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G = C6.D36order 432 = 24·33

2nd non-split extension by C6 of D36 acting via D36/D18=C2

metabelian, supersoluble, monomial

Aliases: C6.13D36, C36.31D6, Dic181S3, C12.10D18, C3⋊C82D9, C12.45S32, C4.9(S3×D9), (C3×C9)⋊2SD16, (C3×C18).2D4, C32(C72⋊C2), (C3×C12).70D6, (C3×C6).30D12, C36⋊S3.1C2, C91(Q82S3), (C3×Dic18)⋊1C2, C18.2(C3⋊D4), (C3×C36).2C22, C6.2(C3⋊D12), C2.5(C3⋊D36), C32.3(C24⋊C2), C3.1(C325SD16), (C9×C3⋊C8)⋊3C2, (C3×C3⋊C8).5S3, SmallGroup(432,63)

Series: Derived Chief Lower central Upper central

C1C3×C36 — C6.D36
C1C3C9C3×C9C3×C18C3×C36C3×Dic18 — C6.D36
C3×C9C3×C18C3×C36 — C6.D36
C1C2C4

Generators and relations for C6.D36
 G = < a,b,c | a36=c2=1, b6=a18, bab-1=cac=a-1, cbc=a27b5 >

Subgroups: 704 in 76 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, SD16, D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×Q8, C3×C9, Dic9, C36, C36, D18, C3×Dic3, C3×C12, C2×C3⋊S3, C24⋊C2, Q82S3, C9⋊S3, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×Dic6, C12⋊S3, C3×Dic9, C3×C36, C2×C9⋊S3, C72⋊C2, C325SD16, C9×C3⋊C8, C3×Dic18, C36⋊S3, C6.D36
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, Q82S3, D36, C3⋊D12, S3×D9, C72⋊C2, C325SD16, C3⋊D36, C6.D36

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

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

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

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

60 conjugacy classes

class 1 2A2B3A3B3C4A4B6A6B6C8A8B9A9B9C9D9E9F12A12B12C12D12E12F12G18A18B18C18D18E18F24A24B24C24D36A···36F36G···36L72A···72L
order1223334466688999999121212121212121818181818182424242436···3636···3672···72
size111082242362246622244422444363622244466662···24···46···6

60 irreducible representations

dim111122222222222224444444
type++++++++++++++++++++
imageC1C2C2C2S3S3D4D6D6SD16D9C3⋊D4D12D18C24⋊C2D36C72⋊C2S32Q82S3C3⋊D12S3×D9C325SD16C3⋊D36C6.D36
kernelC6.D36C9×C3⋊C8C3×Dic18C36⋊S3Dic18C3×C3⋊C8C3×C18C36C3×C12C3×C9C3⋊C8C18C3×C6C12C32C6C3C12C9C6C4C3C2C1
# reps1111111112322346121113236

Matrix representation of C6.D36 in GL6(𝔽73)

2130000
23520000
001000
000100
0000352
00001145
,
47180000
7260000
0017200
001000
0000720
000021
,
52700000
25210000
0072000
0072100
0000720
000021

G:=sub<GL(6,GF(73))| [21,23,0,0,0,0,3,52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,11,0,0,0,0,52,45],[47,7,0,0,0,0,18,26,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,2,0,0,0,0,0,1],[52,25,0,0,0,0,70,21,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,2,0,0,0,0,0,1] >;

C6.D36 in GAP, Magma, Sage, TeX

C_6.D_{36}
% in TeX

G:=Group("C6.D36");
// GroupNames label

G:=SmallGroup(432,63);
// by ID

G=gap.SmallGroup(432,63);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,36,254,58,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^36=c^2=1,b^6=a^18,b*a*b^-1=c*a*c=a^-1,c*b*c=a^27*b^5>;
// generators/relations

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