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G = C62.21D6  order 432 = 24·33

4th non-split extension by C62 of D6 acting via D6/C2=S3

metabelian, supersoluble, monomial

Aliases: C62.21D6, (C6xC12):1S3, (C6xC12):1C6, C6.16(S3xC12), C6.11D12:C3, C6.13(C3xD12), (C3xC6).20D12, C62.7(C2xC6), C32:5(D6:C4), He3:5(C22:C4), (C2xHe3).21D4, C2.2(He3:4D4), C2.2(He3:6D4), (C22xHe3).19C22, (C2xC3:S3):C12, (C2xC4xHe3):1C2, C3.2(C3xD6:C4), (C2xC32:C6):2C4, (C2xC6).41(S3xC6), (C3xC6).4(C2xC12), (C3xC6).16(C4xS3), (C2xC12).6(C3xS3), (C2xC3:Dic3):1C6, (C3xC6).10(C3xD4), C6.15(C3xC3:D4), C2.5(C4xC32:C6), (C2xC32:C12):3C2, (C2xC4):1(C32:C6), (C22xC3:S3).1C6, C32:2(C3xC22:C4), (C3xC6).20(C3:D4), (C2xHe3).20(C2xC4), C22.6(C2xC32:C6), (C22xC32:C6).2C2, SmallGroup(432,141)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C62.21D6
Chief series
C1 — C3 — C32 — C3xC6 — C62 — C22xHe3 — C22xC32:C6 — C62.21D6
Lower central
C32 — C3xC6 — C62.21D6
Upper central
C1 — C22 — C2xC4

Generators and relations for C62.21D6
 G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=b3c5 >

Subgroups: 677 in 135 conjugacy classes, 40 normal (36 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2xC4, C2xC4, C23, C32, C32, Dic3, C12, D6, C2xC6, C2xC6, C22:C4, C3xS3, C3:S3, C3xC6, C3xC6, C2xDic3, C2xC12, C2xC12, C22xS3, C22xC6, He3, C3xDic3, C3:Dic3, C3xC12, S3xC6, C2xC3:S3, C2xC3:S3, C62, C62, D6:C4, C3xC22:C4, C32:C6, C2xHe3, C6xDic3, C2xC3:Dic3, C6xC12, C6xC12, S3xC2xC6, C22xC3:S3, C32:C12, C4xHe3, C2xC32:C6, C2xC32:C6, C22xHe3, C3xD6:C4, C6.11D12, C2xC32:C12, C2xC4xHe3, C22xC32:C6, C62.21D6
Quotients: C1, C2, C3, C4, C22, S3, C6, C2xC4, D4, C12, D6, C2xC6, C22:C4, C3xS3, C4xS3, D12, C3:D4, C2xC12, C3xD4, S3xC6, D6:C4, C3xC22:C4, C32:C6, S3xC12, C3xD12, C3xC3:D4, C2xC32:C6, C3xD6:C4, C4xC32:C6, He3:4D4, He3:6D4, C62.21D6

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E3F4A4B4C4D6A6B6C6D···6I6J···6R6S6T6U6V12A12B12C12D12E···12T12U12V12W12X
order12222233333344446666···66···666661212121212···1212121212
size111118182336662218182223···36···61818181822226···618181818

62 irreducible representations

dim111111111122222222222266666
type+++++++++++
imageC1C2C2C2C3C4C6C6C6C12S3D4D6C3xS3C4xS3D12C3:D4C3xD4S3xC6S3xC12C3xD12C3xC3:D4C32:C6C2xC32:C6C4xC32:C6He3:4D4He3:6D4
kernelC62.21D6C2xC32:C12C2xC4xHe3C22xC32:C6C6.11D12C2xC32:C6C2xC3:Dic3C6xC12C22xC3:S3C2xC3:S3C6xC12C2xHe3C62C2xC12C3xC6C3xC6C3xC6C3xC6C2xC6C6C6C6C2xC4C22C2C2C2
# reps111124222812122224244411222

Matrix representation of C62.21D6 ►in GL8(F13)

40000000
04000000
00100000
00010000
000012100
000012000
000000012
000000112
,
120000000
012000000
00010000
001210000
00000100
000012100
00000001
000000121
,
111000000
112000000
000000114
00000092
001140000
00920000
000011400
00009200
,
122000000
01000000
000000411
00000029
000041100
00002900
004110000
00290000

G:=sub<GL(8,GF(13))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,12,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1],[1,1,0,0,0,0,0,0,11,12,0,0,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,4,2,0,0,11,9,0,0,0,0,0,0,4,2,0,0,0,0],[12,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0,4,2,0,0,0,0,0,0,11,9,0,0,0,0] >;
 

C62.21D6 in GAP, Magma, Sage, TeX 

C_6^2._{21}D_6
 
% in TeX
 
G:=Group("C6^2.21D6");
 
// GroupNames label
 
G:=SmallGroup(432,141);
 
// by ID
 
G=gap.SmallGroup(432,141);
 
# by ID
 
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,4037,2035,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=a^3,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^3*c^5>;
 
// generators/relations
 

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