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G = C6.18D36  order 432 = 24·33

7th non-split extension by C6 of D36 acting via D36/D18=C2

metabelian, supersoluble, monomial

Aliases: C6.18D36, C18.17D12, C62.58D6, C6.4(C4×D9), C9⋊1(D6⋊C4), C18.5(C4×S3), C3⋊1(D18⋊C4), (C2×Dic3)⋊2D9, (C6×Dic9)⋊1C2, (C2×Dic9)⋊1S3, (C2×C6).13D18, (C3×C18).17D4, (C2×C18).13D6, (C3×C6).35D12, C6.6(C9⋊D4), C22.9(S3×D9), (Dic3×C18)⋊2C2, C18.6(C3⋊D4), (C6×C18).7C22, (C6×Dic3).4S3, C2.2(C9⋊D12), C2.3(C3⋊D36), C32.3(D6⋊C4), C6.5(C6.D6), C6.19(C3⋊D12), C2.5(C18.D6), C3.1(C6.D12), (C2×C9⋊S3)⋊1C4, (C2×C6).19S32, (C3×C9)⋊2(C22⋊C4), (C3×C6).41(C4×S3), (C3×C18).12(C2×C4), (C22×C9⋊S3).1C2, (C3×C6).52(C3⋊D4), SmallGroup(432,92)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C18 — C6.18D36
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — Dic3×C18 — C6.18D36
Lower central
C3×C9 — C3×C18 — C6.18D36
Upper central
C1 — C22

Generators and relations for C6.18D36
 G = < a,b,c | a6=b36=c2=1, bab-1=cac=a-1, cbc=a3b-1 >

Subgroups: 1052 in 134 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, D9, C18, C18, C3⋊S3, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C22×S3, C3×C9, Dic9, C36, D18, C2×C18, C2×C18, C3×Dic3, C2×C3⋊S3, C62, D6⋊C4, C9⋊S3, C3×C18, C2×Dic9, C2×C36, C22×D9, C6×Dic3, C6×Dic3, C22×C3⋊S3, C3×Dic9, C9×Dic3, C2×C9⋊S3, C2×C9⋊S3, C6×C18, D18⋊C4, C6.D12, C6×Dic9, Dic3×C18, C22×C9⋊S3, C6.18D36
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, D18, S32, D6⋊C4, C4×D9, D36, C9⋊D4, C6.D6, C3⋊D12, S3×D9, D18⋊C4, C6.D12, C18.D6, C3⋊D36, C9⋊D12, C6.18D36

Smallest permutation representation of C6.18D36
►On 72 points
Generators in S72
(1 44 13 56 25 68)(2 69 26 57 14 45)(3 46 15 58 27 70)(4 71 28 59 16 47)(5 48 17 60 29 72)(6 37 30 61 18 49)(7 50 19 62 31 38)(8 39 32 63 20 51)(9 52 21 64 33 40)(10 41 34 65 22 53)(11 54 23 66 35 42)(12 43 36 67 24 55)

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

66 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6F6G6H6I9A9B9C9D9E9F12A12B12C12D12E12F12G12H18A···18I18J···18R36A···36L
order12222233344446···6666999999121212121212121218···1818···1836···36
size111154542246618182···24442224446666181818182···24···46···6

66 irreducible representations

dim1111122222222222222224444444
type+++++++++++++++++++++
imageC1C2C2C2C4S3S3D4D6D6D9C4×S3D12C3⋊D4C4×S3D12C3⋊D4D18C4×D9D36C9⋊D4S32C6.D6C3⋊D12S3×D9C18.D6C3⋊D36C9⋊D12
kernelC6.18D36C6×Dic9Dic3×C18C22×C9⋊S3C2×C9⋊S3C2×Dic9C6×Dic3C3×C18C2×C18C62C2×Dic3C18C18C18C3×C6C3×C6C3×C6C2×C6C6C6C6C2×C6C6C6C22C2C2C2
# reps1111411211322222236661123333

Matrix representation of C6.18D36 ►in GL6(𝔽37)

3600000
0360000
000100
00363600
0000360
0000036
,
36350000
010000
0036000
001100
0000919
0000358
,
100000
36360000
001000
00363600
0000173
00001520

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,36,0,0,0,0,1,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[36,0,0,0,0,0,35,1,0,0,0,0,0,0,36,1,0,0,0,0,0,1,0,0,0,0,0,0,9,35,0,0,0,0,19,8],[1,36,0,0,0,0,0,36,0,0,0,0,0,0,1,36,0,0,0,0,0,36,0,0,0,0,0,0,17,15,0,0,0,0,3,20] >;
 

C6.18D36 in GAP, Magma, Sage, TeX 

C_6._{18}D_{36}
 
% in TeX
 
G:=Group("C6.18D36");
 
// GroupNames label
 
G:=SmallGroup(432,92);
 
// by ID
 
G=gap.SmallGroup(432,92);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,92,3091,662,4037,7069]);
 
// Polycyclic
 
G:=Group<a,b,c|a^6=b^36=c^2=1,b*a*b^-1=c*a*c=a^-1,c*b*c=a^3*b^-1>;
 
// generators/relations
 

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