D12.34D6 - GroupNames

G = D₁₂.₃₄D₆ order 288 = 2⁵·3²

The non-split extension by D₁₂ of D₆ acting through Inn(D₁₂)

Aliases: D₁₂.₃₄D₆, Dic₆.₃₅D₆, C₃²:₂2_-¹⁺⁴, C₆².₁₃₀C₂³, C₄oD₁₂:₆S₃, C₃:D₄.₂D₆, C₃:₂(Q₈oD₁₂), (S₃xDic₆):₈C₂, (C₄xS₃).₁₄D₆, C₆.₅(S₃xC₂³), (C₃xC₆).₅C₂⁴, D₁₂:₅S₃:₇C₂, D₆.₄D₆:₃C₂, (C₂xC₁₂).₁₆₇D₆, (S₃xC₆).₃C₂³, D₆.₄(C₂²xS₃), (S₃xC₁₂).₃₁C₂², (C₆xC₁₂).₁₆₀C₂², (C₃xC₁₂).₁₁₄C₂³, C₁₂.₁₃₁(C₂²xS₃), D₆:S₃.₄C₂², (C₃xD₁₂).₄₃C₂², C₃:Dic₃.₁₆C₂³, (C₃xDic₃).₄C₂³, Dic₃.₃(C₂²xS₃), (S₃xDic₃).₂C₂², C₃²:₂Q₈.₅C₂², (C₃xDic₆).₄₄C₂², C₃²:₄Q₈.₃₅C₂², C₄.₆₂(C₂xS₃²), (C₂xC₄).₃₆S₃², C₂.₈(C₂²xS₃²), C₂².₁₁(C₂xS₃²), (C₃xC₄oD₁₂):₁₀C₂, (C₂xC₆).₁₃(C₂²xS₃), (C₂xC₃²:₄Q₈):₂₀C₂, (C₃xC₃:D₄).₃C₂², (C₂xC₃:Dic₃).₁₀₃C₂², SmallGroup(288,946)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — D₁₂.₃₄D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — S₃xDic₃ — S₃xDic₆ — D₁₂.₃₄D₆

Lower central

C₃² — C₃xC₆ — D₁₂.₃₄D₆

Upper central

C₁ — C₂ — C₂xC₄

Generators and relations for D₁₂.₃₄D₆
G = < a,b,c,d | a¹²=b²=1, c⁶=d²=a⁶, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=a⁶c⁵ >

Subgroups: 994 in 311 conjugacy classes, 108 normal (10 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₂xC₄, C₂xC₄, D₄, Q₈, C₃², Dic₃, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xQ₈, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, Dic₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, 2_-¹⁺⁴, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, C₆², C₂xDic₆, C₄oD₁₂, C₄oD₁₂, D₄:₂S₃, S₃xQ₈, C₃xC₄oD₄, S₃xDic₃, D₆:S₃, C₃²:₂Q₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₃²:₄Q₈, C₂xC₃:Dic₃, C₆xC₁₂, Q₈oD₁₂, S₃xDic₆, D₁₂:₅S₃, D₆.₄D₆, C₃xC₄oD₁₂, C₂xC₃²:₄Q₈, D₁₂.₃₄D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²xS₃, 2_-¹⁺⁴, S₃², S₃xC₂³, C₂xS₃², Q₈oD₁₂, C₂²xS₃², D₁₂.₃₄D₆

Smallest permutation representation of D₁₂.₃₄D₆
►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 45)(14 44)(15 43)(16 42)(17 41)(18 40)(19 39)(20 38)(21 37)(22 48)(23 47)(24 46)

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

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

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

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

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

45 conjugacy classes

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

45 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+	+	-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	D₆	D₆	D₆	D₆	D₆	2_-¹⁺⁴	S₃²	C₂xS₃²	C₂xS₃²	Q₈oD₁₂	D₁₂.₃₄D₆
kernel	D₁₂.₃₄D₆	S₃xDic₆	D₁₂:₅S₃	D₆.₄D₆	C₃xC₄oD₁₂	C₂xC₃²:₄Q₈	C₄oD₁₂	Dic₆	C₄xS₃	D₁₂	C₃:D₄	C₂xC₁₂	C₃²	C₂xC₄	C₄	C₂²	C₃	C₁
# reps	1	4	4	4	2	1	2	2	4	2	4	2	1	1	2	1	4	4

Matrix representation of D₁₂.₃₄D₆ ►in GL₄(F₁₃) generated by

7	3	0	0
10	10	0	0
0	0	7	3
0	0	10	10

0	0	8	5
0	0	0	5
5	8	0	0
0	8	0	0

6	10	0	0
3	3	0	0
0	0	3	3
0	0	10	6

0	0	10	10
0	0	3	7
6	10	0	0
3	3	0	0

G:=sub<GL(4,GF(13))| [7,10,0,0,3,10,0,0,0,0,7,10,0,0,3,10],[0,0,5,0,0,0,8,8,8,0,0,0,5,5,0,0],[6,3,0,0,10,3,0,0,0,0,3,10,0,0,3,6],[0,0,6,3,0,0,10,3,10,3,0,0,10,7,0,0] >;

D₁₂.₃₄D₆ in GAP, Magma, Sage, TeX

D_{12}._{34}D_6

% in TeX

G:=Group("D12.34D6");

// GroupNames label

G:=SmallGroup(288,946);

// by ID

G=gap.SmallGroup(288,946);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = D12.34D6 order 288 = 25·32

The non-split extension by D12 of D6 acting through Inn(D12)

G = D₁₂.₃₄D₆ order 288 = 2⁵·3²

The non-split extension by D₁₂ of D₆ acting through Inn(D₁₂)