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G = C62.27D6  order 432 = 24·33

10th non-split extension by C62 of D6 acting via D6/C2=S3

metabelian, supersoluble, monomial

Aliases: C62.27D6, C62.5Dic3, (C2×C18)⋊3C12, C18.9(C2×C12), C18.D4⋊C3, (C2×Dic9)⋊2C6, C18.11(C3×D4), (C2×C62).3S3, C22⋊3(C9⋊C12), C23.3(C9⋊C6), (C22×C18).2C6, C6.20(C6×Dic3), C2.3(Dic9⋊C6), C32.(C6.D4), 3- 1+2⋊2(C22⋊C4), (C2×3- 1+2).11D4, (C22×3- 1+2)⋊1C4, (C23×3- 1+2).1C2, (C22×3- 1+2).6C22, (C2×C9⋊C12)⋊2C2, C2.5(C2×C9⋊C12), C9⋊2(C3×C22⋊C4), (C2×C18).6(C2×C6), (C2×C6).48(S3×C6), C6.32(C3×C3⋊D4), C22.7(C2×C9⋊C6), (C3×C6).32(C3⋊D4), (C22×C6).26(C3×S3), (C2×C6).20(C3×Dic3), (C3×C6).15(C2×Dic3), C3.3(C3×C6.D4), (C2×3- 1+2).9(C2×C4), SmallGroup(432,167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — C62.27D6
Chief series
C1 — C3 — C9 — C18 — C2×C18 — C22×3- 1+2 — C2×C9⋊C12 — C62.27D6
Lower central
C9 — C18 — C62.27D6
Upper central
C1 — C22 — C23

Generators and relations for C62.27D6
 G = < a,b,c,d | a6=b6=1, c6=b4, d2=a3b3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b-1c5 >

Subgroups: 342 in 118 conjugacy classes, 46 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C9, C9, C32, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C18, C18, C18, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×C6, C22×C6, 3- 1+2, Dic9, C2×C18, C2×C18, C2×C18, C3×Dic3, C62, C62, C62, C6.D4, C3×C22⋊C4, C2×3- 1+2, C2×3- 1+2, C2×3- 1+2, C2×Dic9, C22×C18, C22×C18, C6×Dic3, C2×C62, C9⋊C12, C22×3- 1+2, C22×3- 1+2, C22×3- 1+2, C18.D4, C3×C6.D4, C2×C9⋊C12, C23×3- 1+2, C62.27D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Dic3, C12, D6, C2×C6, C22⋊C4, C3×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Dic3, S3×C6, C6.D4, C3×C22⋊C4, C9⋊C6, C6×Dic3, C3×C3⋊D4, C9⋊C12, C2×C9⋊C6, C3×C6.D4, C2×C9⋊C12, Dic9⋊C6, C62.27D6

Smallest permutation representation of C62.27D6
►On 72 points
Generators in S72
(1 10)(2 5 8 11 14 17)(3 18 15 12 9 6)(4 13)(7 16)(19 70 31 64 25 58)(20 65)(21 60 27 66 33 72)(22 55 34 67 28 61)(23 68)(24 63 30 69 36 57)(26 71)(29 56)(32 59)(35 62)(37 52 49 46 43 40)(38 47)(39 42 45 48 51 54)(41 50)(44 53)

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

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D6A···6G6H···6M6N6O6P6Q9A9B9C12A···12H18A···18U
order12222233344446···66···6666699912···1218···18
size111122233181818182···23···3666666618···186···6

62 irreducible representations

dim1111111122222222226666
type+++++-++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3×S3C3×D4C3⋊D4C3×Dic3S3×C6C3×C3⋊D4C9⋊C6C9⋊C12C2×C9⋊C6Dic9⋊C6
kernelC62.27D6C2×C9⋊C12C23×3- 1+2C18.D4C22×3- 1+2C2×Dic9C22×C18C2×C18C2×C62C2×3- 1+2C62C62C22×C6C18C3×C6C2×C6C2×C6C6C23C22C22C2
# reps1212442812212444281214

Matrix representation of C62.27D6 ►in GL8(𝔽37)

110000000
011000000
003626360831
000110000
000027000
000003600
000000110
000000027
,
10000000
01000000
002700141414
000270000
000027000
000001100
000000110
000000011
,
3636000000
10000000
0036026322626
002811666
000360000
00000010
00000001
000001000
,
60000000
3131000000
00140673333
000000011
000003600
000036000
00211010232323
000270000

G:=sub<GL(8,GF(37))| [11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,36,0,0,0,0,0,0,0,26,11,0,0,0,0,0,0,36,0,27,0,0,0,0,0,0,0,0,36,0,0,0,0,8,0,0,0,11,0,0,0,31,0,0,0,0,27],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,27,0,0,0,0,0,14,0,0,11,0,0,0,0,14,0,0,0,11,0,0,0,14,0,0,0,0,11],[36,1,0,0,0,0,0,0,36,0,0,0,0,0,0,0,0,0,36,28,0,0,0,0,0,0,0,1,36,0,0,0,0,0,26,1,0,0,0,0,0,0,32,6,0,0,0,10,0,0,26,6,0,1,0,0,0,0,26,6,0,0,1,0],[6,31,0,0,0,0,0,0,0,31,0,0,0,0,0,0,0,0,14,0,0,0,21,0,0,0,0,0,0,0,10,27,0,0,6,0,0,36,10,0,0,0,7,0,36,0,23,0,0,0,33,0,0,0,23,0,0,0,33,11,0,0,23,0] >;
 

C62.27D6 in GAP, Magma, Sage, TeX 

C_6^2._{27}D_6
 
% in TeX
 
G:=Group("C6^2.27D6");
 
// GroupNames label
 
G:=SmallGroup(432,167);
 
// by ID
 
G=gap.SmallGroup(432,167);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,365,10085,2035,292,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=b^4,d^2=a^3*b^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=b^-1*c^5>;
 
// generators/relations
 

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