D12.8D6 - GroupNames

G = D₁₂.₈D₆ order 288 = 2⁵·3²

8^th non-split extension by D₁₂ of D₆ acting via D₆/C₃=C₂²

Aliases: D₁₂.₈D₆, Dic₆.₇D₆, D₄.₉S₃², D₄:S₃:₅S₃, C₃:C₈.₁₄D₆, D₄.S₃:₅S₃, C₆.₅₉(S₃xD₄), (C₃xD₄).₁₃D₆, D₁₂:S₃:₄C₂, C₃:₃(D₈:₃S₃), C₃²:₁₂(C₄oD₈), C₃:Dic₃.₅₈D₄, C₃²:₃Q₁₆:₈C₂, C₁₂.₂₉D₆:₁C₂, C₁₂.D₆:₂C₂, C₃:₃(Q₈.₇D₆), C₁₂.₁₃(C₂²xS₃), (C₃xC₁₂).₁₃C₂³, D₁₂.S₃:₁₂C₂, C₂.₁₉(Dic₃:D₆), (C₃xD₁₂).₁₅C₂², (D₄xC₃²).₉C₂², C₃²:₄Q₈.₉C₂², (C₃xDic₆).₁₄C₂², C₄.₁₃(C₂xS₃²), (C₃xD₄:S₃):₄C₂, (C₃xD₄.S₃):₈C₂, (C₂xC₃:S₃).₂₃D₄, (C₃xC₆).₁₂₈(C₂xD₄), (C₃xC₃:C₈).₁₈C₂², (C₄xC₃:S₃).₁₅C₂², SmallGroup(288,584)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — D₁₂.₈D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃xD₁₂ — D₁₂:S₃ — D₁₂.₈D₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — D₁₂.₈D₆

Upper central

C₁ — C₂ — C₄ — D₄

Generators and relations for D₁₂.₈D₆
G = < a,b,c,d | a¹²=b²=c⁶=1, d²=a⁹, bab=a^-1, cac^-1=a⁷, ad=da, cbc^-1=dbd^-1=a³b, dcd^-1=a³c^-1 >

Subgroups: 602 in 144 conjugacy classes, 38 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₈, C₂xC₄, D₄, D₄, Q₈, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₈, D₈, SD₁₆, Q₁₆, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, Dic₆, Dic₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₃xD₄, C₃xD₄, C₃xQ₈, C₄oD₈, C₃xDic₃, C₃:Dic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, C₂xC₃:S₃, C₆², S₃xC₈, C₂₄:C₂, Dic₁₂, D₄:S₃, D₄.S₃, D₄.S₃, C₃:Q₁₆, C₃xD₈, C₃xSD₁₆, D₄:₂S₃, Q₈:₃S₃, C₃xC₃:C₈, S₃xDic₃, C₃:D₁₂, C₃xDic₆, C₃xD₁₂, C₃²:₄Q₈, C₄xC₃:S₃, C₂xC₃:Dic₃, C₃²:₇D₄, D₄xC₃², D₈:₃S₃, Q₈.₇D₆, C₁₂.₂₉D₆, D₁₂.S₃, C₃²:₃Q₁₆, C₃xD₄:S₃, C₃xD₄.S₃, D₁₂:S₃, C₁₂.D₆, D₁₂.₈D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂²xS₃, C₄oD₈, S₃², S₃xD₄, C₂xS₃², D₈:₃S₃, Q₈.₇D₆, Dic₃:D₆, D₁₂.₈D₆

Smallest permutation representation of D₁₂.₈D₆
►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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 39)(14 38)(15 37)(16 48)(17 47)(18 46)(19 45)(20 44)(21 43)(22 42)(23 41)(24 40)

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

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

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

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

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

33 conjugacy classes

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

33 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+	+		+	+	+	-		+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	D₄	D₆	D₆	D₆	D₆	C₄oD₈	S₃²	S₃xD₄	C₂xS₃²	D₈:₃S₃	Q₈.₇D₆	Dic₃:D₆	D₁₂.₈D₆
kernel	D₁₂.₈D₆	C₁₂.₂₉D₆	D₁₂.S₃	C₃²:₃Q₁₆	C₃xD₄:S₃	C₃xD₄.S₃	D₁₂:S₃	C₁₂.D₆	D₄:S₃	D₄.S₃	C₃:Dic₃	C₂xC₃:S₃	C₃:C₈	Dic₆	D₁₂	C₃xD₄	C₃²	D₄	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	1	1	1	1	1	1	1	1	1	1	1	2	1	1	2	4	1	2	1	2	2	2	1

Matrix representation of D₁₂.₈D₆ ►in GL₆(F₇₃)

27	0	0	0	0	0
36	46	0	0	0	0
0	0	72	1	0	0
0	0	72	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

15	14	0	0	0	0
57	58	0	0	0	0
0	0	1	0	0	0
0	0	1	72	0	0
0	0	0	0	72	0
0	0	0	0	0	72

4	67	0	0	0	0
39	69	0	0	0	0
0	0	72	0	0	0
0	0	0	72	0	0
0	0	0	0	72	1
0	0	0	0	72	0

63	0	0	0	0	0
3	22	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	1	72

G:=sub<GL(6,GF(73))| [27,36,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[15,57,0,0,0,0,14,58,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[4,39,0,0,0,0,67,69,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[63,3,0,0,0,0,0,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

D₁₂.₈D₆ in GAP, Magma, Sage, TeX

D_{12}._8D_6

% in TeX

G:=Group("D12.8D6");

// GroupNames label

G:=SmallGroup(288,584);

// by ID

G=gap.SmallGroup(288,584);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,254,135,675,346,185,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^9,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^3*c^-1>;

// generators/relations

G = D12.8D6 order 288 = 25·32

8th non-split extension by D12 of D6 acting via D6/C3=C22

G = D₁₂.₈D₆ order 288 = 2⁵·3²

8^th non-split extension by D₁₂ of D₆ acting via D₆/C₃=C₂²