C2^4.67D6 - GroupNames

G = C₂⁴.₆₇D₆ order 192 = 2⁶·3

14^th non-split extension by C₂⁴ of D₆ acting via D₆/S₃=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴.₆₇D₆, D₆⋊C₄⋊₈C₂², C₂²≀C₂⋊₁₀S₃, C₂⁴⋊₄S₃⋊₄C₂, (C₂×Dic₃)⋊₂₀D₄, (D₄×Dic₃)⋊₁₀C₂, (C₂×D₄).₁₄₈D₆, C₂²⋊C₄.₄₄D₆, C₂².₃₉(S₃×D₄), C₆.₅₃(C₂²×D₄), C₂³.₁₄D₆⋊₁C₂, Dic₃⋊₄D₄⋊₁C₂, (C₂×C₁₂).₂₅C₂³, (C₂×C₆).₁₃₀C₂⁴, Dic₃⋊C₄⋊₆C₂², C₄⋊Dic₃⋊₂₃C₂², Dic₃.₄₄(C₂×D₄), (C₂³×Dic₃)⋊₅C₂, (C₂²×C₆).₇C₂³, C₂²⋊₄(D₄⋊₂S₃), C₃⋊₃(C₂².₁₉C₂⁴), (C₄×Dic₃)⋊₁₂C₂², (C₂×Dic₆)⋊₁₈C₂², (C₆×D₄).₁₀₉C₂², C₂³.₁₆D₆⋊₁C₂, C₂³.₂₁D₆⋊₈C₂, C₂³.₂₃D₆⋊₂C₂, (C₂³×C₆).₆₆C₂², Dic₃.D₄⋊₁₁C₂, C₆.D₄⋊₁₁C₂², (C₂²×S₃).₅₂C₂³, C₂³.₁₈₅(C₂²×S₃), C₂².₁₅₁(S₃×C₂³), (C₂×Dic₃).₂₁₉C₂³, (C₂²×Dic₃)⋊₁₀C₂², C₂.₂₆(C₂×S₃×D₄), (S₃×C₂×C₄)⋊₄C₂², (C₂×C₆)⋊₉(C₄○D₄), C₆.₇₅(C₂×C₄○D₄), (C₂×C₆).₅₂(C₂×D₄), (C₃×C₂²≀C₂)⋊₂C₂, (C₂×D₄⋊₂S₃)⋊₅C₂, (C₂×C₃⋊D₄)⋊₆C₂², C₂.₂₆(C₂×D₄⋊₂S₃), (C₂×C₄).₂₅(C₂²×S₃), (C₃×C₂²⋊C₄).₁C₂², SmallGroup(192,1145)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₆ — C₂⁴.₆₇D₆

Chief series

C₁ — C₃ — C₆ — C₂×C₆ — C₂×Dic₃ — C₂²×Dic₃ — C₂³×Dic₃ — C₂⁴.₆₇D₆

Lower central

C₃ — C₂×C₆ — C₂⁴.₆₇D₆

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for C₂⁴.₆₇D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=1, f²=d, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, ebe^-1=fbf^-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 784 in 330 conjugacy classes, 111 normal (39 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, Dic₃, Dic₃, C₁₂, D₆, C₂×C₆, C₂×C₆, C₂×C₆, C₄², C₂²⋊C₄, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, Dic₆, C₄×S₃, C₂×Dic₃, C₂×Dic₃, C₂×Dic₃, C₃⋊D₄, C₂×C₁₂, C₂×C₁₂, C₃×D₄, C₂²×S₃, C₂²×C₆, C₂²×C₆, C₂²×C₆, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂³×C₄, C₂×C₄○D₄, C₄×Dic₃, Dic₃⋊C₄, C₄⋊Dic₃, D₆⋊C₄, C₆.D₄, C₆.D₄, C₃×C₂²⋊C₄, C₃×C₂²⋊C₄, C₂×Dic₆, S₃×C₂×C₄, D₄⋊₂S₃, C₂²×Dic₃, C₂²×Dic₃, C₂²×Dic₃, C₂×C₃⋊D₄, C₂×C₃⋊D₄, C₆×D₄, C₆×D₄, C₂³×C₆, C₂².₁₉C₂⁴, C₂³.₁₆D₆, Dic₃.D₄, Dic₃⋊₄D₄, C₂³.₂₁D₆, D₄×Dic₃, C₂³.₂₃D₆, C₂³.₁₄D₆, C₂⁴⋊₄S₃, C₃×C₂²≀C₂, C₂×D₄⋊₂S₃, C₂³×Dic₃, C₂⁴.₆₇D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂×D₄, C₄○D₄, C₂⁴, C₂²×S₃, C₂²×D₄, C₂×C₄○D₄, S₃×D₄, D₄⋊₂S₃, S₃×C₂³, C₂².₁₉C₂⁴, C₂×S₃×D₄, C₂×D₄⋊₂S₃, C₂⁴.₆₇D₆

Smallest permutation representation of C₂⁴.₆₇D₆
►On 48 points

Generators in S₄₈

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

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

(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(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 41)(32 42)(33 37)(34 38)(35 39)(36 40)

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

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

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

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

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

42 conjugacy classes

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

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

C₄○D₄

S₃×D₄

D₄⋊₂S₃

kernel

C₂⁴.₆₇D₆

C₂³.₁₆D₆

Dic₃.D₄

Dic₃⋊₄D₄

C₂³.₂₁D₆

D₄×Dic₃

C₂³.₂₃D₆

C₂³.₁₄D₆

C₂⁴⋊₄S₃

C₃×C₂²≀C₂

C₂×D₄⋊₂S₃

C₂³×Dic₃

C₂²≀C₂

C₂×Dic₃

C₂²⋊C₄

C₂×D₄

C₂⁴

C₂×C₆

C₂²

# reps

Matrix representation of C₂⁴.₆₇D₆ ►in GL₆(𝔽₁₃)

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

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

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	1	0
0	0	0	0	0	1

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

0	8	0	0	0	0
8	0	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	3	7
0	0	0	0	10	10

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,10,0,0,0,0,7,10] >;

C₂⁴.₆₇D₆ in GAP, Magma, Sage, TeX

C_2^4._{67}D_6

% in TeX

G:=Group("C2^4.67D6");

// GroupNames label

G:=SmallGroup(192,1145);

// by ID

G=gap.SmallGroup(192,1145);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,570,185,6278]);

// 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,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,e*b*e^-1=f*b*f^-1=b*c=c*b,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 = C24.67D6 order 192 = 26·3

14th non-split extension by C24 of D6 acting via D6/S3=C2

G = C₂⁴.₆₇D₆ order 192 = 2⁶·3

14^th non-split extension by C₂⁴ of D₆ acting via D₆/S₃=C₂