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## G = C3×C9⋊Q16order 432 = 24·33

### Direct product of C3 and C9⋊Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C3×C9⋊Q16
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×Dic18 — C3×C9⋊Q16
 Lower central C9 — C18 — C36 — C3×C9⋊Q16
 Upper central C1 — C6 — C12 — C3×Q8

Generators and relations for C3×C9⋊Q16
G = < a,b,c,d | a3=b9=c8=1, d2=c4, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 190 in 66 conjugacy classes, 30 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×Q8, C3×C9, Dic9, C36, C36, C3×Dic3, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C3×C18, C9⋊C8, Dic18, Q8×C9, Q8×C9, C3×C3⋊C8, C3×Dic6, Q8×C32, C3×Dic9, C3×C36, C3×C36, C9⋊Q16, C3×C3⋊Q16, C3×C9⋊C8, C3×Dic18, Q8×C3×C9, C3×C9⋊Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, D9, C3×S3, C3⋊D4, C3×D4, D18, S3×C6, C3⋊Q16, C3×Q16, C3×D9, C9⋊D4, C3×C3⋊D4, C6×D9, C9⋊Q16, C3×C3⋊Q16, C3×C9⋊D4, C3×C9⋊Q16

Smallest permutation representation of C3×C9⋊Q16
On 144 points
Generators in S144
(1 7 4)(2 8 5)(3 9 6)(10 16 13)(11 17 14)(12 18 15)(19 25 22)(20 26 23)(21 27 24)(28 34 31)(29 35 32)(30 36 33)(37 43 40)(38 44 41)(39 45 42)(46 52 49)(47 53 50)(48 54 51)(55 61 58)(56 62 59)(57 63 60)(64 70 67)(65 71 68)(66 72 69)(73 76 79)(74 77 80)(75 78 81)(82 85 88)(83 86 89)(84 87 90)(91 94 97)(92 95 98)(93 96 99)(100 103 106)(101 104 107)(102 105 108)(109 112 115)(110 113 116)(111 114 117)(118 121 124)(119 122 125)(120 123 126)(127 130 133)(128 131 134)(129 132 135)(136 139 142)(137 140 143)(138 141 144)
(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)
(1 131 22 122 13 140 31 113)(2 130 23 121 14 139 32 112)(3 129 24 120 15 138 33 111)(4 128 25 119 16 137 34 110)(5 127 26 118 17 136 35 109)(6 135 27 126 18 144 36 117)(7 134 19 125 10 143 28 116)(8 133 20 124 11 142 29 115)(9 132 21 123 12 141 30 114)(37 98 64 80 46 107 55 89)(38 97 65 79 47 106 56 88)(39 96 66 78 48 105 57 87)(40 95 67 77 49 104 58 86)(41 94 68 76 50 103 59 85)(42 93 69 75 51 102 60 84)(43 92 70 74 52 101 61 83)(44 91 71 73 53 100 62 82)(45 99 72 81 54 108 63 90)
(1 49 13 40)(2 50 14 41)(3 51 15 42)(4 52 16 43)(5 53 17 44)(6 54 18 45)(7 46 10 37)(8 47 11 38)(9 48 12 39)(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)(73 136 82 127)(74 137 83 128)(75 138 84 129)(76 139 85 130)(77 140 86 131)(78 141 87 132)(79 142 88 133)(80 143 89 134)(81 144 90 135)(91 109 100 118)(92 110 101 119)(93 111 102 120)(94 112 103 121)(95 113 104 122)(96 114 105 123)(97 115 106 124)(98 116 107 125)(99 117 108 126)

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

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

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

81 conjugacy classes

 class 1 2 3A 3B 3C 3D 3E 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 9A ··· 9I 12A 12B 12C ··· 12M 12N 12O 18A ··· 18I 24A 24B 24C 24D 36A ··· 36AA order 1 2 3 3 3 3 3 4 4 4 6 6 6 6 6 8 8 9 ··· 9 12 12 12 ··· 12 12 12 18 ··· 18 24 24 24 24 36 ··· 36 size 1 1 1 1 2 2 2 2 4 36 1 1 2 2 2 18 18 2 ··· 2 2 2 4 ··· 4 36 36 2 ··· 2 18 18 18 18 4 ··· 4

81 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 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 D4 D6 Q16 D9 C3×S3 C3×D4 C3⋊D4 D18 S3×C6 C3×Q16 C3×D9 C9⋊D4 C3×C3⋊D4 C6×D9 C3×C9⋊D4 C3⋊Q16 C9⋊Q16 C3×C3⋊Q16 C3×C9⋊Q16 kernel C3×C9⋊Q16 C3×C9⋊C8 C3×Dic18 Q8×C3×C9 C9⋊Q16 C9⋊C8 Dic18 Q8×C9 Q8×C32 C3×C18 C3×C12 C3×C9 C3×Q8 C3×Q8 C18 C3×C6 C12 C12 C9 Q8 C6 C6 C4 C2 C32 C3 C3 C1 # reps 1 1 1 1 2 2 2 2 1 1 1 2 3 2 2 2 3 2 4 6 6 4 6 12 1 3 2 6

Matrix representation of C3×C9⋊Q16 in GL4(𝔽73) generated by

 8 0 0 0 0 8 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 55 0 0 0 0 1 0 0 0 0 1
,
 0 72 0 0 1 0 0 0 0 0 32 59 0 0 47 0
,
 72 0 0 0 0 1 0 0 0 0 22 11 0 0 9 51
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,55,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,72,0,0,0,0,0,32,47,0,0,59,0],[72,0,0,0,0,1,0,0,0,0,22,9,0,0,11,51] >;

C3×C9⋊Q16 in GAP, Magma, Sage, TeX

C_3\times C_9\rtimes Q_{16}
% in TeX

G:=Group("C3xC9:Q16");
// GroupNames label

G:=SmallGroup(432,156);
// by ID

G=gap.SmallGroup(432,156);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^9=c^8=1,d^2=c^4,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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