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G = C62.4D6  order 432 = 24·33

4th non-split extension by C62 of D6 acting faithfully

non-abelian, supersoluble, monomial

Aliases: C62.4D6, (C3×C6).6D12, C32⋊1(D6⋊C4), He3⋊3(C22⋊C4), (C2×He3).15D4, C6.15(S3×Dic3), C6.5(D6⋊S3), C3.2(D6⋊Dic3), C2.2(He3⋊2D4), C2.2(He3⋊3D4), C32⋊(C6.D4), C6.34(C3⋊D12), C22.6(C32⋊D6), (C22×He3).4C22, (C2×C6).50S32, (C3×C6).8(C4×S3), (C2×C32⋊C6)⋊1C4, (C2×C3⋊S3)⋊1Dic3, (C2×C3⋊Dic3)⋊1S3, (C2×C32⋊C12)⋊1C2, (C2×He3⋊3C4)⋊1C2, (C22×C3⋊S3).1S3, (C3×C6).5(C2×Dic3), C2.5(C6.S32), (C3×C6).10(C3⋊D4), (C2×He3).15(C2×C4), (C22×C32⋊C6).1C2, SmallGroup(432,97)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3 — C2×He3 — C62.4D6
Chief series
C1 — C3 — C32 — He3 — C2×He3 — C22×He3 — C22×C32⋊C6 — C62.4D6
Lower central
He3 — C2×He3 — C62.4D6
Upper central
C1 — C22

Generators and relations for C62.4D6
 G = < a,b,c,d | a6=b6=1, c6=a3, d2=a3b3, ab=ba, cac-1=dad-1=a-1b4, cbc-1=b-1, bd=db, dcd-1=b3c5 >

Subgroups: 775 in 135 conjugacy classes, 35 normal (31 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, C6, C6, C2×C4, C23, C32, C32, Dic3, C12, D6, C2×C6, C2×C6, C22⋊C4, C3×S3, C3⋊S3, C3×C6, C3×C6, C2×Dic3, C2×C12, C22×S3, C22×C6, He3, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, D6⋊C4, C6.D4, C32⋊C6, C2×He3, C6×Dic3, C2×C3⋊Dic3, S3×C2×C6, C22×C3⋊S3, C32⋊C12, He3⋊3C4, C2×C32⋊C6, C2×C32⋊C6, C22×He3, D6⋊Dic3, C6.D12, C2×C32⋊C12, C2×He3⋊3C4, C22×C32⋊C6, C62.4D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C4×S3, D12, C2×Dic3, C3⋊D4, S32, D6⋊C4, C6.D4, S3×Dic3, D6⋊S3, C3⋊D12, C32⋊D6, D6⋊Dic3, C6.S32, He3⋊2D4, He3⋊3D4, C62.4D6

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D4A4B4C4D6A6B6C6D···6I6J6K6L6M6N6O6P12A···12H
order122222333344446666···6666666612···12
size1111181826612181818182226···61212121818181818···18

38 irreducible representations

dim111112222222244446666
type+++++++-+++--+++
imageC1C2C2C2C4S3S3D4Dic3D6C4×S3D12C3⋊D4S32S3×Dic3D6⋊S3C3⋊D12C32⋊D6C6.S32He3⋊2D4He3⋊3D4
kernelC62.4D6C2×C32⋊C12C2×He3⋊3C4C22×C32⋊C6C2×C32⋊C6C2×C3⋊Dic3C22×C3⋊S3C2×He3C2×C3⋊S3C62C3×C6C3×C6C3×C6C2×C6C6C6C6C22C2C2C2
# reps111141122222611112222

Matrix representation of C62.4D6 ►in GL10(𝔽13)

3000000000
0300000000
8090000000
0809000000
00000000120
00000000012
00001200000
00000120000
00000012000
00000001200
,
12000000000
01200000000
00120000000
00012000000
0000110000
00001200000
0000001100
00000012000
0000000011
00000000120
,
0403000000
991010000000
0809000000
5544000000
00001190000
00001120000
0000000042
00000000119
00000011200
0000004200
,
09010000000
90100000000
01004000000
10040000000
0000240000
00009110000
00000000911
00000000211
00000021100
0000002400

G:=sub<GL(10,GF(13))| [3,0,8,0,0,0,0,0,0,0,0,3,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0],[12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,0,1,0],[0,9,0,5,0,0,0,0,0,0,4,9,8,5,0,0,0,0,0,0,0,10,0,4,0,0,0,0,0,0,3,10,9,4,0,0,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,2,9,0,0],[0,9,0,10,0,0,0,0,0,0,9,0,10,0,0,0,0,0,0,0,0,10,0,4,0,0,0,0,0,0,10,0,4,0,0,0,0,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,4,11,0,0,0,0,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,11,4,0,0,0,0,0,0,9,2,0,0,0,0,0,0,0,0,11,11,0,0] >;
 

C62.4D6 in GAP, Magma, Sage, TeX 

C_6^2._4D_6
 
% in TeX
 
G:=Group("C6^2.4D6");
 
// GroupNames label
 
G:=SmallGroup(432,97);
 
// by ID
 
G=gap.SmallGroup(432,97);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,571,4037,537,14118,7069]);
 
// 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^-1*b^4,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^3*c^5>;
 
// generators/relations
 

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