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G = C36.40D6  order 432 = 24·33

11st non-split extension by C36 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: C36.40D6, C12.40D18, C9⋊C8⋊4S3, C3⋊C8⋊5D9, C12.62S32, C6.2(C4×D9), C18.2(C4×S3), C4.25(S3×D9), C9⋊1(C8⋊S3), (C3×C9)⋊2M4(2), C3⋊1(C8⋊D9), C9⋊Dic3.2C4, (C3×C12).158D6, (C3×C36).39C22, C6.2(C6.D6), C32.3(C8⋊S3), C2.3(C18.D6), C3.1(C12.31D6), (C3×C9⋊C8)⋊8C2, (C9×C3⋊C8)⋊7C2, (C4×C9⋊S3).4C2, (C2×C9⋊S3).2C4, (C3×C3⋊C8).10S3, (C3×C18).4(C2×C4), (C3×C6).36(C4×S3), SmallGroup(432,61)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C18 — C36.40D6
Chief series
C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×C3⋊C8 — C36.40D6
Lower central
C3×C9 — C3×C18 — C36.40D6
Upper central
C1 — C4

Generators and relations for C36.40D6
 G = < a,b,c | a36=c2=1, b6=a27, bab-1=cac=a17, cbc=b5 >

Subgroups: 500 in 76 conjugacy classes, 27 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, M4(2), D9, C18, C18, C3⋊S3, C3×C6, C3⋊C8, C3⋊C8, C24, C4×S3, C3×C9, Dic9, C36, C36, D18, C3⋊Dic3, C3×C12, C2×C3⋊S3, C8⋊S3, C9⋊S3, C3×C18, C9⋊C8, C72, C4×D9, C3×C3⋊C8, C3×C3⋊C8, C4×C3⋊S3, C9⋊Dic3, C3×C36, C2×C9⋊S3, C8⋊D9, C12.31D6, C3×C9⋊C8, C9×C3⋊C8, C4×C9⋊S3, C36.40D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D6, M4(2), D9, C4×S3, D18, S32, C8⋊S3, C4×D9, C6.D6, S3×D9, C8⋊D9, C12.31D6, C18.D6, C36.40D6

Smallest permutation representation of C36.40D6
►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 59 22 56 7 53 28 50 13 47 34 44 19 41 4 38 25 71 10 68 31 65 16 62)(2 40 23 37 8 70 29 67 14 64 35 61 20 58 5 55 26 52 11 49 32 46 17 43)(3 57 24 54 9 51 30 48 15 45 36 42 21 39 6 72 27 69 12 66 33 63 18 60)

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

66 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C8A8B8C8D9A9B9C9D9E9F12A12B12C12D12E12F18A18B18C18D18E18F24A24B24C24D24E24F24G24H36A···36F36G···36L72A···72L
order1223334446668888999999121212121212181818181818242424242424242436···3636···3672···72
size115422411542246618182224442222442224446666181818182···24···46···6

66 irreducible representations

dim1111112222222222222444444
type++++++++++++++
imageC1C2C2C2C4C4S3S3D6D6M4(2)D9C4×S3C4×S3D18C8⋊S3C8⋊S3C4×D9C8⋊D9S32C6.D6S3×D9C12.31D6C18.D6C36.40D6
kernelC36.40D6C3×C9⋊C8C9×C3⋊C8C4×C9⋊S3C9⋊Dic3C2×C9⋊S3C9⋊C8C3×C3⋊C8C36C3×C12C3×C9C3⋊C8C18C3×C6C12C9C32C6C3C12C6C4C3C2C1
# reps11112211112322344612113236

Matrix representation of C36.40D6 ►in GL6(𝔽73)

4600000
0460000
001000
000100
00007042
00003128
,
4910000
54240000
0017200
001000
0000460
00002727
,
100000
48720000
0072000
0072100
0000720
000011

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,70,31,0,0,0,0,42,28],[49,54,0,0,0,0,1,24,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,27,0,0,0,0,0,27],[1,48,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,0,1] >;
 

C36.40D6 in GAP, Magma, Sage, TeX 

C_{36}._{40}D_6
 
% in TeX
 
G:=Group("C36.40D6");
 
// GroupNames label
 
G:=SmallGroup(432,61);
 
// by ID
 
G=gap.SmallGroup(432,61);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,28,141,36,58,3091,662,4037,7069]);
 
// Polycyclic
 
G:=Group<a,b,c|a^36=c^2=1,b^6=a^27,b*a*b^-1=c*a*c=a^17,c*b*c=b^5>;
 
// generators/relations
 

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