D12.2Dic3 - GroupNames

G = D₁₂.₂Dic₃ order 288 = 2⁵·3²

The non-split extension by D₁₂ of Dic₃ acting through Inn(D₁₂)

Aliases: D₁₂.₂Dic₃, Dic₆.₂Dic₃, C₃:C₈.₃₁D₆, C₃:₅(C₈oD₁₂), C₁₂.₃₀(C₄xS₃), C₄oD₁₂.₆S₃, (C₄xS₃).₃₁D₆, (C₃xD₁₂).₁C₄, C₃²:₄(C₈oD₄), C₄.₄(S₃xDic₃), (C₂xC₁₂).₁₀₇D₆, C₆².₄₄(C₂xC₄), (C₃xDic₆).₁C₄, D₆.₁(C₂xDic₃), C₃:D₄.₂Dic₃, C₁₂.₅₈D₆:₈C₂, D₆.Dic₃:₁₀C₂, (S₃xC₁₂).₈C₂², C₃:₁(D₄.Dic₃), (C₆xC₁₂).₆₆C₂², C₁₂.₂₅(C₂xDic₃), C₂².₁(S₃xDic₃), C₆.₃(C₂²xDic₃), (C₃xC₁₂).₁₄₃C₂³, C₁₂.₁₄₂(C₂²xS₃), Dic₃.₁(C₂xDic₃), C₃²:₄C₈.₂₀C₂², (C₂xC₃:C₈):₃S₃, (S₃xC₃:C₈):₉C₂, (C₆xC₃:C₈):₁₆C₂, C₄.₈₉(C₂xS₃²), (C₂xC₄).₆₂S₃², C₆.₈₃(S₃xC₂xC₄), C₂.₅(C₂xS₃xDic₃), (S₃xC₆).₄(C₂xC₄), (C₂xC₆).₁₆(C₄xS₃), (C₃xC₃:D₄).₁C₄, (C₃xC₁₂).₅₆(C₂xC₄), (C₃xC₄oD₁₂).₅C₂, (C₃xC₃:C₈).₃₈C₂², (C₃xC₆).₃₉(C₂²xC₄), (C₃xDic₃).₄(C₂xC₄), (C₂xC₆).₁₈(C₂xDic₃), SmallGroup(288,462)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — D₁₂.₂Dic₃

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — S₃xC₁₂ — S₃xC₃:C₈ — D₁₂.₂Dic₃

Lower central

C₃² — C₃xC₆ — D₁₂.₂Dic₃

Upper central

C₁ — C₄ — C₂xC₄

Generators and relations for D₁₂.₂Dic₃
G = < a,b,c,d | a¹²=b²=1, c⁶=a⁶, d²=a⁶c³, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁵ >

Subgroups: 338 in 134 conjugacy classes, 60 normal (38 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, C₃:C₈, C₂₄, Dic₆, C₄xS₃, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₈oD₄, C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², S₃xC₈, C₈:S₃, C₂xC₃:C₈, C₂xC₃:C₈, C₄.Dic₃, C₂xC₂₄, C₄oD₁₂, C₃xC₄oD₄, C₃xC₃:C₈, C₃²:₄C₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, C₈oD₁₂, D₄.Dic₃, S₃xC₃:C₈, D₆.Dic₃, C₆xC₃:C₈, C₁₂.₅₈D₆, C₃xC₄oD₁₂, D₁₂.₂Dic₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂xC₄, C₂³, Dic₃, D₆, C₂²xC₄, C₄xS₃, C₂xDic₃, C₂²xS₃, C₈oD₄, S₃², S₃xC₂xC₄, C₂²xDic₃, S₃xDic₃, C₂xS₃², C₈oD₁₂, D₄.Dic₃, C₂xS₃xDic₃, D₁₂.₂Dic₃

Smallest permutation representation of D₁₂.₂Dic₃
►On 48 points

Generators in S₄₈

(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)

(1 29)(2 28)(3 27)(4 26)(5 25)(6 36)(7 35)(8 34)(9 33)(10 32)(11 31)(12 30)(13 47)(14 46)(15 45)(16 44)(17 43)(18 42)(19 41)(20 40)(21 39)(22 38)(23 37)(24 48)

(1 2 3 4 5 6 7 8 9 10 11 12)(13 18 23 16 21 14 19 24 17 22 15 20)(25 36 35 34 33 32 31 30 29 28 27 26)(37 44 39 46 41 48 43 38 45 40 47 42)

(1 16 10 13 7 22 4 19)(2 17 11 14 8 23 5 20)(3 18 12 15 9 24 6 21)(25 40 28 43 31 46 34 37)(26 41 29 44 32 47 35 38)(27 42 30 45 33 48 36 39)

