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G = C3×C6×C3⋊C8order 432 = 24·33

Direct product of C3×C6 and C3⋊C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3 — C3×C6×C3⋊C8
 Chief series C1 — C3 — C6 — C12 — C3×C12 — C32×C12 — C32×C3⋊C8 — C3×C6×C3⋊C8
 Lower central C3 — C3×C6×C3⋊C8
 Upper central C1 — C6×C12

Generators and relations for C3×C6×C3⋊C8
G = < a,b,c,d | a3=b6=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 296 in 196 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C2 [×2], C3, C3 [×4], C3 [×4], C4 [×2], C22, C6, C6 [×14], C6 [×12], C8 [×2], C2×C4, C32, C32 [×4], C32 [×4], C12 [×2], C12 [×8], C12 [×8], C2×C6, C2×C6 [×4], C2×C6 [×4], C2×C8, C3×C6, C3×C6 [×14], C3×C6 [×12], C3⋊C8 [×2], C24 [×8], C2×C12, C2×C12 [×4], C2×C12 [×4], C33, C3×C12 [×2], C3×C12 [×8], C3×C12 [×8], C62, C62 [×4], C62 [×4], C2×C3⋊C8, C2×C24 [×4], C32×C6, C32×C6 [×2], C3×C3⋊C8 [×8], C3×C24 [×2], C6×C12, C6×C12 [×4], C6×C12 [×4], C32×C12 [×2], C3×C62, C6×C3⋊C8 [×4], C6×C24, C32×C3⋊C8 [×2], C3×C6×C12, C3×C6×C3⋊C8
Quotients: C1, C2 [×3], C3 [×4], C4 [×2], C22, S3, C6 [×12], C8 [×2], C2×C4, C32, Dic3 [×2], C12 [×8], D6, C2×C6 [×4], C2×C8, C3×S3 [×4], C3×C6 [×3], C3⋊C8 [×2], C24 [×8], C2×Dic3, C2×C12 [×4], C3×Dic3 [×8], C3×C12 [×2], S3×C6 [×4], C62, C2×C3⋊C8, C2×C24 [×4], S3×C32, C3×C3⋊C8 [×8], C3×C24 [×2], C6×Dic3 [×4], C6×C12, C32×Dic3 [×2], S3×C3×C6, C6×C3⋊C8 [×4], C6×C24, C32×C3⋊C8 [×2], Dic3×C3×C6, C3×C6×C3⋊C8

Smallest permutation representation of C3×C6×C3⋊C8
On 144 points
Generators in S144
(1 33 78)(2 34 79)(3 35 80)(4 36 73)(5 37 74)(6 38 75)(7 39 76)(8 40 77)(9 109 137)(10 110 138)(11 111 139)(12 112 140)(13 105 141)(14 106 142)(15 107 143)(16 108 144)(17 104 127)(18 97 128)(19 98 121)(20 99 122)(21 100 123)(22 101 124)(23 102 125)(24 103 126)(25 134 50)(26 135 51)(27 136 52)(28 129 53)(29 130 54)(30 131 55)(31 132 56)(32 133 49)(41 117 82)(42 118 83)(43 119 84)(44 120 85)(45 113 86)(46 114 87)(47 115 88)(48 116 81)(57 95 65)(58 96 66)(59 89 67)(60 90 68)(61 91 69)(62 92 70)(63 93 71)(64 94 72)
(1 94 41 50 143 23)(2 95 42 51 144 24)(3 96 43 52 137 17)(4 89 44 53 138 18)(5 90 45 54 139 19)(6 91 46 55 140 20)(7 92 47 56 141 21)(8 93 48 49 142 22)(9 104 35 66 119 27)(10 97 36 67 120 28)(11 98 37 68 113 29)(12 99 38 69 114 30)(13 100 39 70 115 31)(14 101 40 71 116 32)(15 102 33 72 117 25)(16 103 34 65 118 26)(57 83 135 108 126 79)(58 84 136 109 127 80)(59 85 129 110 128 73)(60 86 130 111 121 74)(61 87 131 112 122 75)(62 88 132 105 123 76)(63 81 133 106 124 77)(64 82 134 107 125 78)
(1 41 143)(2 144 42)(3 43 137)(4 138 44)(5 45 139)(6 140 46)(7 47 141)(8 142 48)(9 35 119)(10 120 36)(11 37 113)(12 114 38)(13 39 115)(14 116 40)(15 33 117)(16 118 34)(17 96 52)(18 53 89)(19 90 54)(20 55 91)(21 92 56)(22 49 93)(23 94 50)(24 51 95)(25 102 72)(26 65 103)(27 104 66)(28 67 97)(29 98 68)(30 69 99)(31 100 70)(32 71 101)(57 126 135)(58 136 127)(59 128 129)(60 130 121)(61 122 131)(62 132 123)(63 124 133)(64 134 125)(73 110 85)(74 86 111)(75 112 87)(76 88 105)(77 106 81)(78 82 107)(79 108 83)(80 84 109)
(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)

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

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

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

216 conjugacy classes

 class 1 2A 2B 2C 3A ··· 3H 3I ··· 3Q 4A 4B 4C 4D 6A ··· 6X 6Y ··· 6AY 8A ··· 8H 12A ··· 12AF 12AG ··· 12BP 24A ··· 24BL order 1 2 2 2 3 ··· 3 3 ··· 3 4 4 4 4 6 ··· 6 6 ··· 6 8 ··· 8 12 ··· 12 12 ··· 12 24 ··· 24 size 1 1 1 1 1 ··· 1 2 ··· 2 1 1 1 1 1 ··· 1 2 ··· 2 3 ··· 3 1 ··· 1 2 ··· 2 3 ··· 3

216 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 type + + + + - + - image C1 C2 C2 C3 C4 C4 C6 C6 C8 C12 C12 C24 S3 Dic3 D6 Dic3 C3×S3 C3⋊C8 C3×Dic3 S3×C6 C3×Dic3 C3×C3⋊C8 kernel C3×C6×C3⋊C8 C32×C3⋊C8 C3×C6×C12 C6×C3⋊C8 C32×C12 C3×C62 C3×C3⋊C8 C6×C12 C32×C6 C3×C12 C62 C3×C6 C6×C12 C3×C12 C3×C12 C62 C2×C12 C3×C6 C12 C12 C2×C6 C6 # reps 1 2 1 8 2 2 16 8 8 16 16 64 1 1 1 1 8 4 8 8 8 32

Matrix representation of C3×C6×C3⋊C8 in GL3(𝔽73) generated by

 8 0 0 0 1 0 0 0 1
,
 65 0 0 0 64 0 0 0 64
,
 1 0 0 0 8 0 0 0 64
,
 72 0 0 0 0 1 0 46 0
G:=sub<GL(3,GF(73))| [8,0,0,0,1,0,0,0,1],[65,0,0,0,64,0,0,0,64],[1,0,0,0,8,0,0,0,64],[72,0,0,0,0,46,0,1,0] >;

C3×C6×C3⋊C8 in GAP, Magma, Sage, TeX

C_3\times C_6\times C_3\rtimes C_8
% in TeX

G:=Group("C3xC6xC3:C8");
// GroupNames label

G:=SmallGroup(432,469);
// by ID

G=gap.SmallGroup(432,469);
# by ID

G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,102,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^6=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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