D12.2Dic3 - GroupNames

G = D₁₂.₂Dic₃ order 288 = 2⁵·3²

The non-split extension by D₁₂ of Dic₃ acting through Inn(D₁₂)

Aliases: D₁₂.₂Dic₃, Dic₆.₂Dic₃, C₃⋊C₈.₃₁D₆, C₃⋊₅(C₈○D₁₂), C₁₂.₃₀(C₄×S₃), C₄○D₁₂.₆S₃, (C₄×S₃).₃₁D₆, (C₃×D₁₂).₁C₄, C₃²⋊₄(C₈○D₄), C₄.₄(S₃×Dic₃), (C₂×C₁₂).₁₀₇D₆, C₆².₄₄(C₂×C₄), (C₃×Dic₆).₁C₄, D₆.₁(C₂×Dic₃), C₃⋊D₄.₂Dic₃, C₁₂.₅₈D₆⋊₈C₂, D₆.Dic₃⋊₁₀C₂, (S₃×C₁₂).₈C₂², C₃⋊₁(D₄.Dic₃), (C₆×C₁₂).₆₆C₂², C₁₂.₂₅(C₂×Dic₃), C₂².₁(S₃×Dic₃), C₆.₃(C₂²×Dic₃), (C₃×C₁₂).₁₄₃C₂³, C₁₂.₁₄₂(C₂²×S₃), Dic₃.₁(C₂×Dic₃), C₃²⋊₄C₈.₂₀C₂², (C₂×C₃⋊C₈)⋊₃S₃, (S₃×C₃⋊C₈)⋊₉C₂, (C₆×C₃⋊C₈)⋊₁₆C₂, C₄.₈₉(C₂×S₃²), (C₂×C₄).₆₂S₃², C₆.₈₃(S₃×C₂×C₄), C₂.₅(C₂×S₃×Dic₃), (S₃×C₆).₄(C₂×C₄), (C₂×C₆).₁₆(C₄×S₃), (C₃×C₃⋊D₄).₁C₄, (C₃×C₁₂).₅₆(C₂×C₄), (C₃×C₄○D₁₂).₅C₂, (C₃×C₃⋊C₈).₃₈C₂², (C₃×C₆).₃₉(C₂²×C₄), (C₃×Dic₃).₄(C₂×C₄), (C₂×C₆).₁₈(C₂×Dic₃), SmallGroup(288,462)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃×C₆ — D₁₂.₂Dic₃

Chief series

C₁ — C₃ — C₃² — C₃×C₆ — C₃×C₁₂ — S₃×C₁₂ — S₃×C₃⋊C₈ — D₁₂.₂Dic₃

Lower central

C₃² — C₃×C₆ — D₁₂.₂Dic₃

Upper central

C₁ — C₄ — C₂×C₄

Generators and relations for D₁₂.₂Dic₃
G = < a,b,c,d | a¹²=b²=1, c⁶=a⁶, d²=a⁶c³, bab=a^-1, ac=ca, ad=da, bc=cb, bd=db, dcd^-1=c⁵ >

Subgroups: 338 in 134 conjugacy classes, 60 normal (38 characteristic)
C₁, C₂, C₂^[×3], C₃^[×2], C₃, C₄^[×2], C₄^[×2], C₂², C₂²^[×2], S₃^[×2], C₆^[×2], C₆^[×6], C₈^[×4], C₂×C₄, C₂×C₄^[×2], D₄^[×3], Q₈, C₃², Dic₃^[×2], C₁₂^[×4], C₁₂^[×4], D₆^[×2], C₂×C₆^[×2], C₂×C₆^[×3], C₂×C₈^[×3], M₄(2)^[×3], C₄○D₄, C₃×S₃^[×2], C₃×C₆, C₃×C₆, C₃⋊C₈^[×2], C₃⋊C₈^[×6], C₂₄^[×2], Dic₆, C₄×S₃^[×2], D₁₂, C₃⋊D₄^[×2], C₂×C₁₂^[×2], C₂×C₁₂^[×3], C₃×D₄^[×3], C₃×Q₈, C₈○D₄, C₃×Dic₃^[×2], C₃×C₁₂^[×2], S₃×C₆^[×2], C₆², S₃×C₈^[×2], C₈⋊S₃^[×2], C₂×C₃⋊C₈, C₂×C₃⋊C₈^[×2], C₄.Dic₃^[×5], C₂×C₂₄, C₄○D₁₂, C₃×C₄○D₄, C₃×C₃⋊C₈^[×2], C₃²⋊₄C₈^[×2], C₃×Dic₆, S₃×C₁₂^[×2], C₃×D₁₂, C₃×C₃⋊D₄^[×2], C₆×C₁₂, C₈○D₁₂, D₄.Dic₃, S₃×C₃⋊C₈^[×2], D₆.Dic₃^[×2], C₆×C₃⋊C₈, C₁₂.₅₈D₆, C₃×C₄○D₁₂, D₁₂.₂Dic₃
Quotients: C₁, C₂^[×7], C₄^[×4], C₂²^[×7], S₃^[×2], C₂×C₄^[×6], C₂³, Dic₃^[×4], D₆^[×6], C₂²×C₄, C₄×S₃^[×2], C₂×Dic₃^[×6], C₂²×S₃^[×2], C₈○D₄, S₃², S₃×C₂×C₄, C₂²×Dic₃, S₃×Dic₃^[×2], C₂×S₃², C₈○D₁₂, D₄.Dic₃, C₂×S₃×Dic₃, D₁₂.₂Dic₃

