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G = C36⋊S3order 216 = 23·33

1st semidirect product of C36 and S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: C361S3, C31D36, C121D9, C91D12, C18.16D6, C6.16D18, C32.4D12, C4⋊(C9⋊S3), (C3×C9)⋊6D4, (C3×C36)⋊3C2, (C3×C6).52D6, C12.2(C3⋊S3), C3.(C12⋊S3), (C3×C12).10S3, (C3×C18).20C22, (C2×C9⋊S3)⋊3C2, C2.4(C2×C9⋊S3), C6.10(C2×C3⋊S3), SmallGroup(216,65)

Series: Derived Chief Lower central Upper central

C1C3×C18 — C36⋊S3
C1C3C32C3×C9C3×C18C2×C9⋊S3 — C36⋊S3
C3×C9C3×C18 — C36⋊S3
C1C2C4

Generators and relations for C36⋊S3
 G = < a,b,c | a36=b3=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 562 in 80 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C3, C3 [×3], C4, C22 [×2], S3 [×8], C6, C6 [×3], D4, C9 [×3], C32, C12, C12 [×3], D6 [×8], D9 [×6], C18 [×3], C3⋊S3 [×2], C3×C6, D12 [×4], C3×C9, C36 [×3], D18 [×6], C3×C12, C2×C3⋊S3 [×2], C9⋊S3 [×2], C3×C18, D36 [×3], C12⋊S3, C3×C36, C2×C9⋊S3 [×2], C36⋊S3
Quotients: C1, C2 [×3], C22, S3 [×4], D4, D6 [×4], D9 [×3], C3⋊S3, D12 [×4], D18 [×3], C2×C3⋊S3, C9⋊S3, D36 [×3], C12⋊S3, C2×C9⋊S3, C36⋊S3

Smallest permutation representation of C36⋊S3
On 108 points
Generators in S108
(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 90 67)(2 91 68)(3 92 69)(4 93 70)(5 94 71)(6 95 72)(7 96 37)(8 97 38)(9 98 39)(10 99 40)(11 100 41)(12 101 42)(13 102 43)(14 103 44)(15 104 45)(16 105 46)(17 106 47)(18 107 48)(19 108 49)(20 73 50)(21 74 51)(22 75 52)(23 76 53)(24 77 54)(25 78 55)(26 79 56)(27 80 57)(28 81 58)(29 82 59)(30 83 60)(31 84 61)(32 85 62)(33 86 63)(34 87 64)(35 88 65)(36 89 66)
(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 25)(14 24)(15 23)(16 22)(17 21)(18 20)(37 84)(38 83)(39 82)(40 81)(41 80)(42 79)(43 78)(44 77)(45 76)(46 75)(47 74)(48 73)(49 108)(50 107)(51 106)(52 105)(53 104)(54 103)(55 102)(56 101)(57 100)(58 99)(59 98)(60 97)(61 96)(62 95)(63 94)(64 93)(65 92)(66 91)(67 90)(68 89)(69 88)(70 87)(71 86)(72 85)

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

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

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

C36⋊S3 is a maximal subgroup of
C6.D36  C3⋊D72  C9⋊D24  C18.D12  C24⋊D9  C721S3  C36.18D6  C36.20D6  Dic65D9  Dic9.D6  S3×D36  D9×D12  C36.70D6  D4×C9⋊S3  C36.29D6
C36⋊S3 is a maximal quotient of
C24.D9  C24⋊D9  C721S3  C36⋊Dic3  C6.11D36

57 conjugacy classes

class 1 2A2B2C3A3B3C3D 4 6A6B6C6D9A···9I12A···12H18A···18I36A···36R
order12223333466669···912···1218···1836···36
size1154542222222222···22···22···22···2

57 irreducible representations

dim1112222222222
type+++++++++++++
imageC1C2C2S3S3D4D6D6D9D12D12D18D36
kernelC36⋊S3C3×C36C2×C9⋊S3C36C3×C12C3×C9C18C3×C6C12C9C32C6C3
# reps11231131962918

Matrix representation of C36⋊S3 in GL4(𝔽37) generated by

61700
202600
00425
001229
,
1000
0100
00036
00136
,
111700
62600
00361
0001
G:=sub<GL(4,GF(37))| [6,20,0,0,17,26,0,0,0,0,4,12,0,0,25,29],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,36,36],[11,6,0,0,17,26,0,0,0,0,36,0,0,0,1,1] >;

C36⋊S3 in GAP, Magma, Sage, TeX

C_{36}\rtimes S_3
% in TeX

G:=Group("C36:S3");
// GroupNames label

G:=SmallGroup(216,65);
// by ID

G=gap.SmallGroup(216,65);
# by ID

G:=PCGroup([6,-2,-2,-2,-3,-3,-3,73,31,2115,453,1444,5189]);
// Polycyclic

G:=Group<a,b,c|a^36=b^3=c^2=1,a*b=b*a,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations

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