A4xC3:C8 - GroupNames

G = A₄×C₃⋊C₈ order 288 = 2⁵·3²

Direct product of A₄ and C₃⋊C₈

direct product, metabelian, soluble, monomial, A-group

Aliases: A₄×C₃⋊C₈, C₃⋊(C₈×A₄), (C₂×C₆)⋊C₂₄, (C₃×A₄)⋊₃C₈, C₆.₁(C₄×A₄), C₄.₄(S₃×A₄), (C₄×A₄).₄S₃, (C₆×A₄).₃C₄, C₁₂.₄(C₂×A₄), (C₂²×C₆).C₁₂, (C₁₂×A₄).₆C₂, C₂.₁(Dic₃×A₄), (C₂²×C₁₂).₂C₆, (C₂×A₄).₂Dic₃, C₂³.₃(C₃×Dic₃), (C₂²×C₃⋊C₈)⋊C₃, C₂²⋊₂(C₃×C₃⋊C₈), (C₂²×C₄).₄(C₃×S₃), SmallGroup(288,408)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — A₄×C₃⋊C₈

Chief series

C₁ — C₃ — C₂×C₆ — C₂²×C₆ — C₂²×C₁₂ — C₁₂×A₄ — A₄×C₃⋊C₈

Lower central

C₂×C₆ — A₄×C₃⋊C₈

Upper central

C₁ — C₄

Generators and relations for A₄×C₃⋊C₈
G = < a,b,c,d,e | a²=b²=c³=d³=e⁸=1, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 178 in 58 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₈, C₂×C₄, C₂³, C₃², C₁₂, C₁₂, A₄, A₄, C₂×C₆, C₂×C₆, C₂×C₈, C₂²×C₄, C₃×C₆, C₃⋊C₈, C₃⋊C₈, C₂₄, C₂×C₁₂, C₂×A₄, C₂×A₄, C₂²×C₆, C₂²×C₈, C₃×C₁₂, C₃×A₄, C₂×C₃⋊C₈, C₄×A₄, C₄×A₄, C₂²×C₁₂, C₃×C₃⋊C₈, C₆×A₄, C₈×A₄, C₂²×C₃⋊C₈, C₁₂×A₄, A₄×C₃⋊C₈
Quotients: C₁, C₂, C₃, C₄, S₃, C₆, C₈, Dic₃, C₁₂, A₄, C₃×S₃, C₃⋊C₈, C₂₄, C₂×A₄, C₃×Dic₃, C₄×A₄, C₃×C₃⋊C₈, S₃×A₄, C₈×A₄, Dic₃×A₄, A₄×C₃⋊C₈

Smallest permutation representation of A₄×C₃⋊C₈
►On 72 points

Generators in S₇₂

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(25 29)(26 30)(27 31)(28 32)(41 45)(42 46)(43 47)(44 48)(57 61)(58 62)(59 63)(60 64)(65 69)(66 70)(67 71)(68 72)

(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)

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

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

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

48 conjugacy classes

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

48 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	3	3	3	3	6	6	6
type	+	+							+	-					+	+			+	-
image	C₁	C₂	C₃	C₄	C₆	C₈	C₁₂	C₂₄	S₃	Dic₃	C₃×S₃	C₃⋊C₈	C₃×Dic₃	C₃×C₃⋊C₈	A₄	C₂×A₄	C₄×A₄	C₈×A₄	S₃×A₄	Dic₃×A₄	A₄×C₃⋊C₈
kernel	A₄×C₃⋊C₈	C₁₂×A₄	C₂²×C₃⋊C₈	C₆×A₄	C₂²×C₁₂	C₃×A₄	C₂²×C₆	C₂×C₆	C₄×A₄	C₂×A₄	C₂²×C₄	A₄	C₂³	C₂²	C₃⋊C₈	C₁₂	C₆	C₃	C₄	C₂	C₁
# reps	1	1	2	2	2	4	4	8	1	1	2	2	2	4	1	1	2	4	1	1	2

Matrix representation of A₄×C₃⋊C₈ ►in GL₅(𝔽₇₃)

1	0	0	0	0
0	1	0	0	0
0	0	72	64	0
0	0	0	1	0
0	0	0	0	72

1	0	0	0	0
0	1	0	0	0
0	0	72	0	65
0	0	0	72	0
0	0	0	0	1

1	0	0	0	0
0	1	0	0	0
0	0	64	0	0
0	0	2	9	8
0	0	0	72	0

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

5	42	0	0	0
37	68	0	0	0
0	0	72	0	0
0	0	0	72	0
0	0	0	0	72

G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,64,1,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,65,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,64,2,0,0,0,0,9,72,0,0,0,8,0],[72,1,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[5,37,0,0,0,42,68,0,0,0,0,0,72,0,0,0,0,0,72,0,0,0,0,0,72] >;

A₄×C₃⋊C₈ in GAP, Magma, Sage, TeX

A_4\times C_3\rtimes C_8

% in TeX

G:=Group("A4xC3:C8");

// GroupNames label

G:=SmallGroup(288,408);

// by ID

G=gap.SmallGroup(288,408);

# by ID

G:=PCGroup([7,-2,-3,-2,-2,-2,2,-3,42,58,1271,516,9414]);

// Polycyclic

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

// generators/relations

G = A4×C3⋊C8 order 288 = 25·32

Direct product of A4 and C3⋊C8

G = A₄×C₃⋊C₈ order 288 = 2⁵·3²

Direct product of A₄ and C₃⋊C₈