D8xC3:S3 - GroupNames

G = D₈xC₃:S₃ order 288 = 2⁵·3²

Direct product of D₈ and C₃:S₃

direct product, metabelian, supersoluble, monomial

Aliases: D₈xC₃:S₃, C₂₄:₅D₆, C₃:₄(S₃xD₈), (C₃xD₄):₃D₆, (C₃xD₈):₂S₃, C₃²:₅D₈:₈C₂, C₃²:₁₂(C₂xD₈), (C₃xC₂₄):₉C₂², C₆.₁₁₈(S₃xD₄), (C₃²xD₈):₅C₂, C₃²:₇D₈:₅C₂, C₃:Dic₃.₄₈D₄, C₁₂:S₃:₇C₂², C₁₂.₈₇(C₂²xS₃), (C₃xC₁₂).₉₁C₂³, (D₄xC₃²):₇C₂², C₃²:₄C₈:₂₀C₂², C₈:₄(C₂xC₃:S₃), (C₈xC₃:S₃):₃C₂, (D₄xC₃:S₃):₃C₂, D₄:₁(C₂xC₃:S₃), C₂.₁₅(D₄xC₃:S₃), (C₂xC₃:S₃).₇₂D₄, C₄.₁(C₂²xC₃:S₃), (C₃xC₆).₂₃₉(C₂xD₄), (C₄xC₃:S₃).₇₀C₂², SmallGroup(288,767)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — D₈xC₃:S₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₄xC₃:S₃ — D₄xC₃:S₃ — D₈xC₃:S₃

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — D₈xC₃:S₃

Upper central

C₁ — C₂ — C₄ — D₈

Generators and relations for D₈xC₃:S₃
G = < a,b,c,d,e | a⁸=b²=c³=d³=e²=1, bab=a^-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece=c^-1, ede=d^-1 >

Subgroups: 1236 in 228 conjugacy classes, 57 normal (17 characteristic)
C₁, C₂, C₂, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₈, C₂xC₄, D₄, D₄, C₂³, C₃², Dic₃, C₁₂, D₆, C₂xC₆, C₂xC₈, D₈, D₈, C₂xD₄, C₃:S₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, C₄xS₃, D₁₂, C₃:D₄, C₃xD₄, C₂²xS₃, C₂xD₈, C₃:Dic₃, C₃xC₁₂, C₂xC₃:S₃, C₂xC₃:S₃, C₆², S₃xC₈, D₂₄, D₄:S₃, C₃xD₈, S₃xD₄, C₃²:₄C₈, C₃xC₂₄, C₄xC₃:S₃, C₁₂:S₃, C₃²:₇D₄, D₄xC₃², C₂²xC₃:S₃, S₃xD₈, C₈xC₃:S₃, C₃²:₅D₈, C₃²:₇D₈, C₃²xD₈, D₄xC₃:S₃, D₈xC₃:S₃
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, D₈, C₂xD₄, C₃:S₃, C₂²xS₃, C₂xD₈, C₂xC₃:S₃, S₃xD₄, C₂²xC₃:S₃, S₃xD₈, D₄xC₃:S₃, D₈xC₃:S₃

Smallest permutation representation of D₈xC₃:S₃
►On 72 points

Generators in S₇₂

(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)

(2 8)(3 7)(4 6)(10 16)(11 15)(12 14)(17 23)(18 22)(19 21)(25 29)(26 28)(30 32)(33 35)(36 40)(37 39)(41 47)(42 46)(43 45)(50 56)(51 55)(52 54)(58 64)(59 63)(60 62)(65 69)(66 68)(70 72)

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

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	3D	4A	4B	6A	6B	6C	6D	6E	···	6L	8A	8B	8C	8D	12A	12B	12C	12D	24A	···	24H
order	1	2	2	2	2	2	2	2	3	3	3	3	4	4	6	6	6	6	6	···	6	8	8	8	8	12	12	12	12	24	···	24
size	1	1	4	4	9	9	36	36	2	2	2	2	2	18	2	2	2	2	8	···	8	2	2	18	18	4	4	4	4	4	···	4

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

D₈

S₃xD₄

S₃xD₈

kernel

D₈xC₃:S₃

C₈xC₃:S₃

C₃²:₅D₈

C₃²:₇D₈

C₃²xD₈

D₄xC₃:S₃

C₃xD₈

C₃:Dic₃

C₂xC₃:S₃

C₂₄

C₃xD₄

C₃:S₃

C₆

C₃

# reps

Matrix representation of D₈xC₃:S₃ ►in GL₆(F₇₃)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	0	48
0	0	0	0	38	41

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	48	72

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	70	0	0	0	0
1	71	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

72	3	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	72	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,38,0,0,0,0,48,41],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,48,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,70,71,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

D₈xC₃:S₃ in GAP, Magma, Sage, TeX

D_8\times C_3\rtimes S_3

% in TeX

G:=Group("D8xC3:S3");

// GroupNames label

G:=SmallGroup(288,767);

// by ID

G=gap.SmallGroup(288,767);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,135,346,185,80,2693,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^8=b^2=c^3=d^3=e^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e=c^-1,e*d*e=d^-1>;

// generators/relations

G = D8xC3:S3 order 288 = 25·32

Direct product of D8 and C3:S3

G = D₈xC₃:S₃ order 288 = 2⁵·3²

Direct product of D₈ and C₃:S₃