Copied to
clipboard

G = C62.13D6  order 432 = 24·33

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

metabelian, supersoluble, monomial

Aliases: C62.13D6, C62.(C2×C6), (D4×He3)⋊4C2, C12.16(S3×C6), He3⋊9(C4○D4), C12.D6⋊C3, He3⋊6D4⋊6C2, He3⋊3Q8⋊8C2, (C3×C12).25D6, (D4×C32)⋊3C6, (D4×C32)⋊4S3, C32⋊7D4⋊2C6, C32⋊4Q8⋊3C6, D4⋊2(C32⋊C6), C32⋊5(D4⋊2S3), (C4×He3).21C22, (C2×He3).24C23, C32⋊C12.12C22, (C22×He3).13C22, (C4×C3⋊S3)⋊2C6, C6.34(S3×C2×C6), (C4×C32⋊C6)⋊6C2, (C2×C6).11(S3×C6), (C3×C12).5(C2×C6), (C2×C3⋊Dic3)⋊3C6, C32⋊2(C3×C4○D4), C4.5(C2×C32⋊C6), (C2×C32⋊C12)⋊9C2, (C3×D4).10(C3×S3), C3.2(C3×D4⋊2S3), C3⋊Dic3.3(C2×C6), (C3×C6).6(C22×C6), (C3×C6).28(C22×S3), C22.1(C2×C32⋊C6), C2.7(C22×C32⋊C6), (C2×C32⋊C6).11C22, (C2×C3⋊S3).2(C2×C6), SmallGroup(432,361)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C62.13D6
Chief series
C1 — C3 — C32 — C3×C6 — C2×He3 — C2×C32⋊C6 — C4×C32⋊C6 — C62.13D6
Lower central
C32 — C3×C6 — 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, C2×C4, D4, D4, Q8, C32, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×D4, C3×Q8, He3, C3×Dic3, C3⋊Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C62, C62, D4⋊2S3, C3×C4○D4, C32⋊C6, C2×He3, C2×He3, C3×Dic6, S3×C12, C6×Dic3, C3×C3⋊D4, C32⋊4Q8, C4×C3⋊S3, C2×C3⋊Dic3, C32⋊7D4, D4×C32, D4×C32, C32⋊C12, C32⋊C12, C4×He3, C2×C32⋊C6, C22×He3, C3×D4⋊2S3, C12.D6, He3⋊3Q8, C4×C32⋊C6, C2×C32⋊C12, He3⋊6D4, D4×He3, C62.13D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C22×S3, C22×C6, S3×C6, D4⋊2S3, C3×C4○D4, C32⋊C6, S3×C2×C6, C2×C32⋊C6, C3×D4⋊2S3, C22×C32⋊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++++++-+++-+++
imageC1C2C2C2C2C2C3C6C6C6C6C6C62.13D6S3D6D6C4○D4C3×S3S3×C6S3×C6C3×C4○D4D4⋊2S3C3×D4⋊2S3C32⋊C6C2×C32⋊C6C2×C32⋊C6
kernelC62.13D6He3⋊3Q8C4×C32⋊C6C2×C32⋊C12He3⋊6D4D4×He3C12.D6C32⋊4Q8C4×C3⋊S3C2×C3⋊Dic3C32⋊7D4D4×C32C1D4×C32C3×C12C62He3C3×D4C12C2×C6C32C32C3D4C4C22
# 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
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