C12.95S3^2 - GroupNames

G = C₁₂.₉₅S₃² order 432 = 2⁴·3³

15^th non-split extension by C₁₂ of S₃² acting via S₃²/C₃⋊S₃=C₂

Aliases: C₁₂.₉₅S₃², C₃³⋊₅Q₈⋊₉C₂, (C₃×C₁₂).₁₇₁D₆, C₃³⋊₂₀(C₄○D₄), C₃³⋊₉D₄⋊₁₃C₂, C₃⋊Dic₃.₄₇D₆, C₃⋊₅(D₆.D₆), C₃²⋊₁₄(C₄○D₁₂), C₄.₈(C₃²⋊₄D₆), (C₃²×C₆).₆₇C₂³, (C₃²×C₁₂).₇₄C₂², C₆.₉₆(C₂×S₃²), (C₁₂×C₃⋊S₃)⋊₂C₂, (C₄×C₃⋊S₃)⋊₁₂S₃, (C₂×C₃⋊S₃).₄₇D₆, (C₆×C₃⋊S₃).₅₈C₂², C₂.₅(C₂×C₃²⋊₄D₆), (C₃×C₆).₁₁₇(C₂²×S₃), (C₃×C₃⋊Dic₃).₄₆C₂², SmallGroup(432,689)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃²×C₆ — C₁₂.₉₅S₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₆×C₃⋊S₃ — C₃³⋊₉D₄ — C₁₂.₉₅S₃²

Lower central

C₃³ — C₃²×C₆ — C₁₂.₉₅S₃²

Upper central

C₁ — C₄

Generators and relations for C₁₂.₉₅S₃²
G = < a,b,c,d,e | a¹²=b³=c²=d³=1, e²=a⁶, ab=ba, cac=eae^-1=a⁵, ad=da, cbc=b^-1, bd=db, be=eb, cd=dc, ece^-1=a⁶c, ede^-1=d^-1 >

Subgroups: 1064 in 214 conjugacy classes, 47 normal (8 characteristic)
C₁, C₂, C₂, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₂×C₄, D₄, Q₈, C₃², C₃², Dic₃, C₁₂, C₁₂, D₆, C₂×C₆, C₄○D₄, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, Dic₆, C₄×S₃, D₁₂, C₃⋊D₄, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, C₄○D₁₂, C₃×C₃⋊S₃, C₃²×C₆, D₆⋊S₃, C₃⋊D₁₂, C₃²⋊₂Q₈, S₃×C₁₂, C₄×C₃⋊S₃, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₆×C₃⋊S₃, D₆.D₆, C₃³⋊₉D₄, C₃³⋊₅Q₈, C₁₂×C₃⋊S₃, C₁₂.₉₅S₃²
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂²×S₃, S₃², C₄○D₁₂, C₂×S₃², C₃²⋊₄D₆, D₆.D₆, C₂×C₃²⋊₄D₆, C₁₂.₉₅S₃²

Smallest permutation representation of C₁₂.₉₅S₃²
►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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)

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

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

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

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

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

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

54 conjugacy classes

class	1	2A	2B	2C	2D	3A	3B	3C	3D	···	3H	4A	4B	4C	4D	4E	6A	6B	6C	6D	···	6H	6I	···	6N	12A	···	12F	12G	···	12P	12Q	···	12V
order	1	2	2	2	2	3	3	3	3	···	3	4	4	4	4	4	6	6	6	6	···	6	6	···	6	12	···	12	12	···	12	12	···	12
size	1	1	18	18	18	2	2	2	4	···	4	1	1	18	18	18	2	2	2	4	···	4	18	···	18	2	···	2	4	···	4	18	···	18

54 irreducible representations

dim	1	1	1	1	2	2	2	2	2	2	4	4	4	4	4	4
type	+	+	+	+	+	+	+	+			+	+
image	C₁	C₂	C₂	C₂	S₃	D₆	D₆	D₆	C₄○D₄	C₄○D₁₂	S₃²	C₂×S₃²	C₃²⋊₄D₆	D₆.D₆	C₂×C₃²⋊₄D₆	C₁₂.₉₅S₃²
kernel	C₁₂.₉₅S₃²	C₃³⋊₉D₄	C₃³⋊₅Q₈	C₁₂×C₃⋊S₃	C₄×C₃⋊S₃	C₃⋊Dic₃	C₃×C₁₂	C₂×C₃⋊S₃	C₃³	C₃²	C₁₂	C₆	C₄	C₃	C₂	C₁
# reps	1	3	1	3	3	3	3	3	2	12	3	3	2	6	2	4

Matrix representation of C₁₂.₉₅S₃² ►in GL₆(𝔽₁₃)

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	12	12	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
12	12	0	0	0	0
0	0	1	0	0	0
0	0	12	12	0	0
0	0	0	0	11	4
0	0	0	0	9	2

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	0	12
0	0	0	0	1	12

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

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

C₁₂.₉₅S₃² in GAP, Magma, Sage, TeX

C_{12}._{95}S_3^2

% in TeX

G:=Group("C12.95S3^2");

// GroupNames label

G:=SmallGroup(432,689);

// by ID

G=gap.SmallGroup(432,689);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,58,1124,571,2028,14118]);

// Polycyclic

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

// generators/relations

G = C12.95S32 order 432 = 24·33

15th non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

G = C₁₂.₉₅S₃² order 432 = 2⁴·3³

15^th non-split extension by C₁₂ of S₃² acting via S₃²/C₃⋊S₃=C₂