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## G = C3×Q8.D9order 432 = 24·33

### Direct product of C3 and Q8.D9

Aliases: C3×Q8.D9, C32.2CSU2(𝔽3), Q8.(C3×D9), C6.2(C3×S4), Q8⋊C9.3C6, (C3×C6).11S4, (C3×Q8).4D9, C6.8(C3.S4), (Q8×C32).6S3, C3.2(C3×CSU2(𝔽3)), C2.2(C3×C3.S4), (C3×Q8⋊C9).1C2, (C3×Q8).2(C3×S3), SmallGroup(432,244)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2 — Q8 — Q8⋊C9 — C3×Q8.D9
 Chief series C1 — C2 — Q8 — C3×Q8 — Q8⋊C9 — C3×Q8⋊C9 — C3×Q8.D9
 Lower central Q8⋊C9 — C3×Q8.D9
 Upper central C1 — C6

Generators and relations for C3×Q8.D9
G = < a,b,c,d,e | a3=b4=d9=1, c2=e2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe-1=b-1c, dcd-1=bc, ece-1=b2c, ede-1=d-1 >

Smallest permutation representation of C3×Q8.D9
On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(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)(136 139 142)(137 140 143)(138 141 144)
(1 89 101 73)(2 93 102 120)(3 134 103 112)(4 83 104 76)(5 96 105 123)(6 128 106 115)(7 86 107 79)(8 99 108 126)(9 131 100 109)(10 62 32 69)(11 24 33 51)(12 140 34 43)(13 56 35 72)(14 27 36 54)(15 143 28 37)(16 59 29 66)(17 21 30 48)(18 137 31 40)(19 58 46 65)(20 38 47 144)(22 61 49 68)(23 41 50 138)(25 55 52 71)(26 44 53 141)(39 60 136 67)(42 63 139 70)(45 57 142 64)(74 111 90 133)(75 94 82 121)(77 114 84 127)(78 97 85 124)(80 117 87 130)(81 91 88 118)(92 110 119 132)(95 113 122 135)(98 116 125 129)
(1 92 101 119)(2 133 102 111)(3 82 103 75)(4 95 104 122)(5 127 105 114)(6 85 106 78)(7 98 107 125)(8 130 108 117)(9 88 100 81)(10 23 32 50)(11 139 33 42)(12 55 34 71)(13 26 35 53)(14 142 36 45)(15 58 28 65)(16 20 29 47)(17 136 30 39)(18 61 31 68)(19 37 46 143)(21 60 48 67)(22 40 49 137)(24 63 51 70)(25 43 52 140)(27 57 54 64)(38 59 144 66)(41 62 138 69)(44 56 141 72)(73 110 89 132)(74 93 90 120)(76 113 83 135)(77 96 84 123)(79 116 86 129)(80 99 87 126)(91 109 118 131)(94 112 121 134)(97 115 124 128)
(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 43 101 140)(2 42 102 139)(3 41 103 138)(4 40 104 137)(5 39 105 136)(6 38 106 144)(7 37 107 143)(8 45 108 142)(9 44 100 141)(10 134 32 112)(11 133 33 111)(12 132 34 110)(13 131 35 109)(14 130 36 117)(15 129 28 116)(16 128 29 115)(17 127 30 114)(18 135 31 113)(19 125 46 98)(20 124 47 97)(21 123 48 96)(22 122 49 95)(23 121 50 94)(24 120 51 93)(25 119 52 92)(26 118 53 91)(27 126 54 99)(55 89 71 73)(56 88 72 81)(57 87 64 80)(58 86 65 79)(59 85 66 78)(60 84 67 77)(61 83 68 76)(62 82 69 75)(63 90 70 74)

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

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

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

45 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 8A 8B 9A ··· 9I 12A 12B 12C 12D 12E 12F 12G 18A ··· 18I 24A 24B 24C 24D order 1 2 3 3 3 3 3 4 4 6 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 12 12 18 ··· 18 24 24 24 24 size 1 1 1 1 2 2 2 6 36 1 1 2 2 2 18 18 8 ··· 8 6 6 12 12 12 36 36 8 ··· 8 18 18 18 18

45 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 3 3 4 4 4 4 6 6 type + + + + - + - - + image C1 C2 C3 C6 S3 D9 C3×S3 CSU2(𝔽3) C3×D9 C3×CSU2(𝔽3) S4 C3×S4 CSU2(𝔽3) Q8.D9 C3×CSU2(𝔽3) C3×Q8.D9 C3.S4 C3×C3.S4 kernel C3×Q8.D9 C3×Q8⋊C9 Q8.D9 Q8⋊C9 Q8×C32 C3×Q8 C3×Q8 C32 Q8 C3 C3×C6 C6 C32 C3 C3 C1 C6 C2 # reps 1 1 2 2 1 3 2 2 6 4 2 4 1 3 2 6 1 2

Matrix representation of C3×Q8.D9 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 1 2 0 0 72 72
,
 1 0 0 0 0 1 0 0 0 0 11 71 0 0 61 62
,
 37 0 0 0 38 2 0 0 0 0 60 61 0 0 7 12
,
 4 4 0 0 51 69 0 0 0 0 46 60 0 0 0 27
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,1,72,0,0,2,72],[1,0,0,0,0,1,0,0,0,0,11,61,0,0,71,62],[37,38,0,0,0,2,0,0,0,0,60,7,0,0,61,12],[4,51,0,0,4,69,0,0,0,0,46,0,0,0,60,27] >;

C3×Q8.D9 in GAP, Magma, Sage, TeX

C_3\times Q_8.D_9
% in TeX

G:=Group("C3xQ8.D9");
// GroupNames label

G:=SmallGroup(432,244);
// by ID

G=gap.SmallGroup(432,244);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,1512,632,142,1011,3784,1908,172,2273,1153,285,124]);
// Polycyclic

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

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