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G = C4×C9⋊S3order 216 = 23·33

Direct product of C4 and C9⋊S3

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C4×C9⋊S3, C362S3, C122D9, C18.15D6, C6.15D18, C92(C4×S3), C32(C4×D9), (C3×C36)⋊5C2, C9⋊Dic35C2, (C3×C6).51D6, C12.5(C3⋊S3), (C3×C12).18S3, C32.5(C4×S3), (C3×C18).19C22, C3.(C4×C3⋊S3), (C3×C9)⋊6(C2×C4), C2.1(C2×C9⋊S3), C6.9(C2×C3⋊S3), (C2×C9⋊S3).2C2, SmallGroup(216,64)

Series: Derived Chief Lower central Upper central

C1C3×C9 — C4×C9⋊S3
C1C3C32C3×C9C3×C18C2×C9⋊S3 — C4×C9⋊S3
C3×C9 — C4×C9⋊S3
C1C4

Generators and relations for C4×C9⋊S3
 G = < a,b,c,d | a4=b9=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 410 in 80 conjugacy classes, 35 normal (15 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C4, C22, S3 [×8], C6, C6 [×3], C2×C4, C9 [×3], C32, Dic3 [×4], C12, C12 [×3], D6 [×4], D9 [×6], C18 [×3], C3⋊S3 [×2], C3×C6, C4×S3 [×4], C3×C9, Dic9 [×3], C36 [×3], D18 [×3], C3⋊Dic3, C3×C12, C2×C3⋊S3, C9⋊S3 [×2], C3×C18, C4×D9 [×3], C4×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C4×C9⋊S3
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D6 [×4], D9 [×3], C3⋊S3, C4×S3 [×4], D18 [×3], C2×C3⋊S3, C9⋊S3, C4×D9 [×3], C4×C3⋊S3, C2×C9⋊S3, C4×C9⋊S3

Smallest permutation representation of C4×C9⋊S3
On 108 points
Generators in S108
(1 100 46 73)(2 101 47 74)(3 102 48 75)(4 103 49 76)(5 104 50 77)(6 105 51 78)(7 106 52 79)(8 107 53 80)(9 108 54 81)(10 59 86 32)(11 60 87 33)(12 61 88 34)(13 62 89 35)(14 63 90 36)(15 55 82 28)(16 56 83 29)(17 57 84 30)(18 58 85 31)(19 71 98 44)(20 72 99 45)(21 64 91 37)(22 65 92 38)(23 66 93 39)(24 67 94 40)(25 68 95 41)(26 69 96 42)(27 70 97 43)
(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)
(1 32 37)(2 33 38)(3 34 39)(4 35 40)(5 36 41)(6 28 42)(7 29 43)(8 30 44)(9 31 45)(10 21 100)(11 22 101)(12 23 102)(13 24 103)(14 25 104)(15 26 105)(16 27 106)(17 19 107)(18 20 108)(46 59 64)(47 60 65)(48 61 66)(49 62 67)(50 63 68)(51 55 69)(52 56 70)(53 57 71)(54 58 72)(73 86 91)(74 87 92)(75 88 93)(76 89 94)(77 90 95)(78 82 96)(79 83 97)(80 84 98)(81 85 99)
(1 46)(2 54)(3 53)(4 52)(5 51)(6 50)(7 49)(8 48)(9 47)(10 91)(11 99)(12 98)(13 97)(14 96)(15 95)(16 94)(17 93)(18 92)(19 88)(20 87)(21 86)(22 85)(23 84)(24 83)(25 82)(26 90)(27 89)(28 68)(29 67)(30 66)(31 65)(32 64)(33 72)(34 71)(35 70)(36 69)(37 59)(38 58)(39 57)(40 56)(41 55)(42 63)(43 62)(44 61)(45 60)(73 100)(74 108)(75 107)(76 106)(77 105)(78 104)(79 103)(80 102)(81 101)

G:=sub<Sym(108)| (1,100,46,73)(2,101,47,74)(3,102,48,75)(4,103,49,76)(5,104,50,77)(6,105,51,78)(7,106,52,79)(8,107,53,80)(9,108,54,81)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (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), (1,32,37)(2,33,38)(3,34,39)(4,35,40)(5,36,41)(6,28,42)(7,29,43)(8,30,44)(9,31,45)(10,21,100)(11,22,101)(12,23,102)(13,24,103)(14,25,104)(15,26,105)(16,27,106)(17,19,107)(18,20,108)(46,59,64)(47,60,65)(48,61,66)(49,62,67)(50,63,68)(51,55,69)(52,56,70)(53,57,71)(54,58,72)(73,86,91)(74,87,92)(75,88,93)(76,89,94)(77,90,95)(78,82,96)(79,83,97)(80,84,98)(81,85,99), (1,46)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,91)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,90)(27,89)(28,68)(29,67)(30,66)(31,65)(32,64)(33,72)(34,71)(35,70)(36,69)(37,59)(38,58)(39,57)(40,56)(41,55)(42,63)(43,62)(44,61)(45,60)(73,100)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101)>;

