D12:20D6 - GroupNames

G = D₁₂:₂₀D₆ order 288 = 2⁵·3²

4^th semidirect product of D₁₂ and D₆ acting via D₆/C₆=C₂

Aliases: D₁₂:₂₀D₆, Dic₆:₁₈D₆, C₆².₄₂D₄, C₄oD₁₂:₃S₃, (C₂xD₁₂):₈S₃, (C₆xD₁₂):₂C₂, (C₃xC₁₂).₆₁D₄, C₃²:₂D₈:₅C₂, C₃:₄(D₄:D₆), (C₂xC₁₂).₁₁₂D₆, C₃²:₆(C₈:C₂²), Dic₆:S₃:₅C₂, C₃:₄(D₁₂:₆C₂²), C₁₂.₅₈D₆:₉C₂, (C₃xD₁₂):₂₀C₂², C₁₂.₄₃(C₃:D₄), (C₃xC₁₂).₅₉C₂³, (C₆xC₁₂).₇₂C₂², C₁₂.₇₉(C₂²xS₃), C₃²:₄C₈:₂C₂², C₄.₁₃(D₆:S₃), (C₃xDic₆):₁₈C₂², C₂².₄(D₆:S₃), (C₂xC₄).₅S₃², C₄.₇₄(C₂xS₃²), (C₃xC₄oD₁₂):₂C₂, (C₃xC₆).₆₃(C₂xD₄), C₆.₇₁(C₂xC₃:D₄), C₂.₆(C₂xD₆:S₃), (C₂xC₆).₅₅(C₃:D₄), SmallGroup(288,471)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — D₁₂:₂₀D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃xD₁₂ — C₃²:₂D₈ — D₁₂:₂₀D₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — D₁₂:₂₀D₆

Upper central

C₁ — C₂ — C₂xC₄

Generators and relations for D₁₂:₂₀D₆
G = < a,b,c,d | a¹²=b²=c⁶=d²=1, bab=a^-1, ac=ca, dad=a⁷, cbc^-1=a⁶b, dbd=a³b, dcd=c^-1 >

Subgroups: 530 in 146 conjugacy classes, 44 normal (34 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², C₂², S₃, C₆, C₆, C₈, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₃², Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, M₄(2), D₈, SD₁₆, C₂xD₄, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, Dic₆, C₄xS₃, D₁₂, D₁₂, D₁₂, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₂²xS₃, C₂²xC₆, C₈:C₂², C₃xDic₃, C₃xC₁₂, S₃xC₆, C₆², C₄.Dic₃, D₄:S₃, D₄.S₃, Q₈:₂S₃, C₂xD₁₂, C₄oD₁₂, C₆xD₄, C₃xC₄oD₄, C₃²:₄C₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xD₁₂, C₃xD₁₂, C₃xC₃:D₄, C₆xC₁₂, S₃xC₂xC₆, D₁₂:₆C₂², D₄:D₆, C₃²:₂D₈, Dic₆:S₃, C₁₂.₅₈D₆, C₆xD₁₂, C₃xC₄oD₁₂, D₁₂:₂₀D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₃:D₄, C₂²xS₃, C₈:C₂², S₃², C₂xC₃:D₄, D₆:S₃, C₂xS₃², D₁₂:₆C₂², D₄:D₆, C₂xD₆:S₃, D₁₂:₂₀D₆

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

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	2D	2E	3A	3B	3C	4A	4B	4C	6A	6B	6C	6D	6E	6F	6G	6H	6I	···	6N	8A	8B	12A	12B	12C	···	12I	12J	12K
order	1	2	2	2	2	2	3	3	3	4	4	4	6	6	6	6	6	6	6	6	6	···	6	8	8	12	12	12	···	12	12	12
size	1	1	2	12	12	12	2	2	4	2	2	12	2	2	2	2	4	4	4	4	12	···	12	36	36	2	2	4	···	4	12	12

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+	+	+	+	+	+			+	+	-	+	-		+
image	C₁	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₄	D₄	D₆	D₆	D₆	C₃:D₄	C₃:D₄	C₈:C₂²	S₃²	D₆:S₃	C₂xS₃²	D₆:S₃	D₁₂:₆C₂²	D₄:D₆	D₁₂:₂₀D₆
kernel	D₁₂:₂₀D₆	C₃²:₂D₈	Dic₆:S₃	C₁₂.₅₈D₆	C₆xD₁₂	C₃xC₄oD₁₂	C₂xD₁₂	C₄oD₁₂	C₃xC₁₂	C₆²	Dic₆	D₁₂	C₂xC₁₂	C₁₂	C₂xC₆	C₃²	C₂xC₄	C₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	2	2	1	1	1	1	1	1	1	1	3	2	4	4	1	1	1	1	1	2	2	4

Matrix representation of D₁₂:₂₀D₆ ►in GL₄(F₇₃) generated by

49	0	0	0
0	3	0	0
0	0	24	0
0	0	0	70

0	3	0	0
49	0	0	0
0	0	0	65
0	0	9	0

8	0	0	0
0	65	0	0
0	0	64	0
0	0	0	9

0	0	64	0
0	0	0	9
8	0	0	0
0	65	0	0

G:=sub<GL(4,GF(73))| [49,0,0,0,0,3,0,0,0,0,24,0,0,0,0,70],[0,49,0,0,3,0,0,0,0,0,0,9,0,0,65,0],[8,0,0,0,0,65,0,0,0,0,64,0,0,0,0,9],[0,0,8,0,0,0,0,65,64,0,0,0,0,9,0,0] >;

D₁₂:₂₀D₆ in GAP, Magma, Sage, TeX

D_{12}\rtimes_{20}D_6

% in TeX

G:=Group("D12:20D6");

// GroupNames label

G:=SmallGroup(288,471);

// by ID

G=gap.SmallGroup(288,471);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,219,675,346,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = D12:20D6 order 288 = 25·32

4th semidirect product of D12 and D6 acting via D6/C6=C2

G = D₁₂:₂₀D₆ order 288 = 2⁵·3²

4^th semidirect product of D₁₂ and D₆ acting via D₆/C₆=C₂