C2^4.43D6 - GroupNames

G = C₂⁴.₄₃D₆ order 192 = 2⁶·3

32^nd non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₄₃D₆, C₆.₂₄2₊¹⁺⁴, C₂²⋊C₄.₁D₆, C₂²≀C₂.₃S₃, (D₄×Dic₃)⋊₁₁C₂, (C₂×D₄).₁₄₉D₆, (C₂×C₆).₁₃₁C₂⁴, (C₂×C₁₂).₂₆C₂³, Dic₃⋊C₄⋊₇C₂², C₄⋊Dic₃⋊₂₄C₂², C₂³.₁₂D₆⋊₁₁C₂, C₂.₂₆(D₄⋊₆D₆), (C₄×Dic₃)⋊₁₃C₂², (C₂×Dic₆)⋊₁₉C₂², (C₆×D₄).₁₁₀C₂², C₂³.₂₃D₆⋊₃C₂, C₂³.₁₆D₆⋊₂C₂, C₂³.₈D₆⋊₁₁C₂, (C₂³×C₆).₆₇C₂², Dic₃.D₄⋊₁₂C₂, C₃⋊₄(C₂².₄₅C₂⁴), C₆.D₄⋊₁₂C₂², C₂².₁₅₂(S₃×C₂³), (C₂²×C₆).₁₈₀C₂³, C₂³.₁₈₆(C₂²×S₃), C₂².₁₇(D₄⋊₂S₃), (C₂×Dic₃).₂₂₀C₂³, (C₂²×Dic₃)⋊₁₁C₂², C₆.₇₆(C₂×C₄○D₄), (C₃×C₂²≀C₂).₂C₂, (C₂×C₆).₄₃(C₄○D₄), C₂.₂₇(C₂×D₄⋊₂S₃), (C₂×C₄).₂₆(C₂²×S₃), (C₂×C₆.D₄)⋊₁₉C₂, (C₃×C₂²⋊C₄).₂C₂², SmallGroup(192,1146)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₄₃D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×Dic₃ — C₂²×Dic₃ — D₄×Dic₃ — C₂⁴.₄₃D₆

Lower central

C₃ — C₂×C₆ — C₂⁴.₄₃D₆

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for C₂⁴.₄₃D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=1, f²=d, ab=ba, eae^-1=faf^-1=ac=ca, ad=da, fbf^-1=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 576 in 248 conjugacy classes, 99 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², C₆, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, C₁₂, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, Dic₆, C₂×Dic₃, C₂×Dic₃, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₄×Dic₃, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂×Dic₆, C₂²×Dic₃, C₂²×Dic₃, C₆×D₄, C₆×D₄, C₂³×C₆, C₂².₄₅C₂⁴, C₂³.₁₆D₆, Dic₃.D₄, C₂³.₈D₆, D₄×Dic₃, C₂³.₂₃D₆, C₂³.₂₃D₆, C₂³.₁₂D₆, C₂×C₆.D₄, C₃×C₂²≀C₂, C₂⁴.₄₃D₆
Quotients: C₁, C₂, C₂², S₃, C₂³, D₆, C₄○D₄, C₂⁴, C₂²×S₃, C₂×C₄○D₄, 2₊¹⁺⁴, D₄⋊₂S₃, S₃×C₂³, C₂².₄₅C₂⁴, C₂×D₄⋊₂S₃, D₄⋊₆D₆, C₂⁴.₄₃D₆

Smallest permutation representation of C₂⁴.₄₃D₆
►On 48 points

Generators in S₄₈

(2 7)(4 9)(6 11)(14 48)(16 44)(18 46)(20 25)(22 27)(24 29)(32 40)(34 42)(36 38)

(1 19)(3 21)(5 23)(8 26)(10 28)(12 30)(13 47)(14 40)(15 43)(16 42)(17 45)(18 38)(31 39)(32 48)(33 41)(34 44)(35 37)(36 46)

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₆

C₄○D₄

2₊¹⁺⁴

D₄⋊₂S₃

D₄⋊₆D₆

kernel

C₂⁴.₄₃D₆

C₂³.₁₆D₆

Dic₃.D₄

C₂³.₈D₆

D₄×Dic₃

C₂³.₂₃D₆

C₂³.₁₂D₆

C₂×C₆.D₄

C₃×C₂²≀C₂

C₂²≀C₂

C₂²⋊C₄

C₂×D₄

C₂⁴

C₂×C₆

C₆

C₂²

C₂

# reps

Matrix representation of C₂⁴.₄₃D₆ ►in GL₆(𝔽₁₃)

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	1	0
0	0	0	0	0	12

12	0	0	0	0	0
10	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	12	0
0	0	0	0	0	1

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	12	0
0	0	0	0	0	12

12	0	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	12	0
0	0	0	0	0	12

1	8	0	0	0	0
0	12	0	0	0	0
0	0	9	0	0	0
0	0	3	3	0	0
0	0	0	0	0	1
0	0	0	0	1	0

5	0	0	0	0	0
0	5	0	0	0	0
0	0	10	6	0	0
0	0	3	3	0	0
0	0	0	0	0	8
0	0	0	0	8	0

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[12,10,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,8,12,0,0,0,0,0,0,9,3,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[5,0,0,0,0,0,0,5,0,0,0,0,0,0,10,3,0,0,0,0,6,3,0,0,0,0,0,0,0,8,0,0,0,0,8,0] >;

C₂⁴.₄₃D₆ in GAP, Magma, Sage, TeX

C_2^4._{43}D_6

% in TeX

G:=Group("C2^4.43D6");

// GroupNames label

G:=SmallGroup(192,1146);

// by ID

G=gap.SmallGroup(192,1146);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,219,1571,297,6278]);

// Polycyclic

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

// generators/relations

G = C24.43D6 order 192 = 26·3

32nd non-split extension by C24 of D6 acting via D6/C3=C22

G = C₂⁴.₄₃D₆ order 192 = 2⁶·3

32^nd non-split extension by C₂⁴ of D₆ acting via D₆/C₃=C₂²