G:=Group( (1,100,46,73)(2,101,47,74)(3,102,48,75)(4,103,49,76)(5,104,50,77)(6,105,51,78)(7,106,52,79)(8,107,53,80)(9,108,54,81)(10,59,86,32)(11,60,87,33)(12,61,88,34)(13,62,89,35)(14,63,90,36)(15,55,82,28)(16,56,83,29)(17,57,84,30)(18,58,85,31)(19,71,98,44)(20,72,99,45)(21,64,91,37)(22,65,92,38)(23,66,93,39)(24,67,94,40)(25,68,95,41)(26,69,96,42)(27,70,97,43), (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), (1,32,37)(2,33,38)(3,34,39)(4,35,40)(5,36,41)(6,28,42)(7,29,43)(8,30,44)(9,31,45)(10,21,100)(11,22,101)(12,23,102)(13,24,103)(14,25,104)(15,26,105)(16,27,106)(17,19,107)(18,20,108)(46,59,64)(47,60,65)(48,61,66)(49,62,67)(50,63,68)(51,55,69)(52,56,70)(53,57,71)(54,58,72)(73,86,91)(74,87,92)(75,88,93)(76,89,94)(77,90,95)(78,82,96)(79,83,97)(80,84,98)(81,85,99), (1,46)(2,54)(3,53)(4,52)(5,51)(6,50)(7,49)(8,48)(9,47)(10,91)(11,99)(12,98)(13,97)(14,96)(15,95)(16,94)(17,93)(18,92)(19,88)(20,87)(21,86)(22,85)(23,84)(24,83)(25,82)(26,90)(27,89)(28,68)(29,67)(30,66)(31,65)(32,64)(33,72)(34,71)(35,70)(36,69)(37,59)(38,58)(39,57)(40,56)(41,55)(42,63)(43,62)(44,61)(45,60)(73,100)(74,108)(75,107)(76,106)(77,105)(78,104)(79,103)(80,102)(81,101) );

G=PermutationGroup([(1,100,46,73),(2,101,47,74),(3,102,48,75),(4,103,49,76),(5,104,50,77),(6,105,51,78),(7,106,52,79),(8,107,53,80),(9,108,54,81),(10,59,86,32),(11,60,87,33),(12,61,88,34),(13,62,89,35),(14,63,90,36),(15,55,82,28),(16,56,83,29),(17,57,84,30),(18,58,85,31),(19,71,98,44),(20,72,99,45),(21,64,91,37),(22,65,92,38),(23,66,93,39),(24,67,94,40),(25,68,95,41),(26,69,96,42),(27,70,97,43)], [(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)], [(1,32,37),(2,33,38),(3,34,39),(4,35,40),(5,36,41),(6,28,42),(7,29,43),(8,30,44),(9,31,45),(10,21,100),(11,22,101),(12,23,102),(13,24,103),(14,25,104),(15,26,105),(16,27,106),(17,19,107),(18,20,108),(46,59,64),(47,60,65),(48,61,66),(49,62,67),(50,63,68),(51,55,69),(52,56,70),(53,57,71),(54,58,72),(73,86,91),(74,87,92),(75,88,93),(76,89,94),(77,90,95),(78,82,96),(79,83,97),(80,84,98),(81,85,99)], [(1,46),(2,54),(3,53),(4,52),(5,51),(6,50),(7,49),(8,48),(9,47),(10,91),(11,99),(12,98),(13,97),(14,96),(15,95),(16,94),(17,93),(18,92),(19,88),(20,87),(21,86),(22,85),(23,84),(24,83),(25,82),(26,90),(27,89),(28,68),(29,67),(30,66),(31,65),(32,64),(33,72),(34,71),(35,70),(36,69),(37,59),(38,58),(39,57),(40,56),(41,55),(42,63),(43,62),(44,61),(45,60),(73,100),(74,108),(75,107),(76,106),(77,105),(78,104),(79,103),(80,102),(81,101)])

C4×C9⋊S3 is a maximal subgroup of
C36.38D6  C36.40D6  C72⋊S3  D18.D6  Dic18⋊S3  D12⋊D9  D6.D18  C4×S3×D9  C36⋊D6  C36.70D6  C36.27D6  C36.29D6
C4×C9⋊S3 is a maximal quotient of
C72⋊S3  C6.Dic18  C6.11D36

60 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D6A6B6C6D9A···9I12A···12H18A···18I36A···36R
order12223333444466669···912···1218···1836···36
size112727222211272722222···22···22···22···2

60 irreducible representations

dim11111222222222
type++++++++++
imageC1C2C2C2C4S3S3D6D6D9C4×S3C4×S3D18C4×D9
kernelC4×C9⋊S3C9⋊Dic3C3×C36C2×C9⋊S3C9⋊S3C36C3×C12C18C3×C6C12C9C32C6C3
# reps111143131962918

Matrix representation of C4×C9⋊S3 in GL4(𝔽37) generated by

31000
03100
00310
00031
,
61100
261700
00611
002617
,
36100
36000
00036
00136
,
261700
61100
0001
0010
G:=sub<GL(4,GF(37))| [31,0,0,0,0,31,0,0,0,0,31,0,0,0,0,31],[6,26,0,0,11,17,0,0,0,0,6,26,0,0,11,17],[36,36,0,0,1,0,0,0,0,0,0,1,0,0,36,36],[26,6,0,0,17,11,0,0,0,0,0,1,0,0,1,0] >;

C4×C9⋊S3 in GAP, Magma, Sage, TeX

C_4\times C_9\rtimes S_3
% in TeX

G:=Group("C4xC9:S3");
// GroupNames label

G:=SmallGroup(216,64);
// by ID

G=gap.SmallGroup(216,64);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,31,2115,453,1444,5189]);
// Polycyclic

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

׿
×
𝔽