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## G = S3×Q8⋊C9order 432 = 24·33

### Direct product of S3 and Q8⋊C9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — C3×Q8 — S3×Q8⋊C9
 Chief series C1 — C2 — C6 — C3×Q8 — Q8×C32 — C3×Q8⋊C9 — S3×Q8⋊C9
 Lower central C3×Q8 — S3×Q8⋊C9
 Upper central C1 — C6

Generators and relations for S3×Q8⋊C9
G = < a,b,c,d,e | a3=b2=c4=e9=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ece-1=d, ede-1=cd >

Subgroups: 228 in 65 conjugacy classes, 23 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C2×C4, Q8, Q8, C9, C32, Dic3, C12, D6, C2×C6, C2×Q8, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×C9, C2×C18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, S3×C9, C3×C18, Q8⋊C9, Q8⋊C9, C3×Dic6, S3×C12, Q8×C32, S3×C18, C2×Q8⋊C9, C3×S3×Q8, C3×Q8⋊C9, S3×Q8⋊C9
Quotients: C1, C2, C3, S3, C6, C9, A4, C18, C3×S3, SL2(𝔽3), C2×A4, C3.A4, C2×SL2(𝔽3), S3×C9, Q8⋊C9, C2×C3.A4, S3×A4, C2×Q8⋊C9, S3×SL2(𝔽3), S3×C3.A4, S3×Q8⋊C9

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

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

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

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

63 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 6H 6I 9A ··· 9F 9G ··· 9L 12A 12B 12C 12D 12E 12F 12G 18A ··· 18F 18G ··· 18L 18M ··· 18X order 1 2 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 6 6 9 ··· 9 9 ··· 9 12 12 12 12 12 12 12 18 ··· 18 18 ··· 18 18 ··· 18 size 1 1 3 3 1 1 2 2 2 6 18 1 1 2 2 2 3 3 3 3 4 ··· 4 8 ··· 8 6 6 12 12 12 18 18 4 ··· 4 8 ··· 8 12 ··· 12

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 3 3 3 3 4 4 4 6 6 type + + + - + + - + image C1 C2 C3 C6 C9 C18 S3 C3×S3 SL2(𝔽3) SL2(𝔽3) S3×C9 Q8⋊C9 A4 C2×A4 C3.A4 C2×C3.A4 S3×SL2(𝔽3) S3×SL2(𝔽3) S3×Q8⋊C9 S3×A4 S3×C3.A4 kernel S3×Q8⋊C9 C3×Q8⋊C9 C3×S3×Q8 Q8×C32 S3×Q8 C3×Q8 Q8⋊C9 C3×Q8 C3×S3 C3×S3 Q8 S3 S3×C6 C3×C6 D6 C6 C3 C3 C1 C6 C2 # reps 1 1 2 2 6 6 1 2 2 4 6 12 1 1 2 2 1 2 6 1 2

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

 26 0 0 0 0 10 0 0 0 0 1 0 0 0 0 1
,
 0 1 0 0 1 0 0 0 0 0 36 0 0 0 0 36
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 36 0
,
 1 0 0 0 0 1 0 0 0 0 5 14 0 0 14 32
,
 16 0 0 0 0 16 0 0 0 0 4 27 0 0 28 32
G:=sub<GL(4,GF(37))| [26,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,36,0,0,0,0,36],[1,0,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,5,14,0,0,14,32],[16,0,0,0,0,16,0,0,0,0,4,28,0,0,27,32] >;

S3×Q8⋊C9 in GAP, Magma, Sage, TeX

S_3\times Q_8\rtimes C_9
% in TeX

G:=Group("S3xQ8:C9");
// GroupNames label

G:=SmallGroup(432,268);
// by ID

G=gap.SmallGroup(432,268);
# by ID

G:=PCGroup([7,-2,-3,-3,-2,2,-3,-2,50,766,360,326,515,242,6053]);
// Polycyclic

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

׿
×
𝔽