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## G = C3×Q8⋊D9order 432 = 24·33

### Direct product of C3 and Q8⋊D9

Aliases: C3×Q8⋊D9, C32.2GL2(𝔽3), Q8⋊(C3×D9), Q8⋊C95C6, C6.4(C3×S4), (C3×Q8)⋊2D9, (C3×C6).12S4, C6.9(C3.S4), (Q8×C32).8S3, C3.2(C3×GL2(𝔽3)), (C3×Q8⋊C9)⋊3C2, C2.3(C3×C3.S4), (C3×Q8).4(C3×S3), SmallGroup(432,246)

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=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, dbd-1=c, ebe=b-1c, dcd-1=bc, ece=b2c, ede=d-1 >

Subgroups: 314 in 57 conjugacy classes, 16 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C32, C12, D6, C2×C6, SD16, D9, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×C9, D18, C3×C12, S3×C6, Q82S3, C3×SD16, C3×D9, C3×C18, Q8⋊C9, Q8⋊C9, C3×C3⋊C8, C3×D12, Q8×C32, C6×D9, Q8⋊D9, C3×Q82S3, C3×Q8⋊C9, C3×Q8⋊D9
Quotients: C1, C2, C3, S3, C6, D9, C3×S3, S4, GL2(𝔽3), C3×D9, C3.S4, C3×S4, Q8⋊D9, C3×GL2(𝔽3), C3×C3.S4, C3×Q8⋊D9

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

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

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

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

45 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A ··· 9I 12A 12B 12C 12D 12E 18A ··· 18I 24A 24B 24C 24D order 1 2 2 3 3 3 3 3 4 6 6 6 6 6 6 6 8 8 9 ··· 9 12 12 12 12 12 18 ··· 18 24 24 24 24 size 1 1 36 1 1 2 2 2 6 1 1 2 2 2 36 36 18 18 8 ··· 8 6 6 12 12 12 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 GL2(𝔽3) C3×D9 C3×GL2(𝔽3) S4 C3×S4 GL2(𝔽3) Q8⋊D9 C3×GL2(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

 1 0 0 0 0 1 0 0 0 0 8 0 0 0 0 8
,
 1 0 0 0 0 1 0 0 0 0 7 60 0 0 60 66
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 72 0
,
 55 0 0 0 0 4 0 0 0 0 39 46 0 0 47 33
,
 0 69 0 0 18 0 0 0 0 0 26 36 0 0 36 47
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,7,60,0,0,60,66],[1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,0],[55,0,0,0,0,4,0,0,0,0,39,47,0,0,46,33],[0,18,0,0,69,0,0,0,0,0,26,36,0,0,36,47] >;

C3×Q8⋊D9 in GAP, Magma, Sage, TeX

C_3\times Q_8\rtimes D_9
% in TeX

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

G:=SmallGroup(432,246);
// by ID

G=gap.SmallGroup(432,246);
# by ID

G:=PCGroup([7,-2,-3,-3,-3,-2,2,-2,632,142,1011,3784,1908,172,2273,1153,285,124]);
// 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,d*b*d^-1=c,e*b*e=b^-1*c,d*c*d^-1=b*c,e*c*e=b^2*c,e*d*e=d^-1>;
// generators/relations

׿
×
𝔽