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G = C2×D36⋊C3  order 432 = 24·33

Direct product of C2 and D36⋊C3

direct product, metabelian, supersoluble, monomial

Aliases: C2×D36⋊C3, D36⋊5C6, C62.39D6, (C2×D36)⋊C3, C9⋊1(C6×D4), (C2×C36)⋊2C6, C36⋊2(C2×C6), C18⋊1(C3×D4), D18⋊1(C2×C6), C3.3(C6×D12), C32.(C2×D12), C12.79(S3×C6), (C6×C12).14S3, (C3×C12).52D6, C6.18(C3×D12), (C3×C6).28D12, (C22×D9)⋊1C6, C18.3(C22×C6), 3- 1+2⋊1(C2×D4), (C2×3- 1+2)⋊1D4, (C4×3- 1+2)⋊2C22, (C2×3- 1+2).3C23, (C22×3- 1+2).9C22, C4⋊2(C2×C9⋊C6), C6.29(S3×C2×C6), (C2×C4)⋊2(C9⋊C6), (C22×C9⋊C6)⋊1C2, (C2×C9⋊C6)⋊1C22, (C2×C18).9(C2×C6), (C2×C6).59(S3×C6), C2.4(C22×C9⋊C6), (C2×C12).19(C3×S3), C22.10(C2×C9⋊C6), (C3×C6).25(C22×S3), (C2×C4×3- 1+2)⋊2C2, SmallGroup(432,354)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — C2×D36⋊C3
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C22×C9⋊C6 — C2×D36⋊C3
Lower central
C9 — C18 — C2×D36⋊C3
Upper central
C1 — C22 — C2×C4

Generators and relations for C2×D36⋊C3
 G = < a,b,c,d | a2=b36=c2=d3=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b13, dcd-1=b12c >

Subgroups: 782 in 170 conjugacy classes, 62 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, D4, C23, C9, C9, C32, C12, C12, D6, C2×C6, C2×C6, C2×D4, D9, C18, C18, C18, C3×S3, C3×C6, C3×C6, D12, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, 3- 1+2, C36, C36, D18, D18, C2×C18, C2×C18, C3×C12, S3×C6, C62, C2×D12, C6×D4, C9⋊C6, C2×3- 1+2, C2×3- 1+2, D36, C2×C36, C2×C36, C22×D9, C3×D12, C6×C12, S3×C2×C6, C4×3- 1+2, C2×C9⋊C6, C2×C9⋊C6, C22×3- 1+2, C2×D36, C6×D12, D36⋊C3, C2×C4×3- 1+2, C22×C9⋊C6, C2×D36⋊C3
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, S3×C6, C2×D12, C6×D4, C9⋊C6, C3×D12, S3×C2×C6, C2×C9⋊C6, C6×D12, D36⋊C3, C22×C9⋊C6, C2×D36⋊C3

Smallest permutation representation of C2×D36⋊C3
►On 72 points
Generators in S72
(1 63)(2 64)(3 65)(4 66)(5 67)(6 68)(7 69)(8 70)(9 71)(10 72)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)(27 53)(28 54)(29 55)(30 56)(31 57)(32 58)(33 59)(34 60)(35 61)(36 62)

(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)

(1 53)(2 52)(3 51)(4 50)(5 49)(6 48)(7 47)(8 46)(9 45)(10 44)(11 43)(12 42)(13 41)(14 40)(15 39)(16 38)(17 37)(18 72)(19 71)(20 70)(21 69)(22 68)(23 67)(24 66)(25 65)(26 64)(27 63)(28 62)(29 61)(30 60)(31 59)(32 58)(33 57)(34 56)(35 55)(36 54)

(2 26 14)(3 15 27)(5 29 17)(6 18 30)(8 32 20)(9 21 33)(11 35 23)(12 24 36)(37 61 49)(38 50 62)(40 64 52)(41 53 65)(43 67 55)(44 56 68)(46 70 58)(47 59 71)
 
G:=sub<Sym(72)| (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54), (2,26,14)(3,15,27)(5,29,17)(6,18,30)(8,32,20)(9,21,33)(11,35,23)(12,24,36)(37,61,49)(38,50,62)(40,64,52)(41,53,65)(43,67,55)(44,56,68)(46,70,58)(47,59,71)>;
 
