C2^5.4S3 - GroupNames

G = C₂⁵.₄S₃ order 192 = 2⁶·3

1^st non-split extension by C₂⁵ of S₃ acting via S₃/C₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁵.₄S₃, C₂⁴⋊₇Dic₃, C₂⁴.₇₇D₆, (C₂³×C₆)⋊₇C₄, (C₂⁴×C₆).₃C₂, C₃⋊₂(C₂⁴⋊₃C₄), C₆.₈₆C₂²≀C₂, (C₂²×C₆).₂₀₂D₄, C₂.₃(C₂⁴⋊₄S₃), C₂³.₉₇(C₃⋊D₄), C₂³.₃₉(C₂×Dic₃), C₂²⋊₃(C₆.D₄), (C₂³×C₆).₁₀₁C₂², (C₂²×C₆).₃₇₂C₂³, C₂³.₃₁₈(C₂²×S₃), (C₂²×Dic₃)⋊₄C₂², C₂².₅₅(C₂²×Dic₃), (C₂×C₆)⋊₈(C₂²⋊C₄), (C₂×C₆).₅₆₉(C₂×D₄), C₆.₈₈(C₂×C₂²⋊C₄), C₂².₉₈(C₂×C₃⋊D₄), (C₂×C₆.D₄)⋊₁₃C₂, (C₂×C₆).₂₀₃(C₂²×C₄), (C₂²×C₆).₁₃₈(C₂×C₄), C₂.₂₄(C₂×C₆.D₄), SmallGroup(192,806)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁵.₄S₃

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂²×C₆ — C₂²×Dic₃ — C₂×C₆.D₄ — C₂⁵.₄S₃

Lower central

C₃ — C₂×C₆ — C₂⁵.₄S₃

Upper central

C₁ — C₂³ — C₂⁵

Generators and relations for C₂⁵.₄S₃
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f³=1, g²=c, ab=ba, ac=ca, ad=da, gag^-1=ae=ea, af=fa, bc=cb, gbg^-1=bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, gfg^-1=f^-1 >

Subgroups: 1048 in 506 conjugacy classes, 119 normal (9 characteristic)
C₁, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₂×C₄, C₂³, C₂³, C₂³, Dic₃, C₂×C₆, C₂×C₆, C₂×C₆, C₂²⋊C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂×Dic₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₂⁵, C₆.D₄, C₂²×Dic₃, C₂³×C₆, C₂³×C₆, C₂⁴⋊₃C₄, C₂×C₆.D₄, C₂⁴×C₆, C₂⁵.₄S₃
Quotients: C₁, C₂, C₄, C₂², S₃, C₂×C₄, D₄, C₂³, Dic₃, D₆, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×Dic₃, C₃⋊D₄, C₂²×S₃, C₂×C₂²⋊C₄, C₂²≀C₂, C₆.D₄, C₂²×Dic₃, C₂×C₃⋊D₄, C₂⁴⋊₃C₄, C₂×C₆.D₄, C₂⁴⋊₄S₃, C₂⁵.₄S₃

Smallest permutation representation of C₂⁵.₄S₃
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

(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)

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

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

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

60 conjugacy classes

class	1	2A	···	2G	2H	···	2S	3	4A	···	4H	6A	···	6AE
order	1	2	···	2	2	···	2	3	4	···	4	6	···	6
size	1	1	···	1	2	···	2	2	12	···	12	2	···	2

60 irreducible representations

dim	1	1	1	1	2	2	2	2	2
type	+	+	+		+	+	-	+
image	C₁	C₂	C₂	C₄	S₃	D₄	Dic₃	D₆	C₃⋊D₄
kernel	C₂⁵.₄S₃	C₂×C₆.D₄	C₂⁴×C₆	C₂³×C₆	C₂⁵	C₂²×C₆	C₂⁴	C₂⁴	C₂³
# reps	1	6	1	8	1	12	4	3	24

Matrix representation of C₂⁵.₄S₃ ►in GL₅(𝔽₁₃)

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

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

12	0	0	0	0
0	12	0	0	0
0	0	12	0	0
0	0	0	12	0
0	0	0	0	12

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

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

1	0	0	0	0
0	9	0	0	0
0	0	3	0	0
0	0	0	9	0
0	0	0	0	3

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

G:=sub<GL(5,GF(13))| [1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,1,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[1,0,0,0,0,0,9,0,0,0,0,0,3,0,0,0,0,0,9,0,0,0,0,0,3],[8,0,0,0,0,0,0,12,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,1,0] >;

C₂⁵.₄S₃ in GAP, Magma, Sage, TeX

C_2^5._4S_3

% in TeX

G:=Group("C2^5.4S3");

// GroupNames label

G:=SmallGroup(192,806);

// by ID

G=gap.SmallGroup(192,806);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,477,422,6278]);

// Polycyclic

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

// generators/relations

G = C25.4S3 order 192 = 26·3

1st non-split extension by C25 of S3 acting via S3/C3=C2

G = C₂⁵.₄S₃ order 192 = 2⁶·3

1^st non-split extension by C₂⁵ of S₃ acting via S₃/C₃=C₂