C12.93S3^2 - GroupNames

G = C₁₂.₉₃S₃² order 432 = 2⁴·3³

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

metabelian, supersoluble, monomial, A-group

Aliases: C₁₂.₉₃S₃², C₃²⋊₉(S₃×C₈), C₃³⋊₁₃(C₂×C₈), C₃²⋊₄C₈⋊₁₅S₃, (C₃×C₁₂).₁₆₈D₆, C₆.₁₇(S₃×Dic₃), C₃⋊Dic₃.₅Dic₃, C₃⋊₂(C₁₂.₂₉D₆), C₆.₁₈(C₆.D₆), C₄.₆(C₃²⋊₄D₆), (C₃²×C₁₂).₇₀C₂², C₃⋊₂(S₃×C₃⋊C₈), C₃⋊S₃⋊₃(C₃⋊C₈), (C₃×C₃⋊S₃)⋊₆C₈, C₃²⋊₇(C₂×C₃⋊C₈), (C₆×C₃⋊S₃).₇C₄, (C₄×C₃⋊S₃).₇S₃, (C₃×C₆).₅₃(C₄×S₃), (C₁₂×C₃⋊S₃).₁₁C₂, (C₂×C₃⋊S₃).₅Dic₃, (C₃×C₃⋊Dic₃).₉C₄, C₂.₁(C₃³⋊₉(C₂×C₄)), (C₃×C₃²⋊₄C₈)⋊₁₃C₂, (C₃²×C₆).₄₄(C₂×C₄), (C₃×C₆).₄₀(C₂×Dic₃), SmallGroup(432,455)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₃³ — C₁₂.₉₃S₃²

Chief series

C₁ — C₃ — C₃² — C₃³ — C₃²×C₆ — C₃²×C₁₂ — C₁₂×C₃⋊S₃ — C₁₂.₉₃S₃²

Lower central

C₃³ — C₁₂.₉₃S₃²

Upper central

C₁ — C₄

Generators and relations for C₁₂.₉₃S₃²
G = < a,b,c,d,e | a³=b³=c⁸=d³=e²=1, ab=ba, cac^-1=a^-1, ad=da, ae=ea, cbc^-1=ebe=b^-1, bd=db, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 456 in 126 conjugacy classes, 39 normal (19 characteristic)
C₁, C₂, C₂, C₃, C₃, C₃, C₄, C₄, C₂², S₃, C₆, C₆, C₆, C₈, C₂×C₄, C₃², C₃², C₃², Dic₃, C₁₂, C₁₂, C₁₂, D₆, C₂×C₆, C₂×C₈, C₃×S₃, C₃⋊S₃, C₃×C₆, C₃×C₆, C₃×C₆, C₃⋊C₈, C₂₄, C₄×S₃, C₂×C₁₂, C₃³, C₃×Dic₃, C₃⋊Dic₃, C₃×C₁₂, C₃×C₁₂, C₃×C₁₂, S₃×C₆, C₂×C₃⋊S₃, S₃×C₈, C₂×C₃⋊C₈, C₃×C₃⋊S₃, C₃²×C₆, C₃×C₃⋊C₈, C₃²⋊₄C₈, S₃×C₁₂, C₄×C₃⋊S₃, C₃×C₃⋊Dic₃, C₃²×C₁₂, C₆×C₃⋊S₃, S₃×C₃⋊C₈, C₁₂.₂₉D₆, C₃×C₃²⋊₄C₈, C₁₂×C₃⋊S₃, C₁₂.₉₃S₃²
Quotients: C₁, C₂, C₄, C₂², S₃, C₈, C₂×C₄, Dic₃, D₆, C₂×C₈, C₃⋊C₈, C₄×S₃, C₂×Dic₃, S₃², S₃×C₈, C₂×C₃⋊C₈, S₃×Dic₃, C₆.D₆, C₃²⋊₄D₆, S₃×C₃⋊C₈, C₁₂.₂₉D₆, C₃³⋊₉(C₂×C₄), C₁₂.₉₃S₃²

Smallest permutation representation of C₁₂.₉₃S₃²
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

60 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	3D	···	3H	4A	4B	4C	4D	6A	6B	6C	6D	···	6H	6I	6J	8A	···	8H	12A	···	12F	12G	···	12P	12Q	12R	24A	···	24H
order	1	2	2	2	3	3	3	3	···	3	4	4	4	4	6	6	6	6	···	6	6	6	8	···	8	12	···	12	12	···	12	12	12	24	···	24
size	1	1	9	9	2	2	2	4	···	4	1	1	9	9	2	2	2	4	···	4	18	18	9	···	9	2	···	2	4	···	4	18	18	18	···	18

60 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4
type	+	+	+				+	+	-	+	-				+	-	+
image	C₁	C₂	C₂	C₄	C₄	C₈	S₃	S₃	Dic₃	D₆	Dic₃	C₃⋊C₈	C₄×S₃	S₃×C₈	S₃²	S₃×Dic₃	C₆.D₆	C₃²⋊₄D₆	S₃×C₃⋊C₈	C₁₂.₂₉D₆	C₃³⋊₉(C₂×C₄)	C₁₂.₉₃S₃²
kernel	C₁₂.₉₃S₃²	C₃×C₃²⋊₄C₈	C₁₂×C₃⋊S₃	C₃×C₃⋊Dic₃	C₆×C₃⋊S₃	C₃×C₃⋊S₃	C₃²⋊₄C₈	C₄×C₃⋊S₃	C₃⋊Dic₃	C₃×C₁₂	C₂×C₃⋊S₃	C₃⋊S₃	C₃×C₆	C₃²	C₁₂	C₆	C₆	C₄	C₃	C₃	C₂	C₁
# reps	1	2	1	2	2	8	2	1	1	3	1	4	4	8	3	2	1	2	4	2	2	4

Matrix representation of C₁₂.₉₃S₃² ►in GL₆(𝔽₇₃)

72	72	0	0	0	0
1	0	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	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	72	0	0
0	0	1	72	0	0
0	0	0	0	1	0
0	0	0	0	0	1

51	0	0	0	0	0
22	22	0	0	0	0
0	0	0	72	0	0
0	0	72	0	0	0
0	0	0	0	46	0
0	0	0	0	0	46

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	72	72
0	0	0	0	1	0

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	72
0	0	0	0	72	0

G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,22,0,0,0,0,0,22,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,46,0,0,0,0,0,0,46],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;

C₁₂.₉₃S₃² in GAP, Magma, Sage, TeX

C_{12}._{93}S_3^2

% in TeX

G:=Group("C12.93S3^2");

// GroupNames label

G:=SmallGroup(432,455);

// by ID

G=gap.SmallGroup(432,455);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,36,58,1124,571,2028,14118]);

// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^8=d^3=e^2=1,a*b=b*a,c*a*c^-1=a^-1,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C12.93S32 order 432 = 24·33

13rd non-split extension by C12 of S32 acting via S32/C3⋊S3=C2

G = C₁₂.₉₃S₃² order 432 = 2⁴·3³

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