Smallest permutation representation of D₁₂.₂Dic₃
►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 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	4A	4B	4C	4D	4E	6A	6B	6C	6D	6E	6F	6G	6H	6I	6J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J	12A	···	12F	12G	···	12K	12L	12M	24A	···	24H
order	1	2	2	2	2	3	3	3	4	4	4	4	4	6	6	6	6	6	6	6	6	6	6	8	8	8	8	8	8	8	8	8	8	12	···	12	12	···	12	12	12	24	···	24
size	1	1	2	6	6	2	2	4	1	1	2	6	6	2	2	2	2	4	4	4	4	12	12	3	3	3	3	6	6	18	18	18	18	2	···	2	4	···	4	12	12	6	···	6

54 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+				+	+	+	-	+	-	-	+					+	-	+	-
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	S₃	S₃	D₆	Dic₃	D₆	Dic₃	Dic₃	D₆	C₄×S₃	C₄×S₃	C₈○D₄	C₈○D₁₂	S₃²	S₃×Dic₃	C₂×S₃²	S₃×Dic₃	D₄.Dic₃	D₁₂.₂Dic₃
kernel	D₁₂.₂Dic₃	S₃×C₃⋊C₈	D₆.Dic₃	C₆×C₃⋊C₈	C₁₂.₅₈D₆	C₃×C₄○D₁₂	C₃×Dic₆	C₃×D₁₂	C₃×C₃⋊D₄	C₂×C₃⋊C₈	C₄○D₁₂	C₃⋊C₈	Dic₆	C₄×S₃	D₁₂	C₃⋊D₄	C₂×C₁₂	C₁₂	C₂×C₆	C₃²	C₃	C₂×C₄	C₄	C₄	C₂²	C₃	C₁
# reps	1	2	2	1	1	1	2	2	4	1	1	2	1	2	1	2	2	2	2	4	8	1	1	1	1	2	4

Matrix representation of D₁₂.₂Dic₃ ►in GL₆(𝔽₇₃)

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

1	0	0	0	0	0
1	72	0	0	0	0
0	0	27	19	0	0
0	0	27	46	0	0
0	0	0	0	72	0
0	0	0	0	0	72

72	0	0	0	0	0
0	72	0	0	0	0
0	0	46	0	0	0
0	0	0	46	0	0
0	0	0	0	0	1
0	0	0	0	72	1

27	0	0	0	0	0
0	27	0	0	0	0
0	0	22	0	0	0
0	0	0	22	0	0
0	0	0	0	0	46
0	0	0	0	46	0

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,46,46,0,0,0,0,0,27,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,27,27,0,0,0,0,19,46,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[27,0,0,0,0,0,0,27,0,0,0,0,0,0,22,0,0,0,0,0,0,22,0,0,0,0,0,0,0,46,0,0,0,0,46,0] >;

D₁₂.₂Dic₃ in GAP, Magma, Sage, TeX

D_{12}._2{\rm Dic}_3

% in TeX

G:=Group("D12.2Dic3");

// GroupNames label

G:=SmallGroup(288,462);

// by ID

G=gap.SmallGroup(288,462);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,64,422,80,1356,9414]);

// Polycyclic

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

// generators/relations

G = D12.2Dic3 order 288 = 25·32

The non-split extension by D12 of Dic3 acting through Inn(D12)

G = D₁₂.₂Dic₃ order 288 = 2⁵·3²

The non-split extension by D₁₂ of Dic₃ acting through Inn(D₁₂)