G:=sub<Sym(48)| (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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48), (1,2,3,4,5,6,7,8,9,10,11,12)(13,18,23,16,21,14,19,24,17,22,15,20)(25,36,35,34,33,32,31,30,29,28,27,26)(37,44,39,46,41,48,43,38,45,40,47,42), (1,16,10,13,7,22,4,19)(2,17,11,14,8,23,5,20)(3,18,12,15,9,24,6,21)(25,40,28,43,31,46,34,37)(26,41,29,44,32,47,35,38)(27,42,30,45,33,48,36,39)>;

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), (1,29)(2,28)(3,27)(4,26)(5,25)(6,36)(7,35)(8,34)(9,33)(10,32)(11,31)(12,30)(13,47)(14,46)(15,45)(16,44)(17,43)(18,42)(19,41)(20,40)(21,39)(22,38)(23,37)(24,48), (1,2,3,4,5,6,7,8,9,10,11,12)(13,18,23,16,21,14,19,24,17,22,15,20)(25,36,35,34,33,32,31,30,29,28,27,26)(37,44,39,46,41,48,43,38,45,40,47,42), (1,16,10,13,7,22,4,19)(2,17,11,14,8,23,5,20)(3,18,12,15,9,24,6,21)(25,40,28,43,31,46,34,37)(26,41,29,44,32,47,35,38)(27,42,30,45,33,48,36,39) );

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)], [(1,29),(2,28),(3,27),(4,26),(5,25),(6,36),(7,35),(8,34),(9,33),(10,32),(11,31),(12,30),(13,47),(14,46),(15,45),(16,44),(17,43),(18,42),(19,41),(20,40),(21,39),(22,38),(23,37),(24,48)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,18,23,16,21,14,19,24,17,22,15,20),(25,36,35,34,33,32,31,30,29,28,27,26),(37,44,39,46,41,48,43,38,45,40,47,42)], [(1,16,10,13,7,22,4,19),(2,17,11,14,8,23,5,20),(3,18,12,15,9,24,6,21),(25,40,28,43,31,46,34,37),(26,41,29,44,32,47,35,38),(27,42,30,45,33,48,36,39)]])

54 conjugacy classes

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

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+				+	+	+	-	+	-	-	+					+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	S₃	D₆	Dic₃	D₆	Dic₃	Dic₃	D₆	C₄xS₃	C₄xS₃	C₈oD₄	C₈oD₁₂	S₃²	S₃xDic₃	C₂xS₃²	S₃xDic₃	D₄.Dic₃	D₁₂.₂Dic₃
kernel	D₁₂.₂Dic₃	S₃xC₃:C₈	D₆.Dic₃	C₆xC₃:C₈	C₁₂.₅₈D₆	C₃xC₄oD₁₂	C₃xDic₆	C₃xD₁₂	C₃xC₃:D₄	C₂xC₃:C₈	C₄oD₁₂	C₃:C₈	Dic₆	C₄xS₃	D₁₂	C₃:D₄	C₂xC₁₂	C₁₂	C₂xC₆	C₃²	C₃	C₂xC₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	2	2	1	1	1	2	2	4	1	1	2	1	2	1	2	2	2	2	4	8	1	1	1	1	2	4

Matrix representation of D₁₂.₂Dic₃ ►in GL₆(F₇₃)

1	72	0	0	0	0
1	0	0	0	0	0
0	0	46	0	0	0
0	0	46	27	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
1	72	0	0	0	0
0	0	27	19	0	0
0	0	27	46	0	0
0	0	0	0	72	0
0	0	0	0	0	72

72	0	0	0	0	0
0	72	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	0	1
0	0	0	0	72	1

27	0	0	0	0	0
0	27	0	0	0	0
0	0	22	0	0	0
0	0	0	22	0	0
0	0	0	0	0	46
0	0	0	0	46	0

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,27,27,0,0,0,0,19,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

D₁₂.₂Dic₃ in GAP, Magma, Sage, TeX

D_{12}._2{\rm Dic}_3

% in TeX

G:=Group("D12.2Dic3");

// GroupNames label

G:=SmallGroup(288,462);

// by ID

G=gap.SmallGroup(288,462);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^6*c^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;

// generators/relations

G = D12.2Dic3 order 288 = 25·32

The non-split extension by D12 of Dic3 acting through Inn(D12)

G = D₁₂.₂Dic₃ order 288 = 2⁵·3²

The non-split extension by D₁₂ of Dic₃ acting through Inn(D₁₂)