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## G = C6×C9⋊C8order 432 = 24·33

### Direct product of C6 and C9⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C9 — C6×C9⋊C8
 Chief series C1 — C3 — C9 — C18 — C36 — C3×C36 — C3×C9⋊C8 — C6×C9⋊C8
 Lower central C9 — C6×C9⋊C8
 Upper central C1 — C2×C12

Generators and relations for C6×C9⋊C8
G = < a,b,c | a6=b9=c8=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 158 in 82 conjugacy classes, 54 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C9, C9, C32, C12, C12, C2×C6, C2×C6, C2×C8, C18, C18, C18, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C3×C9, C36, C36, C2×C18, C2×C18, C3×C12, C62, C2×C3⋊C8, C2×C24, C3×C18, C3×C18, C9⋊C8, C2×C36, C2×C36, C3×C3⋊C8, C6×C12, C3×C36, C6×C18, C2×C9⋊C8, C6×C3⋊C8, C3×C9⋊C8, C6×C36, C6×C9⋊C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, D9, C3×S3, C3⋊C8, C24, C2×Dic3, C2×C12, Dic9, D18, C3×Dic3, S3×C6, C2×C3⋊C8, C2×C24, C3×D9, C9⋊C8, C2×Dic9, C3×C3⋊C8, C6×Dic3, C3×Dic9, C6×D9, C2×C9⋊C8, C6×C3⋊C8, C3×C9⋊C8, C6×Dic9, C6×C9⋊C8

Smallest permutation representation of C6×C9⋊C8
On 144 points
Generators in S144
(1 43 7 40 4 37)(2 44 8 41 5 38)(3 45 9 42 6 39)(10 49 16 46 13 52)(11 50 17 47 14 53)(12 51 18 48 15 54)(19 58 25 55 22 61)(20 59 26 56 23 62)(21 60 27 57 24 63)(28 67 34 64 31 70)(29 68 35 65 32 71)(30 69 36 66 33 72)(73 115 76 109 79 112)(74 116 77 110 80 113)(75 117 78 111 81 114)(82 124 85 118 88 121)(83 125 86 119 89 122)(84 126 87 120 90 123)(91 133 94 127 97 130)(92 134 95 128 98 131)(93 135 96 129 99 132)(100 142 103 136 106 139)(101 143 104 137 107 140)(102 144 105 138 108 141)
(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 31 122 13 131 22 113)(2 139 32 121 14 130 23 112)(3 138 33 120 15 129 24 111)(4 137 34 119 16 128 25 110)(5 136 35 118 17 127 26 109)(6 144 36 126 18 135 27 117)(7 143 28 125 10 134 19 116)(8 142 29 124 11 133 20 115)(9 141 30 123 12 132 21 114)(37 107 64 89 46 98 55 80)(38 106 65 88 47 97 56 79)(39 105 66 87 48 96 57 78)(40 104 67 86 49 95 58 77)(41 103 68 85 50 94 59 76)(42 102 69 84 51 93 60 75)(43 101 70 83 52 92 61 74)(44 100 71 82 53 91 62 73)(45 108 72 90 54 99 63 81)

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

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

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

144 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 3E 4A 4B 4C 4D 6A ··· 6F 6G ··· 6O 8A ··· 8H 9A ··· 9I 12A ··· 12H 12I ··· 12T 18A ··· 18AA 24A ··· 24P 36A ··· 36AJ order 1 2 2 2 3 3 3 3 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 9 ··· 9 12 ··· 12 12 ··· 12 18 ··· 18 24 ··· 24 36 ··· 36 size 1 1 1 1 1 1 2 2 2 1 1 1 1 1 ··· 1 2 ··· 2 9 ··· 9 2 ··· 2 1 ··· 1 2 ··· 2 2 ··· 2 9 ··· 9 2 ··· 2

144 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 Dic3 D9 C3×S3 C3⋊C8 Dic9 D18 C3×Dic3 S3×C6 Dic9 C3×Dic3 C3×D9 C9⋊C8 C3×C3⋊C8 C3×Dic9 C6×D9 C3×Dic9 C3×C9⋊C8 kernel C6×C9⋊C8 C3×C9⋊C8 C6×C36 C2×C9⋊C8 C3×C36 C6×C18 C9⋊C8 C2×C36 C3×C18 C36 C2×C18 C18 C6×C12 C3×C12 C3×C12 C62 C2×C12 C2×C12 C3×C6 C12 C12 C12 C12 C2×C6 C2×C6 C2×C4 C6 C6 C4 C4 C22 C2 # reps 1 2 1 2 2 2 4 2 8 4 4 16 1 1 1 1 3 2 4 3 3 2 2 3 2 6 12 8 6 6 6 24

Matrix representation of C6×C9⋊C8 in GL4(𝔽73) generated by

 8 0 0 0 0 65 0 0 0 0 64 0 0 0 0 64
,
 1 0 0 0 0 1 0 0 0 0 37 0 0 0 33 2
,
 63 0 0 0 0 72 0 0 0 0 72 63 0 0 0 1
G:=sub<GL(4,GF(73))| [8,0,0,0,0,65,0,0,0,0,64,0,0,0,0,64],[1,0,0,0,0,1,0,0,0,0,37,33,0,0,0,2],[63,0,0,0,0,72,0,0,0,0,72,0,0,0,63,1] >;

C6×C9⋊C8 in GAP, Magma, Sage, TeX

C_6\times C_9\rtimes C_8
% in TeX

G:=Group("C6xC9:C8");
// GroupNames label

G:=SmallGroup(432,124);
// by ID

G=gap.SmallGroup(432,124);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^6=b^9=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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