)(85 128 92 121)(86 129 93 122)(87 130 94 123)(88 131 95 124)(89 132 96 125)(90 133 97 126)

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

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

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

135 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 4E 4F 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 12A ··· 12F 12G ··· 12O 12P ··· 12U 18A ··· 18F 18G ··· 18L 18M ··· 18X 36A ··· 36R 36S ··· 36AJ 36AK ··· 36BB order 1 2 2 2 3 3 3 3 3 4 4 4 4 4 4 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 ··· 12 12 ··· 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 3 3 1 1 2 2 2 2 2 2 6 6 6 1 1 2 2 2 3 3 3 3 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4 6 ··· 6 1 ··· 1 2 ··· 2 3 ··· 3 2 ··· 2 4 ··· 4 6 ··· 6

135 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + - + - image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 Q8 D6 C3×S3 C3×Q8 S3×C6 S3×C9 Q8×C9 S3×C18 S3×Q8 C3×S3×Q8 S3×Q8×C9 kernel S3×Q8×C9 C9×Dic6 S3×C36 Q8×C3×C9 C3×S3×Q8 C3×Dic6 S3×C12 Q8×C32 S3×Q8 Dic6 C4×S3 C3×Q8 Q8×C9 S3×C9 C36 C3×Q8 C3×S3 C12 Q8 S3 C4 C9 C3 C1 # reps 1 3 3 1 2 6 6 2 6 18 18 6 1 2 3 2 4 6 6 12 18 1 2 6

Matrix representation of S3×Q8×C9 in GL4(𝔽37) generated by

 10 0 0 0 0 10 0 0 0 0 16 0 0 0 0 16
,
 1 0 0 0 0 1 0 0 0 0 26 0 0 0 27 10
,
 1 0 0 0 0 1 0 0 0 0 1 9 0 0 0 36
,
 0 1 0 0 36 0 0 0 0 0 1 0 0 0 0 1
,
 11 27 0 0 27 26 0 0 0 0 36 0 0 0 0 36
G:=sub<GL(4,GF(37))| [10,0,0,0,0,10,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,26,27,0,0,0,10],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,9,36],[0,36,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[11,27,0,0,27,26,0,0,0,0,36,0,0,0,0,36] >;

S3×Q8×C9 in GAP, Magma, Sage, TeX

S_3\times Q_8\times C_9
% in TeX

G:=Group("S3xQ8xC9");
// GroupNames label

G:=SmallGroup(432,366);
// by ID

G=gap.SmallGroup(432,366);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,192,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^9=b^3=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