Copied to
clipboard

G = C9×C8⋊S3order 432 = 24·33

Direct product of C9 and C8⋊S3

direct product, metacyclic, supersoluble, monomial

Aliases: C9×C8⋊S3, C727S3, D6.C36, C245C18, C36.77D6, Dic3.C36, C3⋊C84C18, C83(S3×C9), (C3×C72)⋊14C2, C2.3(S3×C36), C6.2(C2×C36), (C3×C9)⋊4M4(2), (C4×S3).2C18, (S3×C36).4C2, (S3×C18).1C4, (S3×C6).1C12, (S3×C12).9C6, C18.22(C4×S3), C4.13(S3×C18), C24.25(C3×S3), C6.34(S3×C12), (C3×C24).23C6, C31(C9×M4(2)), C12.117(S3×C6), C12.13(C2×C18), (C9×Dic3).3C4, (C3×C36).51C22, (C3×Dic3).3C12, C32.2(C3×M4(2)), (C9×C3⋊C8)⋊11C2, (C3×C3⋊C8).7C6, (C3×C8⋊S3).C3, C3.4(C3×C8⋊S3), (C3×C6).38(C2×C12), (C3×C18).15(C2×C4), (C3×C12).88(C2×C6), SmallGroup(432,110)

Series: Derived Chief Lower central Upper central

C1C6 — C9×C8⋊S3
C1C3C6C3×C6C3×C12C3×C36S3×C36 — C9×C8⋊S3
C3C6 — C9×C8⋊S3
C1C36C72

Generators and relations for C9×C8⋊S3
 G = < a,b,c,d | a9=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

Subgroups: 124 in 68 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, C8, C2×C4, C9, C9, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, M4(2), C18, C18 [×2], C3×S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, C3×C9, C36, C36 [×2], C2×C18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C3×C18, C72, C72 [×2], C2×C36, C3×C3⋊C8, C3×C24, S3×C12, C9×Dic3, C3×C36, S3×C18, C9×M4(2), C3×C8⋊S3, C9×C3⋊C8, C3×C72, S3×C36, C9×C8⋊S3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, C9, C12 [×2], D6, C2×C6, M4(2), C18 [×3], C3×S3, C4×S3, C2×C12, C36 [×2], C2×C18, S3×C6, C8⋊S3, C3×M4(2), S3×C9, C2×C36, S3×C12, S3×C18, C9×M4(2), C3×C8⋊S3, S3×C36, C9×C8⋊S3

