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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, (C2xC18):3C12, C18.9(C2xC12), C18.D4:C3, (C2xDic9):2C6, C18.11(C3xD4), (C2xC62).3S3, C22:3(C9:C12), C23.3(C9:C6), (C22xC18).2C6, C6.20(C6xDic3), C2.3(Dic9:C6), C32.(C6.D4), 3- 1+2:2(C22:C4), (C2x3- 1+2).11D4, (C22x3- 1+2):1C4, (C23x3- 1+2).1C2, (C22x3- 1+2).6C22, (C2xC9:C12):2C2, C2.5(C2xC9:C12), C9:2(C3xC22:C4), (C2xC18).6(C2xC6), (C2xC6).48(S3xC6), C6.32(C3xC3:D4), C22.7(C2xC9:C6), (C3xC6).32(C3:D4), (C22xC6).26(C3xS3), (C2xC6).20(C3xDic3), (C3xC6).15(C2xDic3), C3.3(C3xC6.D4), (C2x3- 1+2).9(C2xC4), SmallGroup(432,167)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C18 — C62.27D6
Chief series
C1 — C3 — C9 — C18 — C2xC18 — C22x3- 1+2 — C2xC9: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, C2xC4, C23, C9, C9, C32, Dic3, C12, C2xC6, C2xC6, C2xC6, C22:C4, C18, C18, C18, C3xC6, C3xC6, C3xC6, C2xDic3, C2xC12, C22xC6, C22xC6, 3- 1+2, Dic9, C2xC18, C2xC18, C2xC18, C3xDic3, C62, C62, C62, C6.D4, C3xC22:C4, C2x3- 1+2, C2x3- 1+2, C2x3- 1+2, C2xDic9, C22xC18, C22xC18, C6xDic3, C2xC62, C9:C12, C22x3- 1+2, C22x3- 1+2, C22x3- 1+2, C18.D4, C3xC6.D4, C2xC9:C12, C23x3- 1+2, C62.27D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, Dic3, C12, D6, C2xC6, C22:C4, C3xS3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xDic3, S3xC6, C6.D4, C3xC22:C4, C9:C6, C6xDic3, C3xC3:D4, C9:C12, C2xC9:C6, C3xC6.D4, C2xC9: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+++++-++-+
imageC1C2C2C3C4C6C6C12S3D4Dic3D6C3xS3C3xD4C3:D4C3xDic3S3xC6C3xC3:D4C9:C6C9:C12C2xC9:C6Dic9:C6
kernelC62.27D6C2xC9:C12C23x3- 1+2C18.D4C22x3- 1+2C2xDic9C22xC18C2xC18C2xC62C2x3- 1+2C62C62C22xC6C18C3xC6C2xC6C2xC6C6C23C22C22C2
# 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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