C2^4:8D6 - GroupNames

G = C₂⁴:₈D₆ order 192 = 2⁶·3

3^rd semidirect product of C₂⁴ and D₆ acting via D₆/C₃=C₂²

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂⁴:₈D₆, C₆.₂₆2₊¹⁺⁴, C₃:₂D₄², D₆:₆(C₂xD₄), C₃:D₄:₄D₄, (C₂xD₄):₁₈D₆, C₂²:₄(S₃xD₄), C₂²wrC₂:₄S₃, D₆:D₄:₉C₂, C₂³:₂D₆:₄C₂, C₂²:C₄:₂₄D₆, Dic₃:₃(C₂xD₄), (C₆xD₄):₇C₂², Dic₃:D₄:₁₃C₂, C₁₂:₃D₄:₁₁C₂, D₆:C₄:₁₁C₂², C₆.₅₆(C₂²xD₄), C₂³.₁₄D₆:₂C₂, Dic₃:₄D₄:₂C₂, (C₂xD₁₂):₁₉C₂², (C₂xC₆).₁₃₄C₂⁴, (C₂xC₁₂).₂₈C₂³, Dic₃:C₄:₉C₂², (S₃xC₂³):₇C₂², (C₂³xC₆):₁₀C₂², C₂.₂₈(D₄:₆D₆), (C₄xDic₃):₁₄C₂², C₆.D₄:₁₅C₂², (C₂²xS₃).₅₃C₂³, C₂³.₁₁₈(C₂²xS₃), (C₂²xC₆).₁₈₁C₂³, C₂².₁₅₅(S₃xC₂³), (C₂xDic₃).₂₂₁C₂³, (C₂²xDic₃):₁₃C₂², (C₂xS₃xD₄):₇C₂, (C₂xC₆):₆(C₂xD₄), C₂.₂₉(C₂xS₃xD₄), (S₃xC₂xC₄):₇C₂², (C₃xC₂²wrC₂):₅C₂, (C₂²xC₃:D₄):₈C₂, (C₂xC₃:D₄):₃₉C₂², (C₃xC₂²:C₄):₅C₂², (C₂xC₄).₂₈(C₂²xS₃), SmallGroup(192,1149)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂xC₆ — C₂⁴:₈D₆

Chief series

C₁ — C₃ — C₆ — C₂xC₆ — C₂²xS₃ — S₃xC₂³ — C₂xS₃xD₄ — C₂⁴:₈D₆

Lower central

C₃ — C₂xC₆ — C₂⁴:₈D₆

Upper central

C₁ — C₂² — C₂²wrC₂

Generators and relations for C₂⁴:₈D₆
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁶=f²=1, ab=ba, eae^-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe^-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e^-1 >

Subgroups: 1344 in 428 conjugacy classes, 115 normal (27 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₂xC₄, C₂xC₄, C₂xC₄, D₄, C₂³, C₂³, C₂³, Dic₃, Dic₃, C₁₂, D₆, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₄², C₂²:C₄, C₂²:C₄, C₂²:C₄, C₄:C₄, C₂²xC₄, C₂xD₄, C₂xD₄, C₂xD₄, C₂⁴, C₂⁴, C₄xS₃, D₁₂, C₂xDic₃, C₂xDic₃, C₃:D₄, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₃xD₄, C₂²xS₃, C₂²xS₃, C₂²xC₆, C₂²xC₆, C₂²xC₆, C₄xD₄, C₂²wrC₂, C₂²wrC₂, C₄:D₄, C₄:₁D₄, C₂²xD₄, C₄xDic₃, Dic₃:C₄, D₆:C₄, C₆.D₄, C₃xC₂²:C₄, C₃xC₂²:C₄, S₃xC₂xC₄, C₂xD₁₂, C₂xD₁₂, S₃xD₄, C₂²xDic₃, C₂xC₃:D₄, C₂xC₃:D₄, C₆xD₄, C₆xD₄, S₃xC₂³, S₃xC₂³, C₂³xC₆, D₄², Dic₃:₄D₄, D₆:D₄, Dic₃:D₄, C₂³:₂D₆, C₂³.₁₄D₆, C₁₂:₃D₄, C₃xC₂²wrC₂, C₂xS₃xD₄, C₂²xC₃:D₄, C₂⁴:₈D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, C₂²xS₃, C₂²xD₄, 2₊¹⁺⁴, S₃xD₄, S₃xC₂³, D₄², C₂xS₃xD₄, D₄:₆D₆, C₂⁴:₈D₆

Smallest permutation representation of C₂⁴:₈D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

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

39 conjugacy classes

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

39 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

2₊¹⁺⁴

S₃xD₄

D₄:₆D₆

kernel

C₂⁴:₈D₆

Dic₃:₄D₄

D₆:D₄

Dic₃:D₄

C₂³:₂D₆

C₂³.₁₄D₆

C₁₂:₃D₄

C₃xC₂²wrC₂

C₂xS₃xD₄

C₂²xC₃:D₄

C₂²wrC₂

C₃:D₄

C₂²:C₄

C₂xD₄

C₂⁴

C₆

C₂²

C₂

# reps

Matrix representation of C₂⁴:₈D₆ ►in GL₆(Z)

-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	-2
0	0	0	0	0	-1

1	-2	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	-2
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	-1	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	-1	0
0	0	0	0	0	-1

1	0	0	0	0	0
1	-1	0	0	0	0
0	0	-1	-1	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	1	-1

-1	0	0	0	0	0
0	-1	0	0	0	0
0	0	-1	-1	0	0
0	0	0	1	0	0
0	0	0	0	-1	0
0	0	0	0	-1	1

G:=sub<GL(6,Integers())| [-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,-2,-1],[1,0,0,0,0,0,-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,-2,-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,-1,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,-1],[1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;

C₂⁴:₈D₆ in GAP, Magma, Sage, TeX

C_2^4\rtimes_8D_6

% in TeX

G:=Group("C2^4:8D6");

// GroupNames label

G:=SmallGroup(192,1149);

// by ID

G=gap.SmallGroup(192,1149);

# by ID

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

// Polycyclic

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

// generators/relations

G = C24:8D6 order 192 = 26·3

3rd semidirect product of C24 and D6 acting via D6/C3=C22

G = C₂⁴:₈D₆ order 192 = 2⁶·3

3^rd semidirect product of C₂⁴ and D₆ acting via D₆/C₃=C₂²