D12.29D6 - GroupNames

G = D₁₂.₂₉D₆ order 288 = 2⁵·3²

4^th non-split extension by D₁₂ of D₆ acting via D₆/C₆=C₂

Aliases: D₁₂.₂₉D₆, C₁₂.₇₅D₁₂, C₆².₅₀D₄, Dic₆.₂₈D₆, C₃:C₈.₃D₆, C₄oD₁₂.₅S₃, C₆.₇₀(C₂xD₁₂), (C₃xC₁₂).₆₈D₄, (C₂xC₆).₁₄D₁₂, C₄.Dic₃:₉S₃, (C₂xC₁₂).₁₂₀D₆, C₃:₅(C₈.D₆), C₃²:₃Q₁₆:₅C₂, D₁₂.S₃:₄C₂, C₃:₁(Q₈.₁₄D₆), C₁₂.₄₈(C₃:D₄), (C₆xC₁₂).₈₀C₂², (C₃xC₁₂).₆₇C₂³, C₁₂.₈₄(C₂²xS₃), C₃²:₇(C₈.C₂²), C₄.₁₇(C₃:D₁₂), (C₃xD₁₂).₃₇C₂², (C₃xDic₆).₃₅C₂², C₂².₁₀(C₃:D₁₂), C₃²:₄Q₈.₂₇C₂², (C₂xC₄).₉S₃², C₄.₅₅(C₂xS₃²), C₆.₆(C₂xC₃:D₄), (C₃xC₆).₇₁(C₂xD₄), (C₃xC₃:C₈).₃C₂², (C₃xC₄oD₁₂).₈C₂, (C₃xC₄.Dic₃):₃C₂, C₂.₁₀(C₂xC₃:D₁₂), (C₂xC₆).₂₀(C₃:D₄), (C₂xC₃²:₄Q₈):₁₀C₂, SmallGroup(288,479)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₁₂ — D₁₂.₂₉D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — C₃xC₁₂ — C₃xD₁₂ — D₁₂.S₃ — D₁₂.₂₉D₆

Lower central

C₃² — C₃xC₆ — C₃xC₁₂ — D₁₂.₂₉D₆

Upper central

C₁ — C₂ — C₂xC₄

Generators and relations for D₁₂.₂₉D₆
G = < a,b,c,d | a¹²=b²=1, c⁶=a⁶, d²=a⁹, bab=a^-1, ac=ca, ad=da, bc=cb, dbd^-1=a³b, dcd^-1=a⁶c⁵ >

Subgroups: 514 in 139 conjugacy classes, 44 normal (34 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, M₄(2), SD₁₆, Q₁₆, C₂xQ₈, C₄oD₄, C₃xS₃, C₃xC₆, C₃xC₆, C₃:C₈, C₂₄, Dic₆, Dic₆, C₄xS₃, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₈.C₂², C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, C₆², C₂₄:C₂, Dic₁₂, C₄.Dic₃, D₄.S₃, C₃:Q₁₆, C₃xM₄(2), C₂xDic₆, C₄oD₁₂, C₃xC₄oD₄, C₃xC₃:C₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃xC₃:D₄, C₃²:₄Q₈, C₃²:₄Q₈, C₂xC₃:Dic₃, C₆xC₁₂, C₈.D₆, Q₈.₁₄D₆, D₁₂.S₃, C₃²:₃Q₁₆, C₃xC₄.Dic₃, C₃xC₄oD₁₂, C₂xC₃²:₄Q₈, D₁₂.₂₉D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, D₁₂, C₃:D₄, C₂²xS₃, C₈.C₂², S₃², C₂xD₁₂, C₂xC₃:D₄, C₃:D₁₂, C₂xS₃², C₈.D₆, Q₈.₁₄D₆, C₂xC₃:D₁₂, D₁₂.₂₉D₆

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

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

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

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

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

39 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	4A	4B	4C	4D	4E	6A	6B	6C	···	6G	6H	6I	8A	8B	12A	12B	12C	12D	12E	···	12J	12K	12L	24A	24B	24C	24D
order	1	2	2	2	3	3	3	4	4	4	4	4	6	6	6	···	6	6	6	8	8	12	12	12	12	12	···	12	12	12	24	24	24	24
size	1	1	2	12	2	2	4	2	2	12	36	36	2	2	4	···	4	12	12	12	12	2	2	2	2	4	···	4	12	12	12	12	12	12

39 irreducible representations

dim	1	1	1	1	1	1	2	2	2	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	D₆	D₁₂	C₃:D₄	D₁₂	C₃:D₄	C₈.C₂²	S₃²	C₃:D₁₂	C₂xS₃²	C₃:D₁₂	C₈.D₆	Q₈.₁₄D₆	D₁₂.₂₉D₆
kernel	D₁₂.₂₉D₆	D₁₂.S₃	C₃²:₃Q₁₆	C₃xC₄.Dic₃	C₃xC₄oD₁₂	C₂xC₃²:₄Q₈	C₄.Dic₃	C₄oD₁₂	C₃xC₁₂	C₆²	C₃:C₈	Dic₆	D₁₂	C₂xC₁₂	C₁₂	C₁₂	C₂xC₆	C₂xC₆	C₃²	C₂xC₄	C₄	C₄	C₂²	C₃	C₃	C₁
# reps	1	2	2	1	1	1	1	1	1	1	2	1	1	2	2	2	2	2	1	1	1	1	1	2	2	4

Matrix representation of D₁₂.₂₉D₆ ►in GL₄(F₇₃) generated by

14	66	0	0
7	7	0	0
7	0	7	66
0	66	7	14

22	22	39	41
10	10	41	2
60	70	51	63
70	48	51	63

59	7	0	0
66	66	0	0
0	7	59	66
66	0	7	66

7	7	52	0
14	14	52	52
56	55	59	66
56	56	59	66

G:=sub<GL(4,GF(73))| [14,7,7,0,66,7,0,66,0,0,7,7,0,0,66,14],[22,10,60,70,22,10,70,48,39,41,51,51,41,2,63,63],[59,66,0,66,7,66,7,0,0,0,59,7,0,0,66,66],[7,14,56,56,7,14,55,56,52,52,59,59,0,52,66,66] >;

D₁₂.₂₉D₆ in GAP, Magma, Sage, TeX

D_{12}._{29}D_6

% in TeX

G:=Group("D12.29D6");

// GroupNames label

G:=SmallGroup(288,479);

// by ID

G=gap.SmallGroup(288,479);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,100,675,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=a^6,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^5>;

// generators/relations

G = D12.29D6 order 288 = 25·32

4th non-split extension by D12 of D6 acting via D6/C6=C2

G = D₁₂.₂₉D₆ order 288 = 2⁵·3²

4^th non-split extension by D₁₂ of D₆ acting via D₆/C₆=C₂