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G = (C6×D4).S3  order 288 = 25·32

11st non-split extension by C6×D4 of S3 acting via S3/C3=C2

metabelian, supersoluble, monomial

Aliases: (C6×D4).11S3, (C3×C12).57D4, (C2×C12).94D6, (C2×C62).3C4, C12.58D6⋊5C2, C3⋊2(C12.D4), C12.39(C3⋊D4), C62.104(C2×C4), (C6×C12).61C22, C32⋊8(C4.D4), (C22×C6).8Dic3, C2.4(C62⋊5C4), C4.13(C32⋊7D4), C23.2(C3⋊Dic3), C6.24(C6.D4), (D4×C3×C6).4C2, (C2×D4).2(C3⋊S3), (C2×C6).47(C2×Dic3), C22.2(C2×C3⋊Dic3), (C3×C6).72(C22⋊C4), (C2×C4).3(C2×C3⋊S3), SmallGroup(288,308)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C62 — (C6×D4).S3
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C12.58D6 — (C6×D4).S3
Lower central
C32 — C3×C6 — C62 — (C6×D4).S3
Upper central
C1 — C2 — C2×C4 — C2×D4

Generators and relations for (C6×D4).S3
 G = < a,b,c,d,e | a6=b4=c2=d3=1, e2=b, ab=ba, ac=ca, ad=da, eae-1=a-1b2, cbc=b-1, bd=db, be=eb, cd=dc, ece-1=a3b2c, ede-1=d-1 >

Subgroups: 364 in 138 conjugacy classes, 57 normal (11 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, D4, C23, C32, C12, C2×C6, C2×C6, M4(2), C2×D4, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×D4, C22×C6, C4.D4, C3×C12, C62, C62, C4.Dic3, C6×D4, C32⋊4C8, C6×C12, D4×C32, C2×C62, C12.D4, C12.58D6, D4×C3×C6, (C6×D4).S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, C3⋊S3, C2×Dic3, C3⋊D4, C4.D4, C3⋊Dic3, C2×C3⋊S3, C6.D4, C2×C3⋊Dic3, C32⋊7D4, C12.D4, C62⋊5C4, (C6×D4).S3

Smallest permutation representation of (C6×D4).S3
►On 72 points
Generators in S72
(1 59 53 5 63 49)(2 54 64)(3 61 55 7 57 51)(4 56 58)(6 50 60)(8 52 62)(9 43 22 13 47 18)(10 23 48)(11 45 24 15 41 20)(12 17 42)(14 19 44)(16 21 46)(25 66 33 29 70 37)(26 34 71)(27 68 35 31 72 39)(28 36 65)(30 38 67)(32 40 69)

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

(1 5)(2 6)(11 15)(12 16)(17 21)(20 24)(27 31)(28 32)(35 39)(36 40)(41 45)(42 46)(49 53)(50 54)(59 63)(60 64)(65 69)(68 72)

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

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

51 conjugacy classes

class 1 2A2B2C2D3A3B3C3D4A4B6A···6L6M···6AB8A8B8C8D12A···12H
order122223333446···66···6888812···12
size112442222222···24···4363636364···4

51 irreducible representations

dim11112222244
type++++++-+
imageC1C2C2C4S3D4D6Dic3C3⋊D4C4.D4C12.D4
kernel(C6×D4).S3C12.58D6D4×C3×C6C2×C62C6×D4C3×C12C2×C12C22×C6C12C32C3
# reps121442481618

Matrix representation of (C6×D4).S3 ►in GL8(𝔽73)

720000000
072000000
0037450000
0045370000
000072000
000007200
00003010
00002001
,
720000000
072000000
007200000
000720000
000017100
000017200
000003723
0000722481
,
01000000
10000000
00010000
00100000
000072000
000072100
00003010
0000102572
,
3628000000
2836000000
00100000
00010000
00001000
00000100
00000010
00000001
,
1712000000
6156000000
0057440000
0029160000
0000720480
000000481
00000310
000011250

G:=sub<GL(8,GF(73))| [72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,37,45,0,0,0,0,0,0,45,37,0,0,0,0,0,0,0,0,72,0,3,2,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,1,1,0,72,0,0,0,0,71,72,3,2,0,0,0,0,0,0,72,48,0,0,0,0,0,0,3,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,72,72,3,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,25,0,0,0,0,0,0,0,72],[36,28,0,0,0,0,0,0,28,36,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[17,61,0,0,0,0,0,0,12,56,0,0,0,0,0,0,0,0,57,29,0,0,0,0,0,0,44,16,0,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,0,0,3,1,0,0,0,0,48,48,1,25,0,0,0,0,0,1,0,0] >;
 

(C6×D4).S3 in GAP, Magma, Sage, TeX 

(C_6\times D_4).S_3
 
% in TeX
 
G:=Group("(C6xD4).S3");
 
// GroupNames label
 
G:=SmallGroup(288,308);
 
// by ID
 
G=gap.SmallGroup(288,308);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,219,100,675,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^6=b^4=c^2=d^3=1,e^2=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a^-1*b^2,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=a^3*b^2*c,e*d*e^-1=d^-1>;
 
// generators/relations
 

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