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## G = C3×Q8×D9order 432 = 24·33

### Direct product of C3, Q8 and D9

Series: Derived Chief Lower central Upper central

 Derived series C1 — C18 — C3×Q8×D9
 Chief series C1 — C3 — C9 — C18 — C3×C18 — C6×D9 — C12×D9 — C3×Q8×D9
 Lower central C9 — C18 — C3×Q8×D9
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×Q8×D9
G = < a,b,c,d,e | a3=b4=d9=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 398 in 126 conjugacy classes, 62 normal (22 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, D9, C18, C18, C3×S3, C3×C6, Dic6, C4×S3, C2×C12, C3×Q8, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, S3×Q8, C6×Q8, C3×D9, C3×C18, Dic18, C4×D9, Q8×C9, Q8×C9, C3×Dic6, S3×C12, Q8×C32, C3×Dic9, C3×C36, C6×D9, Q8×D9, C3×S3×Q8, C3×Dic18, C12×D9, Q8×C3×C9, C3×Q8×D9
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, D9, C3×S3, C3×Q8, C22×S3, C22×C6, D18, S3×C6, S3×Q8, C6×Q8, C3×D9, C22×D9, S3×C2×C6, C6×D9, Q8×D9, C3×S3×Q8, C2×C6×D9, C3×Q8×D9

Smallest permutation representation of C3×Q8×D9
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 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 79 76)(74 80 77)(75 81 78)(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 32 14 23)(2 33 15 24)(3 34 16 25)(4 35 17 26)(5 36 18 27)(6 28 10 19)(7 29 11 20)(8 30 12 21)(9 31 13 22)(37 55 46 64)(38 56 47 65)(39 57 48 66)(40 58 49 67)(41 59 50 68)(42 60 51 69)(43 61 52 70)(44 62 53 71)(45 63 54 72)(73 100 82 91)(74 101 83 92)(75 102 84 93)(76 103 85 94)(77 104 86 95)(78 105 87 96)(79 106 88 97)(80 107 89 98)(81 108 90 99)(109 127 118 136)(110 128 119 137)(111 129 120 138)(112 130 121 139)(113 131 122 140)(114 132 123 141)(115 133 124 142)(116 134 125 143)(117 135 126 144)
(1 50 14 41)(2 51 15 42)(3 52 16 43)(4 53 17 44)(5 54 18 45)(6 46 10 37)(7 47 11 38)(8 48 12 39)(9 49 13 40)(19 64 28 55)(20 65 29 56)(21 66 30 57)(22 67 31 58)(23 68 32 59)(24 69 33 60)(25 70 34 61)(26 71 35 62)(27 72 36 63)(73 118 82 109)(74 119 83 110)(75 120 84 111)(76 121 85 112)(77 122 86 113)(78 123 87 114)(79 124 88 115)(80 125 89 116)(81 126 90 117)(91 136 100 127)(92 137 101 128)(93 138 102 129)(94 139 103 130)(95 140 104 131)(96 141 105 132)(97 142 106 133)(98 143 107 134)(99 144 108 135)
(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 75)(2 74)(3 73)(4 81)(5 80)(6 79)(7 78)(8 77)(9 76)(10 88)(11 87)(12 86)(13 85)(14 84)(15 83)(16 82)(17 90)(18 89)(19 97)(20 96)(21 95)(22 94)(23 93)(24 92)(25 91)(26 99)(27 98)(28 106)(29 105)(30 104)(31 103)(32 102)(33 101)(34 100)(35 108)(36 107)(37 115)(38 114)(39 113)(40 112)(41 111)(42 110)(43 109)(44 117)(45 116)(46 124)(47 123)(48 122)(49 121)(50 120)(51 119)(52 118)(53 126)(54 125)(55 133)(56 132)(57 131)(58 130)(59 129)(60 128)(61 127)(62 135)(63 134)(64 142)(65 141)(66 140)(67 139)(68 138)(69 137)(70 136)(71 144)(72 143)

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

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

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

90 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 ··· 9I 12A ··· 12F 12G ··· 12O 12P ··· 12U 18A ··· 18I 36A ··· 36AA 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 12 ··· 12 12 ··· 12 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 9 9 1 1 2 2 2 2 2 2 18 18 18 1 1 2 2 2 9 9 9 9 2 ··· 2 2 ··· 2 4 ··· 4 18 ··· 18 2 ··· 2 4 ··· 4

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + - + + + - - image C1 C2 C2 C2 C3 C6 C6 C6 S3 Q8 D6 D9 C3×S3 C3×Q8 D18 S3×C6 C3×D9 C6×D9 S3×Q8 Q8×D9 C3×S3×Q8 C3×Q8×D9 kernel C3×Q8×D9 C3×Dic18 C12×D9 Q8×C3×C9 Q8×D9 Dic18 C4×D9 Q8×C9 Q8×C32 C3×D9 C3×C12 C3×Q8 C3×Q8 D9 C12 C12 Q8 C4 C32 C3 C3 C1 # reps 1 3 3 1 2 6 6 2 1 2 3 3 2 4 9 6 6 18 1 3 2 6

Matrix representation of C3×Q8×D9 in GL4(𝔽37) generated by

 26 0 0 0 0 26 0 0 0 0 1 0 0 0 0 1
,
 36 0 0 0 0 36 0 0 0 0 0 1 0 0 36 0
,
 36 0 0 0 0 36 0 0 0 0 10 26 0 0 26 27
,
 12 0 0 0 0 34 0 0 0 0 1 0 0 0 0 1
,
 0 34 0 0 12 0 0 0 0 0 36 0 0 0 0 36
G:=sub<GL(4,GF(37))| [26,0,0,0,0,26,0,0,0,0,1,0,0,0,0,1],[36,0,0,0,0,36,0,0,0,0,0,36,0,0,1,0],[36,0,0,0,0,36,0,0,0,0,10,26,0,0,26,27],[12,0,0,0,0,34,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,34,0,0,0,0,0,36,0,0,0,0,36] >;

C3×Q8×D9 in GAP, Magma, Sage, TeX

C_3\times Q_8\times D_9
% in TeX

G:=Group("C3xQ8xD9");
// GroupNames label

G:=SmallGroup(432,364);
// by ID

G=gap.SmallGroup(432,364);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,303,142,10085,292,14118]);
// Polycyclic

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

׿
×
𝔽