A4xDic6 - GroupNames

G = A₄xDic₆ order 288 = 2⁵·3²

Direct product of A₄ and Dic₆

direct product, metabelian, soluble, monomial

Aliases: A₄xDic₆, C₃:(Q₈xA₄), C₄.₁(S₃xA₄), (C₃xA₄):₄Q₈, (C₄xA₄).₃S₃, C₁₂.₁(C₂xA₄), (C₁₂xA₄).₃C₂, (C₂xA₄).₁₃D₆, C₆.₁(C₂²xA₄), (C₂²xDic₆):C₃, C₂³.₁₆(S₃xC₆), (C₂²xC₁₂).₁C₆, Dic₃.₁(C₂xA₄), (Dic₃xA₄).₂C₂, C₂²:₂(C₃xDic₆), (C₂²xDic₃).C₆, (C₆xA₄).₁₈C₂², (C₂xC₆):(C₃xQ₈), C₂.₃(C₂xS₃xA₄), (C₂²xC₄).₃(C₃xS₃), (C₂²xC₆).₁(C₂xC₆), SmallGroup(288,918)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²xC₆ — A₄xDic₆

Chief series

C₁ — C₃ — C₂xC₆ — C₂²xC₆ — C₆xA₄ — Dic₃xA₄ — A₄xDic₆

Lower central

C₂xC₆ — C₂²xC₆ — A₄xDic₆

Upper central

C₁ — C₂ — C₄

Generators and relations for A₄xDic₆
G = < a,b,c,d,e | a²=b²=c³=d¹²=1, e²=d⁶, cac^-1=ab=ba, ad=da, ae=ea, cbc^-1=a, bd=db, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 378 in 98 conjugacy classes, 27 normal (21 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², C₆, C₆, C₂xC₄, Q₈, C₂³, C₃², Dic₃, Dic₃, C₁₂, C₁₂, A₄, A₄, C₂xC₆, C₂xC₆, C₂²xC₄, C₂²xC₄, C₂xQ₈, C₃xC₆, Dic₆, Dic₆, C₂xDic₃, C₂xC₁₂, C₃xQ₈, C₂xA₄, C₂xA₄, C₂²xC₆, C₂²xQ₈, C₃xDic₃, C₃xC₁₂, C₃xA₄, C₄xA₄, C₄xA₄, C₂xDic₆, C₂²xDic₃, C₂²xC₁₂, C₃xDic₆, C₆xA₄, Q₈xA₄, C₂²xDic₆, Dic₃xA₄, C₁₂xA₄, A₄xDic₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, Q₈, A₄, D₆, C₂xC₆, C₃xS₃, Dic₆, C₃xQ₈, C₂xA₄, S₃xC₆, C₂²xA₄, C₃xDic₆, S₃xA₄, Q₈xA₄, C₂xS₃xA₄, A₄xDic₆

Smallest permutation representation of A₄xDic₆
►On 72 points

Generators in S₇₂

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)

(13 19)(14 20)(15 21)(16 22)(17 23)(18 24)(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)

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

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

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

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

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

36 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	3E	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	6F	6G	12A	12B	12C	12D	12E	···	12J	12K	12L	12M	12N
order	1	2	2	2	3	3	3	3	3	4	4	4	4	4	4	6	6	6	6	6	6	6	12	12	12	12	12	···	12	12	12	12	12
size	1	1	3	3	2	4	4	8	8	2	6	6	6	18	18	2	4	4	6	6	8	8	2	2	6	6	8	···	8	24	24	24	24

36 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	3	3	3	6	6	6	6
type	+	+	+				+	-	+		-				+	+	+	+	-	+	-
image	C₁	C₂	C₂	C₃	C₆	C₆	S₃	Q₈	D₆	C₃xS₃	Dic₆	C₃xQ₈	S₃xC₆	C₃xDic₆	A₄	C₂xA₄	C₂xA₄	S₃xA₄	Q₈xA₄	C₂xS₃xA₄	A₄xDic₆
kernel	A₄xDic₆	Dic₃xA₄	C₁₂xA₄	C₂²xDic₆	C₂²xDic₃	C₂²xC₁₂	C₄xA₄	C₃xA₄	C₂xA₄	C₂²xC₄	A₄	C₂xC₆	C₂³	C₂²	Dic₆	Dic₃	C₁₂	C₄	C₃	C₂	C₁
# reps	1	2	1	2	4	2	1	1	1	2	2	2	2	4	1	2	1	1	1	1	2

Matrix representation of A₄xDic₆ ►in GL₅(F₁₃)

1	0	0	0	0
0	1	0	0	0
0	0	0	0	1
0	0	12	12	12
0	0	1	0	0

1	0	0	0	0
0	1	0	0	0
0	0	0	1	0
0	0	1	0	0
0	0	12	12	12

1	0	0	0	0
0	1	0	0	0
0	0	3	0	0
0	0	0	0	3
0	0	10	10	10

6	0	0	0	0
0	11	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	0	1

0	12	0	0	0
1	0	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,12,0,0,0,1,12,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,1,0,12,0,0,0,0,12],[1,0,0,0,0,0,1,0,0,0,0,0,3,0,10,0,0,0,0,10,0,0,0,3,10],[6,0,0,0,0,0,11,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,1,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12] >;

A₄xDic₆ in GAP, Magma, Sage, TeX

A_4\times {\rm Dic}_6

% in TeX

G:=Group("A4xDic6");

// GroupNames label

G:=SmallGroup(288,918);

// by ID

G=gap.SmallGroup(288,918);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-3,84,197,92,648,271,9414]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^12=1,e^2=d^6,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 = A4xDic6 order 288 = 25·32

Direct product of A4 and Dic6

G = A₄xDic₆ order 288 = 2⁵·3²

Direct product of A₄ and Dic₆