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G = D36⋊C6  order 432 = 24·33

1st semidirect product of D36 and C6 acting faithfully

metabelian, supersoluble, monomial

Aliases: D36⋊1C6, 3- 1+2⋊2D8, C9⋊C8⋊2C6, D4⋊D9⋊C3, D4⋊(C9⋊C6), C9⋊2(C3×D8), C9⋊C24⋊2C2, (D4×C9)⋊1C6, D36⋊C3⋊1C2, C12.6(S3×C6), C36.2(C2×C6), C18.8(C3×D4), (C3×C12).11D6, C32.(D4⋊S3), (D4×C32).3S3, C2.5(Dic9⋊C6), (D4×3- 1+2)⋊1C2, (C2×3- 1+2).8D4, (C4×3- 1+2).2C22, C4.2(C2×C9⋊C6), C3.3(C3×D4⋊S3), (C3×D4).6(C3×S3), C6.23(C3×C3⋊D4), (C3×C6).26(C3⋊D4), SmallGroup(432,155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C36 — D36⋊C6
Chief series
C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — D36⋊C3 — D36⋊C6
Lower central
C9 — C18 — C36 — D36⋊C6
Upper central
C1 — C2 — C4 — D4

Generators and relations for D36⋊C6
 G = < a,b,c | a36=b2=c6=1, bab=a-1, cac-1=a7, cbc-1=a15b >

Subgroups: 334 in 74 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, D12, C3×D4, C3×D4, 3- 1+2, C36, C36, D18, C2×C18, C3×C12, S3×C6, C62, D4⋊S3, C3×D8, C9⋊C6, C2×3- 1+2, C2×3- 1+2, C9⋊C8, D36, D4×C9, D4×C9, C3×C3⋊C8, C3×D12, D4×C32, C4×3- 1+2, C2×C9⋊C6, C22×3- 1+2, D4⋊D9, C3×D4⋊S3, C9⋊C24, D36⋊C3, D4×3- 1+2, D36⋊C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, S3×C6, D4⋊S3, C3×D8, C9⋊C6, C3×C3⋊D4, C2×C9⋊C6, C3×D4⋊S3, Dic9⋊C6, D36⋊C6

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

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

41 conjugacy classes

class 1 2A2B2C3A3B3C 4 6A6B6C6D6E6F6G6H6I8A8B9A9B9C12A12B12C18A18B18C18D···18I24A24B24C24D36A36B36C
order122233346666666668899912121218181818···1824242424363636
size1143623322334412123636181866646666612···1218181818121212

41 irreducible representations

dim1111111112222222222244666
type++++++++++++
imageC1C2C2C2C3C6C6C6D36⋊C6S3D4D6D8C3×S3C3×D4C3⋊D4S3×C6C3×D8C3×C3⋊D4D4⋊S3C3×D4⋊S3C9⋊C6C2×C9⋊C6Dic9⋊C6
kernelD36⋊C6C9⋊C24D36⋊C3D4×3- 1+2D4⋊D9C9⋊C8D36D4×C9C1D4×C32C2×3- 1+2C3×C123- 1+2C3×D4C18C3×C6C12C9C6C32C3D4C4C2
# reps111122221111222224412112

Matrix representation of D36⋊C6 ►in GL10(𝔽73)

0102000000
721712000000
072072000000
172172000000
0000646456000
00007200000
0000119000
00009154972100
00001616710172
0000353269810
,
00320000000
003241000000
16000000000
165700000000
00004129542530
0000622102053
00006868537122
00008231434250
00005913751864
0000471252195052
,
1000000000
0100000000
720720000000
072072000000
0000100000
00000640000
0000898000
000034007200
00006742637290
0000246108065

G:=sub<GL(10,GF(73))| [0,72,0,1,0,0,0,0,0,0,1,1,72,72,0,0,0,0,0,0,0,71,0,1,0,0,0,0,0,0,2,2,72,72,0,0,0,0,0,0,0,0,0,0,64,72,1,9,16,35,0,0,0,0,64,0,1,15,16,32,0,0,0,0,56,0,9,49,71,69,0,0,0,0,0,0,0,72,0,8,0,0,0,0,0,0,0,10,1,1,0,0,0,0,0,0,0,0,72,0],[0,0,16,16,0,0,0,0,0,0,0,0,0,57,0,0,0,0,0,0,32,32,0,0,0,0,0,0,0,0,0,41,0,0,0,0,0,0,0,0,0,0,0,0,41,62,68,8,59,47,0,0,0,0,29,21,68,23,13,12,0,0,0,0,54,0,53,14,7,52,0,0,0,0,2,2,71,34,5,19,0,0,0,0,53,0,2,25,18,50,0,0,0,0,0,53,2,0,64,52],[1,0,72,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,1,0,8,34,67,24,0,0,0,0,0,64,9,0,42,61,0,0,0,0,0,0,8,0,63,0,0,0,0,0,0,0,0,72,72,8,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,65] >;
 

D36⋊C6 in GAP, Magma, Sage, TeX 

D_{36}\rtimes C_6
 
% in TeX
 
G:=Group("D36:C6");
 
// GroupNames label
 
G:=SmallGroup(432,155);
 
// by ID
 
G=gap.SmallGroup(432,155);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,1011,514,80,10085,2035,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c|a^36=b^2=c^6=1,b*a*b=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^15*b>;
 
// generators/relations
 

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