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G = D36⋊3C6  order 432 = 24·33

3rd semidirect product of D36 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: D36⋊3C6, (C4×D9)⋊3C6, (Q8×C9)⋊3C6, D36⋊C3⋊3C2, C36.7(C2×C6), Q8⋊4(C9⋊C6), C12.24(S3×C6), Q8⋊3D9⋊1C3, D18.3(C2×C6), (C3×C12).31D6, C9⋊C12.5C22, C18.8(C22×C6), Dic9.5(C2×C6), C32.(Q8⋊3S3), (Q8×C32).14S3, (Q8×3- 1+2)⋊3C2, 3- 1+2⋊3(C4○D4), (C4×3- 1+2).7C22, (C2×3- 1+2).8C23, (C4×C9⋊C6)⋊3C2, C4.7(C2×C9⋊C6), C9⋊3(C3×C4○D4), C6.42(S3×C2×C6), C2.9(C22×C9⋊C6), (C2×C9⋊C6).3C22, (C3×Q8).32(C3×S3), C3.3(C3×Q8⋊3S3), (C3×C6).34(C22×S3), SmallGroup(432,371)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — D36⋊3C6
Chief series
C1 — C3 — C9 — C18 — C2×3- 1+2 — C2×C9⋊C6 — C4×C9⋊C6 — D36⋊3C6
Lower central
C9 — C18 — D36⋊3C6
Upper central
C1 — C2 — Q8

Generators and relations for D36⋊3C6
 G = < a,b,c | a36=b2=c6=1, bab=a-1, cac-1=a29, cbc-1=a10b >

Subgroups: 502 in 126 conjugacy classes, 52 normal (19 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, D9, C18, C18, C3×S3, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, 3- 1+2, Dic9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, Q8⋊3S3, C3×C4○D4, C9⋊C6, C2×3- 1+2, C4×D9, D36, Q8×C9, Q8×C9, S3×C12, C3×D12, Q8×C32, C9⋊C12, C4×3- 1+2, C2×C9⋊C6, Q8⋊3D9, C3×Q8⋊3S3, C4×C9⋊C6, D36⋊C3, Q8×3- 1+2, D36⋊3C6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, Q8⋊3S3, C3×C4○D4, C9⋊C6, S3×C2×C6, C2×C9⋊C6, C3×Q8⋊3S3, C22×C9⋊C6, D36⋊3C6

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

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

50 conjugacy classes

class 1 2A2B2C2D3A3B3C4A4B4C4D4E6A6B6C6D···6I9A9B9C12A12B12C12D···12I12J12K12L12M18A18B18C36A···36I
order12222333444446666···699912121212···121212121218181836···36
size111818182332229923318···186664446···6999966612···12

50 irreducible representations

dim11111111122222224466
type++++++++++
imageC1C2C2C2C3C6C6C6D36⋊3C6S3D6C4○D4C3×S3S3×C6C3×C4○D4Q8⋊3S3C3×Q8⋊3S3C9⋊C6C2×C9⋊C6
kernelD36⋊3C6C4×C9⋊C6D36⋊C3Q8×3- 1+2Q8⋊3D9C4×D9D36Q8×C9C1Q8×C32C3×C123- 1+2C3×Q8C12C9C32C3Q8C4
# reps1331266211322641213

Matrix representation of D36⋊3C6 ►in GL10(𝔽37)

1111000000
360360000000
35353636000000
2010000000
00000000136
0000000010
00003600000
00000360000
00000036000
00000003600
,
360360000000
1111000000
0010000000
003636000000
00000000360
00000000361
00000036000
00000036100
00003600000
00003610000
,
023023000000
230230000000
028014000000
280140000000
00000360000
00003600000
00000000036
00000000360
00000036100
0000000100

G:=sub<GL(10,GF(37))| [1,36,35,2,0,0,0,0,0,0,1,0,35,0,0,0,0,0,0,0,1,36,36,1,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,1,1,0,0,0,0,0,0,0,0,36,0,0,0,0,0],[36,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,1,1,36,0,0,0,0,0,0,0,1,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,36,36,0,0,0,0,0,0,0,0,0,1,0,0,0,0],[0,23,0,28,0,0,0,0,0,0,23,0,28,0,0,0,0,0,0,0,0,23,0,14,0,0,0,0,0,0,23,0,14,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,36,0,0,0] >;
 

D36⋊3C6 in GAP, Magma, Sage, TeX 

D_{36}\rtimes_3C_6
 
% in TeX
 
G:=Group("D36:3C6");
 
// GroupNames label
 
G:=SmallGroup(432,371);
 
// by ID
 
G=gap.SmallGroup(432,371);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,10085,1034,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c|a^36=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^29,c*b*c^-1=a^10*b>;
 
// generators/relations
 

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