Copied to
clipboard

## G = C9×Q8⋊2S3order 432 = 24·33

### Direct product of C9 and Q8⋊2S3

Series: Derived Chief Lower central Upper central

 Derived series C1 — C12 — C9×Q8⋊2S3
 Chief series C1 — C3 — C6 — C3×C6 — C3×C12 — C3×C36 — C9×D12 — C9×Q8⋊2S3
 Lower central C3 — C6 — C12 — C9×Q8⋊2S3
 Upper central C1 — C18 — C36 — Q8×C9

Generators and relations for C9×Q82S3
G = < a,b,c,d,e | a9=b4=d3=e2=1, c2=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=ebe=b-1, bd=db, cd=dc, ece=b-1c, ede=d-1 >

Subgroups: 168 in 72 conjugacy classes, 33 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, C12, C12, D6, C2×C6, SD16, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×Q8, C3×Q8, C3×C9, C36, C36, C2×C18, C3×C12, C3×C12, S3×C6, Q82S3, C3×SD16, S3×C9, C3×C18, C72, D4×C9, Q8×C9, Q8×C9, C3×C3⋊C8, C3×D12, Q8×C32, C3×C36, C3×C36, S3×C18, C9×SD16, C3×Q82S3, C9×C3⋊C8, C9×D12, Q8×C3×C9, C9×Q82S3
Quotients: C1, C2, C3, C22, S3, C6, D4, C9, D6, C2×C6, SD16, C18, C3×S3, C3⋊D4, C3×D4, C2×C18, S3×C6, Q82S3, C3×SD16, S3×C9, D4×C9, C3×C3⋊D4, S3×C18, C9×SD16, C3×Q82S3, C9×C3⋊D4, C9×Q82S3

Smallest permutation representation of C9×Q82S3
On 144 points
Generators in S144
(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 51 42 32)(2 52 43 33)(3 53 44 34)(4 54 45 35)(5 46 37 36)(6 47 38 28)(7 48 39 29)(8 49 40 30)(9 50 41 31)(10 21 138 134)(11 22 139 135)(12 23 140 127)(13 24 141 128)(14 25 142 129)(15 26 143 130)(16 27 144 131)(17 19 136 132)(18 20 137 133)(55 68 88 79)(56 69 89 80)(57 70 90 81)(58 71 82 73)(59 72 83 74)(60 64 84 75)(61 65 85 76)(62 66 86 77)(63 67 87 78)(91 104 124 115)(92 105 125 116)(93 106 126 117)(94 107 118 109)(95 108 119 110)(96 100 120 111)(97 101 121 112)(98 102 122 113)(99 103 123 114)
(1 67 42 78)(2 68 43 79)(3 69 44 80)(4 70 45 81)(5 71 37 73)(6 72 38 74)(7 64 39 75)(8 65 40 76)(9 66 41 77)(10 122 138 98)(11 123 139 99)(12 124 140 91)(13 125 141 92)(14 126 142 93)(15 118 143 94)(16 119 144 95)(17 120 136 96)(18 121 137 97)(19 100 132 111)(20 101 133 112)(21 102 134 113)(22 103 135 114)(23 104 127 115)(24 105 128 116)(25 106 129 117)(26 107 130 109)(27 108 131 110)(28 83 47 59)(29 84 48 60)(30 85 49 61)(31 86 50 62)(32 87 51 63)(33 88 52 55)(34 89 53 56)(35 90 54 57)(36 82 46 58)
(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 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 79 76)(74 80 77)(75 81 78)(82 88 85)(83 89 86)(84 90 87)(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)
(1 114)(2 115)(3 116)(4 117)(5 109)(6 110)(7 111)(8 112)(9 113)(10 77)(11 78)(12 79)(13 80)(14 81)(15 73)(16 74)(17 75)(18 76)(19 84)(20 85)(21 86)(22 87)(23 88)(24 89)(25 90)(26 82)(27 83)(28 95)(29 96)(30 97)(31 98)(32 99)(33 91)(34 92)(35 93)(36 94)(37 107)(38 108)(39 100)(40 101)(41 102)(42 103)(43 104)(44 105)(45 106)(46 118)(47 119)(48 120)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)

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

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

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

108 conjugacy classes

 class 1 2A 2B 3A 3B 3C 3D 3E 4A 4B 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A ··· 9F 9G ··· 9L 12A 12B 12C ··· 12M 18A ··· 18F 18G ··· 18L 18M ··· 18R 24A 24B 24C 24D 36A ··· 36F 36G ··· 36AD 72A ··· 72L order 1 2 2 3 3 3 3 3 4 4 6 6 6 6 6 6 6 8 8 9 ··· 9 9 ··· 9 12 12 12 ··· 12 18 ··· 18 18 ··· 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 12 1 1 2 2 2 2 4 1 1 2 2 2 12 12 6 6 1 ··· 1 2 ··· 2 2 2 4 ··· 4 1 ··· 1 2 ··· 2 12 ··· 12 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

108 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 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 C9 C18 C18 C18 S3 D4 D6 SD16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×SD16 S3×C9 D4×C9 C3×C3⋊D4 S3×C18 C9×SD16 C9×C3⋊D4 Q8⋊2S3 C3×Q8⋊2S3 C9×Q8⋊2S3 kernel C9×Q8⋊2S3 C9×C3⋊C8 C9×D12 Q8×C3×C9 C3×Q8⋊2S3 C3×C3⋊C8 C3×D12 Q8×C32 Q8⋊2S3 C3⋊C8 D12 C3×Q8 Q8×C9 C3×C18 C36 C3×C9 C3×Q8 C18 C3×C6 C12 C32 Q8 C6 C6 C4 C3 C2 C9 C3 C1 # reps 1 1 1 1 2 2 2 2 6 6 6 6 1 1 1 2 2 2 2 2 4 6 6 4 6 12 12 1 2 6

Matrix representation of C9×Q82S3 in GL4(𝔽73) generated by

 16 0 0 0 0 16 0 0 0 0 1 0 0 0 0 1
,
 72 0 0 0 0 72 0 0 0 0 0 1 0 0 72 0
,
 72 0 0 0 25 1 0 0 0 0 60 66 0 0 66 13
,
 64 0 0 0 30 8 0 0 0 0 1 0 0 0 0 1
,
 21 63 0 0 44 52 0 0 0 0 26 36 0 0 36 47
G:=sub<GL(4,GF(73))| [16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,1],[72,0,0,0,0,72,0,0,0,0,0,72,0,0,1,0],[72,25,0,0,0,1,0,0,0,0,60,66,0,0,66,13],[64,30,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[21,44,0,0,63,52,0,0,0,0,26,36,0,0,36,47] >;

C9×Q82S3 in GAP, Magma, Sage, TeX

C_9\times Q_8\rtimes_2S_3
% in TeX

G:=Group("C9xQ8:2S3");
// GroupNames label

G:=SmallGroup(432,158);
// by ID

G=gap.SmallGroup(432,158);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,197,512,142,2355,1186,192,14118]);
// Polycyclic

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

׿
×
𝔽