G:=Group( (1,63)(2,64)(3,65)(4,66)(5,67)(6,68)(7,69)(8,70)(9,71)(10,72)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52)(27,53)(28,54)(29,55)(30,56)(31,57)(32,58)(33,59)(34,60)(35,61)(36,62), (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), (1,53)(2,52)(3,51)(4,50)(5,49)(6,48)(7,47)(8,46)(9,45)(10,44)(11,43)(12,42)(13,41)(14,40)(15,39)(16,38)(17,37)(18,72)(19,71)(20,70)(21,69)(22,68)(23,67)(24,66)(25,65)(26,64)(27,63)(28,62)(29,61)(30,60)(31,59)(32,58)(33,57)(34,56)(35,55)(36,54), (2,26,14)(3,15,27)(5,29,17)(6,18,30)(8,32,20)(9,21,33)(11,35,23)(12,24,36)(37,61,49)(38,50,62)(40,64,52)(41,53,65)(43,67,55)(44,56,68)(46,70,58)(47,59,71) );
 
G=PermutationGroup([[(1,63),(2,64),(3,65),(4,66),(5,67),(6,68),(7,69),(8,70),(9,71),(10,72),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52),(27,53),(28,54),(29,55),(30,56),(31,57),(32,58),(33,59),(34,60),(35,61),(36,62)], [(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)], [(1,53),(2,52),(3,51),(4,50),(5,49),(6,48),(7,47),(8,46),(9,45),(10,44),(11,43),(12,42),(13,41),(14,40),(15,39),(16,38),(17,37),(18,72),(19,71),(20,70),(21,69),(22,68),(23,67),(24,66),(25,65),(26,64),(27,63),(28,62),(29,61),(30,60),(31,59),(32,58),(33,57),(34,56),(35,55),(36,54)], [(2,26,14),(3,15,27),(5,29,17),(6,18,30),(8,32,20),(9,21,33),(11,35,23),(12,24,36),(37,61,49),(38,50,62),(40,64,52),(41,53,65),(43,67,55),(44,56,68),(46,70,58),(47,59,71)]])
 

62 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B6A6B6C6D···6I6J···6Q9A9B9C12A12B12C12D12E12F12G12H18A···18I36A···36L
order12222222333446666···66···6999121212121212121218···1836···36
size111118181818233222223···318···18666222266666···66···6

62 irreducible representations

dim1111111122222222226666
type+++++++++++++
imageC1C2C2C2C3C6C6C6S3D4D6D6C3×S3C3×D4D12S3×C6S3×C6C3×D12C9⋊C6C2×C9⋊C6C2×C9⋊C6D36⋊C3
kernelC2×D36⋊C3D36⋊C3C2×C4×3- 1+2C22×C9⋊C6C2×D36D36C2×C36C22×D9C6×C12C2×3- 1+2C3×C12C62C2×C12C18C3×C6C12C2×C6C6C2×C4C4C22C2
# reps1412282412212444281214

Matrix representation of C2×D36⋊C3 ►in GL8(𝔽37)

360000000
036000000
00100000
00010000
00001000
00000100
00000010
00000001
,
036000000
136000000
000000105
000000325
0032270000
001050000
0000322700
000010500
,
361000000
01000000
000000036
000000360
000003600
000036000
000360000
003600000
,
100000000
010000000
00100000
00010000
0000363600
00001000
00000001
0000003636

G:=sub<GL(8,GF(37))| [36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,0,32,10,0,0,0,0,0,0,27,5,0,0,0,0,0,0,0,0,32,10,0,0,0,0,0,0,27,5,0,0,10,32,0,0,0,0,0,0,5,5,0,0,0,0],[36,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0],[10,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,1,36] >;
 

C2×D36⋊C3 in GAP, Magma, Sage, TeX 

C_2\times D_{36}\rtimes C_3
 
% in TeX
 
G:=Group("C2xD36:C3");
 
// GroupNames label
 
G:=SmallGroup(432,354);
 
// by ID
 
G=gap.SmallGroup(432,354);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,590,142,10085,1034,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^36=c^2=d^3=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^13,d*c*d^-1=b^12*c>;
 
// generators/relations
 

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