Smallest permutation representation of C9×C8⋊S3
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 88 52 67 31 63 37 79)(2 89 53 68 32 55 38 80)(3 90 54 69 33 56 39 81)(4 82 46 70 34 57 40 73)(5 83 47 71 35 58 41 74)(6 84 48 72 36 59 42 75)(7 85 49 64 28 60 43 76)(8 86 50 65 29 61 44 77)(9 87 51 66 30 62 45 78)(10 104 23 91 140 116 127 125)(11 105 24 92 141 117 128 126)(12 106 25 93 142 109 129 118)(13 107 26 94 143 110 130 119)(14 108 27 95 144 111 131 120)(15 100 19 96 136 112 132 121)(16 101 20 97 137 113 133 122)(17 102 21 98 138 114 134 123)(18 103 22 99 139 115 135 124)
(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)
(1 115)(2 116)(3 117)(4 109)(5 110)(6 111)(7 112)(8 113)(9 114)(10 80)(11 81)(12 73)(13 74)(14 75)(15 76)(16 77)(17 78)(18 79)(19 85)(20 86)(21 87)(22 88)(23 89)(24 90)(25 82)(26 83)(27 84)(28 100)(29 101)(30 102)(31 103)(32 104)(33 105)(34 106)(35 107)(36 108)(37 99)(38 91)(39 92)(40 93)(41 94)(42 95)(43 96)(44 97)(45 98)(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), (1,88,52,67,31,63,37,79)(2,89,53,68,32,55,38,80)(3,90,54,69,33,56,39,81)(4,82,46,70,34,57,40,73)(5,83,47,71,35,58,41,74)(6,84,48,72,36,59,42,75)(7,85,49,64,28,60,43,76)(8,86,50,65,29,61,44,77)(9,87,51,66,30,62,45,78)(10,104,23,91,140,116,127,125)(11,105,24,92,141,117,128,126)(12,106,25,93,142,109,129,118)(13,107,26,94,143,110,130,119)(14,108,27,95,144,111,131,120)(15,100,19,96,136,112,132,121)(16,101,20,97,137,113,133,122)(17,102,21,98,138,114,134,123)(18,103,22,99,139,115,135,124), (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), (1,115)(2,116)(3,117)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,80)(11,81)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,82)(26,83)(27,84)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,99)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(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), (1,88,52,67,31,63,37,79)(2,89,53,68,32,55,38,80)(3,90,54,69,33,56,39,81)(4,82,46,70,34,57,40,73)(5,83,47,71,35,58,41,74)(6,84,48,72,36,59,42,75)(7,85,49,64,28,60,43,76)(8,86,50,65,29,61,44,77)(9,87,51,66,30,62,45,78)(10,104,23,91,140,116,127,125)(11,105,24,92,141,117,128,126)(12,106,25,93,142,109,129,118)(13,107,26,94,143,110,130,119)(14,108,27,95,144,111,131,120)(15,100,19,96,136,112,132,121)(16,101,20,97,137,113,133,122)(17,102,21,98,138,114,134,123)(18,103,22,99,139,115,135,124), (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), (1,115)(2,116)(3,117)(4,109)(5,110)(6,111)(7,112)(8,113)(9,114)(10,80)(11,81)(12,73)(13,74)(14,75)(15,76)(16,77)(17,78)(18,79)(19,85)(20,86)(21,87)(22,88)(23,89)(24,90)(25,82)(26,83)(27,84)(28,100)(29,101)(30,102)(31,103)(32,104)(33,105)(34,106)(35,107)(36,108)(37,99)(38,91)(39,92)(40,93)(41,94)(42,95)(43,96)(44,97)(45,98)(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)], [(1,88,52,67,31,63,37,79),(2,89,53,68,32,55,38,80),(3,90,54,69,33,56,39,81),(4,82,46,70,34,57,40,73),(5,83,47,71,35,58,41,74),(6,84,48,72,36,59,42,75),(7,85,49,64,28,60,43,76),(8,86,50,65,29,61,44,77),(9,87,51,66,30,62,45,78),(10,104,23,91,140,116,127,125),(11,105,24,92,141,117,128,126),(12,106,25,93,142,109,129,118),(13,107,26,94,143,110,130,119),(14,108,27,95,144,111,131,120),(15,100,19,96,136,112,132,121),(16,101,20,97,137,113,133,122),(17,102,21,98,138,114,134,123),(18,103,22,99,139,115,135,124)], [(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)], [(1,115),(2,116),(3,117),(4,109),(5,110),(6,111),(7,112),(8,113),(9,114),(10,80),(11,81),(12,73),(13,74),(14,75),(15,76),(16,77),(17,78),(18,79),(19,85),(20,86),(21,87),(22,88),(23,89),(24,90),(25,82),(26,83),(27,84),(28,100),(29,101),(30,102),(31,103),(32,104),(33,105),(34,106),(35,107),(36,108),(37,99),(38,91),(39,92),(40,93),(41,94),(42,95),(43,96),(44,97),(45,98),(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)])

162 conjugacy classes

class 1 2A2B3A3B3C3D3E4A4B4C6A6B6C6D6E6F6G8A8B8C8D9A···9F9G···9L12A12B12C12D12E···12J12K12L18A···18F18G···18L18M···18R24A···24P24Q24R24S24T36A···36L36M···36X36Y···36AD72A···72AJ72AK···72AV
order12233333444666666688889···99···91212121212···12121218···1818···1818···1824···242424242436···3636···3636···3672···7272···72
size11611222116112226622661···12···211112···2661···12···26···62···266661···12···26···62···26···6

162 irreducible representations

dim111111111111111111222222222222222
type++++++
imageC1C2C2C2C3C4C4C6C6C6C9C12C12C18C18C18C36C36S3D6M4(2)C3×S3C4×S3S3×C6C8⋊S3C3×M4(2)S3×C9S3×C12S3×C18C9×M4(2)C3×C8⋊S3S3×C36C9×C8⋊S3
kernelC9×C8⋊S3C9×C3⋊C8C3×C72S3×C36C3×C8⋊S3C9×Dic3S3×C18C3×C3⋊C8C3×C24S3×C12C8⋊S3C3×Dic3S3×C6C3⋊C8C24C4×S3Dic3D6C72C36C3×C9C24C18C12C9C32C8C6C4C3C3C2C1
# reps11112222226446661212112222446461281224

Matrix representation of C9×C8⋊S3 in GL2(𝔽73) generated by

20
02
,
100
063
,
640
08
,
01
10
G:=sub<GL(2,GF(73))| [2,0,0,2],[10,0,0,63],[64,0,0,8],[0,1,1,0] >;

C9×C8⋊S3 in GAP, Magma, Sage, TeX

C_9\times C_8\rtimes S_3
% in TeX

G:=Group("C9xC8:S3");
// GroupNames label

G:=SmallGroup(432,110);
// by ID

G=gap.SmallGroup(432,110);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-3,-2,-3,1037,92,142,192,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^8=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^5,d*c*d=c^-1>;
// generators/relations

׿
×
𝔽