SL(2,3).11D6 - GroupNames

G = SL₂(𝔽₃).₁₁D₆ order 288 = 2⁵·3²

1^st non-split extension by SL₂(𝔽₃) of D₆ acting through Inn(SL₂(𝔽₃))

Aliases: SL₂(𝔽₃).₁₁D₆, C₃⋊D₄.A₄, Q₈⋊₃S₃⋊C₆, (C₆×Q₈)⋊₂C₆, (S₃×Q₈)⋊₁C₆, D₆.₁(C₂×A₄), Q₈.₇(S₃×C₆), C₃⋊₁(D₄.A₄), Q₈.₁₅D₆⋊C₃, C₆.₆(C₂²×A₄), C₂².₅(S₃×A₄), Dic₃.A₄⋊₄C₂, Dic₃.₂(C₂×A₄), (C₆×SL₂(𝔽₃))⋊₅C₂, (S₃×SL₂(𝔽₃))⋊₄C₂, (C₂×SL₂(𝔽₃))⋊₁S₃, (C₃×SL₂(𝔽₃)).₁₁C₂², C₂.₇(C₂×S₃×A₄), (C₂×Q₈)⋊₃(C₃×S₃), (C₂×C₆).₂₀(C₂×A₄), (C₃×Q₈).₂(C₂×C₆), SmallGroup(288,923)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂ — C₃×Q₈ — SL₂(𝔽₃).₁₁D₆

Chief series

C₁ — C₂ — C₆ — C₃×Q₈ — C₃×SL₂(𝔽₃) — S₃×SL₂(𝔽₃) — SL₂(𝔽₃).₁₁D₆

Lower central

C₃×Q₈ — SL₂(𝔽₃).₁₁D₆

Upper central

C₁ — C₂ — C₂²

Generators and relations for SL₂(𝔽₃).₁₁D₆
G = < a,b,c,d,e | a⁴=c³=d⁶=1, b²=e²=a², bab^-1=a^-1, cac^-1=b, ad=da, ae=ea, cbc^-1=ab, bd=db, be=eb, cd=dc, ce=ec, ede^-1=a²d^-1 >

Subgroups: 454 in 99 conjugacy classes, 23 normal (all characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₂², C₂², S₃, C₆, C₆, C₂×C₄, D₄, Q₈, Q₈, C₃², Dic₃, Dic₃, C₁₂, D₆, D₆, C₂×C₆, C₂×C₆, C₂×Q₈, C₂×Q₈, C₄○D₄, C₃×S₃, C₃×C₆, SL₂(𝔽₃), SL₂(𝔽₃), Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₃⋊D₄, C₂×C₁₂, C₃×D₄, C₃×Q₈, C₃×Q₈, 2_-¹⁺⁴, C₃×Dic₃, S₃×C₆, C₆², C₂×SL₂(𝔽₃), C₂×SL₂(𝔽₃), C₄.A₄, C₄○D₁₂, S₃×Q₈, S₃×Q₈, Q₈⋊₃S₃, Q₈⋊₃S₃, C₆×Q₈, C₃×SL₂(𝔽₃), C₃×C₃⋊D₄, D₄.A₄, Q₈.₁₅D₆, Dic₃.A₄, S₃×SL₂(𝔽₃), C₆×SL₂(𝔽₃), SL₂(𝔽₃).₁₁D₆
Quotients: C₁, C₂, C₃, C₂², S₃, C₆, A₄, D₆, C₂×C₆, C₃×S₃, C₂×A₄, S₃×C₆, C₂²×A₄, S₃×A₄, D₄.A₄, C₂×S₃×A₄, SL₂(𝔽₃).₁₁D₆

Smallest permutation representation of SL₂(𝔽₃).₁₁D₆
►On 48 points

Generators in S₄₈

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

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

(7 36 18)(8 31 13)(9 32 14)(10 33 15)(11 34 16)(12 35 17)(19 47 26)(20 48 27)(21 43 28)(22 44 29)(23 45 30)(24 46 25)

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

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

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

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

33 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	3E	4A	4B	4C	4D	6A	6B	6C	6D	6E	6F	···	6M	6N	6O	12A	12B	12C	12D
order	1	2	2	2	2	3	3	3	3	3	4	4	4	4	6	6	6	6	6	6	···	6	6	6	12	12	12	12
size	1	1	2	6	18	2	4	4	8	8	6	6	6	18	2	2	2	4	4	8	···	8	24	24	12	12	24	24

33 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	3	3	3	3	4	4	4	6	6
type	+	+	+	+					+	+			+	+	+	+	-			+	+
image	C₁	C₂	C₂	C₂	C₃	C₆	C₆	C₆	S₃	D₆	C₃×S₃	S₃×C₆	A₄	C₂×A₄	C₂×A₄	C₂×A₄	D₄.A₄	D₄.A₄	SL₂(𝔽₃).₁₁D₆	S₃×A₄	C₂×S₃×A₄
kernel	SL₂(𝔽₃).₁₁D₆	Dic₃.A₄	S₃×SL₂(𝔽₃)	C₆×SL₂(𝔽₃)	Q₈.₁₅D₆	S₃×Q₈	Q₈⋊₃S₃	C₆×Q₈	C₂×SL₂(𝔽₃)	SL₂(𝔽₃)	C₂×Q₈	Q₈	C₃⋊D₄	Dic₃	D₆	C₂×C₆	C₃	C₃	C₁	C₂²	C₂
# reps	1	1	1	1	2	2	2	2	1	1	2	2	1	1	1	1	1	2	6	1	1

Matrix representation of SL₂(𝔽₃).₁₁D₆ ►in GL₄(𝔽₇) generated by

1	5	0	2
0	0	4	3
6	6	4	2
6	1	4	2

3	6	6	3
4	0	4	5
5	5	4	4
2	5	6	0

2	3	1	4
2	5	3	2
5	5	4	4
6	1	4	2

2	6	4	4
5	6	5	3
2	5	3	3
1	1	3	6

3	0	1	2
6	0	1	1
2	5	5	3
1	1	3	6

G:=sub<GL(4,GF(7))| [1,0,6,6,5,0,6,1,0,4,4,4,2,3,2,2],[3,4,5,2,6,0,5,5,6,4,4,6,3,5,4,0],[2,2,5,6,3,5,5,1,1,3,4,4,4,2,4,2],[2,5,2,1,6,6,5,1,4,5,3,3,4,3,3,6],[3,6,2,1,0,0,5,1,1,1,5,3,2,1,3,6] >;

SL₂(𝔽₃).₁₁D₆ in GAP, Magma, Sage, TeX

{\rm SL}_2({\mathbb F}_3)._{11}D_6

% in TeX

G:=Group("SL(2,3).11D6");

// GroupNames label

G:=SmallGroup(288,923);

// by ID

G=gap.SmallGroup(288,923);

# by ID

G:=PCGroup([7,-2,-2,-3,-2,2,-3,-2,2045,269,360,123,515,242,4037]);

// Polycyclic

G:=Group<a,b,c,d,e|a^4=c^3=d^6=1,b^2=e^2=a^2,b*a*b^-1=a^-1,c*a*c^-1=b,a*d=d*a,a*e=e*a,c*b*c^-1=a*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a^2*d^-1>;

// generators/relations

G = SL2(𝔽3).11D6 order 288 = 25·32

1st non-split extension by SL2(𝔽3) of D6 acting through Inn(SL2(𝔽3))

G = SL₂(𝔽₃).₁₁D₆ order 288 = 2⁵·3²

1^st non-split extension by SL₂(𝔽₃) of D₆ acting through Inn(SL₂(𝔽₃))