D12.25D6 - GroupNames

G = D₁₂.₂₅D₆ order 288 = 2⁵·3²

10^th non-split extension by D₁₂ of D₆ acting via D₆/S₃=C₂

Aliases: D₁₂.₂₅D₆, Dic₆.₂₅D₆, C₃²:₄2_-¹⁺⁴, (S₃xQ₈):₇S₃, Q₈.₁₈S₃², Q₈:₃S₃:₇S₃, C₃:₄(Q₈oD₁₂), (C₄xS₃).₁₆D₆, (C₃xQ₈).₄₆D₆, (S₃xDic₆):₁₁C₂, D₆.D₆:₈C₂, (C₃xC₆).₂₂C₂⁴, C₆.₂₂(S₃xC₂³), D₁₂:S₃:₁₁C₂, D₁₂:₅S₃:₁₁C₂, (S₃xC₆).₂₄C₂³, C₁₂.₃₄(C₂²xS₃), (C₃xC₁₂).₃₄C₂³, D₆.₂₂(C₂²xS₃), (S₃xC₁₂).₃₃C₂², C₃:₃(Q₈.₁₅D₆), D₆:S₃.₉C₂², C₃:D₁₂.₃C₂², (C₃xD₁₂).₃₀C₂², C₃:Dic₃.₂₄C₂³, (S₃xDic₃).₄C₂², Dic₃.₁₁(C₂²xS₃), (C₃xDic₆).₃₀C₂², (C₃xDic₃).₁₅C₂³, C₃²:₂Q₈.₁₀C₂², (Q₈xC₃²).₂₁C₂², C₃²:₄Q₈.₂₂C₂², (C₃xS₃xQ₈):₇C₂, C₄.₃₄(C₂xS₃²), (Q₈xC₃:S₃):₆C₂, C₂.₂₄(C₂²xS₃²), (C₃xQ₈:₃S₃):₇C₂, (C₄xC₃:S₃).₄₄C₂², (C₂xC₃:S₃).₄₆C₂³, SmallGroup(288,963)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃xC₆ — D₁₂.₂₅D₆

Chief series

C₁ — C₃ — C₃² — C₃xC₆ — S₃xC₆ — S₃xDic₃ — S₃xDic₆ — D₁₂.₂₅D₆

Lower central

C₃² — C₃xC₆ — D₁₂.₂₅D₆

Upper central

C₁ — C₂ — Q₈

Generators and relations for D₁₂.₂₅D₆
G = < a,b,c,d | a¹²=b²=1, c⁶=d²=a⁶, bab=a^-1, cac^-1=a⁷, ad=da, cbc^-1=a⁶b, bd=db, dcd^-1=a⁶c⁵ >

Subgroups: 1026 in 311 conjugacy classes, 108 normal (24 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂xC₄, D₄, Q₈, Q₈, C₃², Dic₃, Dic₃, Dic₃, C₁₂, C₁₂, D₆, D₆, D₆, C₂xC₆, C₂xQ₈, C₄oD₄, C₃xS₃, C₃:S₃, C₃xC₆, Dic₆, Dic₆, C₄xS₃, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₃xD₄, C₃xQ₈, C₃xQ₈, 2_-¹⁺⁴, C₃xDic₃, C₃xDic₃, C₃:Dic₃, C₃xC₁₂, S₃xC₆, S₃xC₆, C₂xC₃:S₃, C₂xDic₆, C₄oD₁₂, D₄:₂S₃, S₃xQ₈, S₃xQ₈, Q₈:₃S₃, Q₈:₃S₃, C₆xQ₈, C₃xC₄oD₄, S₃xDic₃, D₆:S₃, C₃:D₁₂, C₃:D₁₂, C₃²:₂Q₈, C₃xDic₆, S₃xC₁₂, C₃xD₁₂, C₃²:₄Q₈, C₄xC₃:S₃, Q₈xC₃², Q₈.₁₅D₆, Q₈oD₁₂, S₃xDic₆, D₁₂:₅S₃, D₁₂:S₃, D₆.D₆, C₃xS₃xQ₈, C₃xQ₈:₃S₃, Q₈xC₃:S₃, D₁₂.₂₅D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₂⁴, C₂²xS₃, 2_-¹⁺⁴, S₃², S₃xC₂³, C₂xS₃², Q₈.₁₅D₆, Q₈oD₁₂, C₂²xS₃², D₁₂.₂₅D₆

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

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

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

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

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

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

42 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	6D	6E	6F	6G	6H	12A	···	12F	12G	12H	12I	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	6	6	6	6	18	2	2	4	2	2	2	6	6	6	6	18	18	18	2	2	4	6	6	12	12	12	4	···	4	6	6	8	8	8	12	12	12

42 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	8
type	+	+	+	+	+	+	+	+	+	+	+	+	+	+	-	+	+		-	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	S₃	S₃	D₆	D₆	D₆	D₆	2_-¹⁺⁴	S₃²	C₂xS₃²	Q₈.₁₅D₆	Q₈oD₁₂	D₁₂.₂₅D₆
kernel	D₁₂.₂₅D₆	S₃xDic₆	D₁₂:₅S₃	D₁₂:S₃	D₆.D₆	C₃xS₃xQ₈	C₃xQ₈:₃S₃	Q₈xC₃:S₃	S₃xQ₈	Q₈:₃S₃	Dic₆	C₄xS₃	D₁₂	C₃xQ₈	C₃²	Q₈	C₄	C₃	C₃	C₁
# reps	1	3	3	3	3	1	1	1	1	1	3	6	3	2	1	1	3	2	2	1

Matrix representation of D₁₂.₂₅D₆ ►in GL₆(F₁₃)

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	0	5	8
0	0	0	0	5	0
0	0	5	8	0	0
0	0	5	0	0	0

12	0	0	0	0	0
0	12	0	0	0	0
0	0	0	0	9	2
0	0	0	0	11	4
0	0	4	11	0	0
0	0	2	9	0	0

0	12	0	0	0	0
1	1	0	0	0	0
0	0	5	0	0	0
0	0	0	5	0	0
0	0	0	0	8	0
0	0	0	0	0	8

0	1	0	0	0	0
1	0	0	0	0	0
0	0	0	0	3	7
0	0	0	0	6	10
0	0	3	7	0	0
0	0	6	10	0	0

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,8,0,0,0,5,5,0,0,0,0,8,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,9,11,0,0,0,0,2,4,0,0],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0] >;

D₁₂.₂₅D₆ in GAP, Magma, Sage, TeX

D_{12}._{25}D_6

% in TeX

G:=Group("D12.25D6");

// GroupNames label

G:=SmallGroup(288,963);

// by ID

G=gap.SmallGroup(288,963);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,80,1356,9414]);

// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=a^6*c^5>;

// generators/relations

G = D12.25D6 order 288 = 25·32

10th non-split extension by D12 of D6 acting via D6/S3=C2

G = D₁₂.₂₅D₆ order 288 = 2⁵·3²

10^th non-split extension by D₁₂ of D₆ acting via D₆/S₃=C₂