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G = C62.13D6  order 432 = 24·33

13rd non-split extension by C62 of D6 acting faithfully

metabelian, supersoluble, monomial

Aliases: C62.13D6, C62.(C2xC6), (D4xHe3):4C2, C12.16(S3xC6), He3:9(C4oD4), C12.D6:C3, He3:6D4:6C2, He3:3Q8:8C2, (C3xC12).25D6, (D4xC32):3C6, (D4xC32):4S3, C32:7D4:2C6, C32:4Q8:3C6, D4:2(C32:C6), C32:5(D4:2S3), (C4xHe3).21C22, (C2xHe3).24C23, C32:C12.12C22, (C22xHe3).13C22, (C4xC3:S3):2C6, C6.34(S3xC2xC6), (C4xC32:C6):6C2, (C2xC6).11(S3xC6), (C3xC12).5(C2xC6), (C2xC3:Dic3):3C6, C32:2(C3xC4oD4), C4.5(C2xC32:C6), (C2xC32:C12):9C2, (C3xD4).10(C3xS3), C3.2(C3xD4:2S3), C3:Dic3.3(C2xC6), (C3xC6).6(C22xC6), (C3xC6).28(C22xS3), C22.1(C2xC32:C6), C2.7(C22xC32:C6), (C2xC32:C6).11C22, (C2xC3:S3).2(C2xC6), SmallGroup(432,361)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3xC6 — C62.13D6
Chief series
C1 — C3 — C32 — C3xC6 — C2xHe3 — C2xC32:C6 — C4xC32:C6 — C62.13D6
Lower central
C32 — C3xC6 — C62.13D6
Upper central
C1 — C2 — D4

Generators and relations for C62.13D6
 G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, cac-1=ab-1, dad-1=ab2, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 645 in 156 conjugacy classes, 52 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2xC4, D4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C4oD4, C3xS3, C3:S3, C3xC6, C3xC6, Dic6, C4xS3, C2xDic3, C3:D4, C2xC12, C3xD4, C3xD4, C3xQ8, He3, C3xDic3, C3:Dic3, C3:Dic3, C3xC12, C3xC12, S3xC6, C2xC3:S3, C62, C62, D4:2S3, C3xC4oD4, C32:C6, C2xHe3, C2xHe3, C3xDic6, S3xC12, C6xDic3, C3xC3:D4, C32:4Q8, C4xC3:S3, C2xC3:Dic3, C32:7D4, D4xC32, D4xC32, C32:C12, C32:C12, C4xHe3, C2xC32:C6, C22xHe3, C3xD4:2S3, C12.D6, He3:3Q8, C4xC32:C6, C2xC32:C12, He3:6D4, D4xHe3, C62.13D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2xC6, C4oD4, C3xS3, C22xS3, C22xC6, S3xC6, D4:2S3, C3xC4oD4, C32:C6, S3xC2xC6, C2xC32:C6, C3xD4:2S3, C22xC32:C6, C62.13D6

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

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

50 conjugacy classes

class 1 2A2B2C2D3A3B3C3D3E3F4A4B4C4D4E6A6B6C6D6E6F···6L6M···6R6S6T12A12B12C12D12E12F12G12H12I12J12K12L12M12N
order1222233333344444666666···66···6661212121212121212121212121212
size1122182336662991818233446···612···121818466999912121218181818

50 irreducible representations

dim111111111111122222222244666
type++++++-+++-+++
imageC1C2C2C2C2C2C3C6C6C6C6C6C62.13D6S3D6D6C4oD4C3xS3S3xC6S3xC6C3xC4oD4D4:2S3C3xD4:2S3C32:C6C2xC32:C6C2xC32:C6
kernelC62.13D6He3:3Q8C4xC32:C6C2xC32:C12He3:6D4D4xHe3C12.D6C32:4Q8C4xC3:S3C2xC3:Dic3C32:7D4D4xC32C1D4xC32C3xC12C62He3C3xD4C12C2xC6C32C32C3D4C4C22
# reps11122122244211122224412112

Matrix representation of C62.13D6 ►in GL10(F13)

00100000000
00010000000
10000000000
01000000000
00000000120
00000000012
00001200000
00000120000
00000012000
00000001200
,
12000000000
01200000000
00120000000
00012000000
00000120000
00001120000
00000001200
00000011200
00000000012
00000000112
,
0001000000
001212000000
01200000000
1100000000
00001200000
00000120000
0000000100
00000012100
00000000112
0000000010
,
0800000000
8000000000
0008000000
0080000000
000010100000
0000730000
00000010600
0000003300
00000000610
0000000037

G:=sub<GL(10,GF(13))| [0,0,10,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,12],[0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,12,0],[0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,0,0,10,7,0,0,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,0,0,10,3,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,0,0,6,3,0,0,0,0,0,0,0,0,10,7] >;
 

C62.13D6 in GAP, Magma, Sage, TeX 

C_6^2._{13}D_6
 
% in TeX
 
G:=Group("C6^2.13D6");
 
// GroupNames label
 
G:=SmallGroup(432,361);
 
// by ID
 
G=gap.SmallGroup(432,361);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,4037,1034,14118]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^6=b^6=1,c^6=d^2=b^3,a*b=b*a,c*a*c^-1=a*b^-1,d*a*d^-1=a*b^2,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
 
// generators/relations
 

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