C2xC8:D6 - GroupNames

G = C₂xC₈:D₆ order 192 = 2⁶·3

Direct product of C₂ and C₈:D₆

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂xC₈:D₆, C₂₄:₂C₂³, D₁₂:₅C₂³, D₂₄:₉C₂², M₄(2):₁₇D₆, C₁₂.₅₈C₂⁴, Dic₆:₅C₂³, C₂³.₅₇D₁₂, (C₂xC₈):₄D₆, C₈:₂(C₂²xS₃), (C₂xD₂₄):₁₄C₂, C₆:₁(C₈:C₂²), (C₂xC₂₄):₇C₂², C₄.₄₈(C₂xD₁₂), (C₂xC₄).₅₇D₁₂, C₂₄:C₂:₈C₂², (C₂xC₁₂).₂₀₃D₄, C₁₂.₂₃₈(C₂xD₄), (C₆xM₄(2)):₃C₂, (C₂xM₄(2)):₃S₃, C₄.₅₅(S₃xC₂³), C₆.₂₅(C₂²xD₄), C₄oD₁₂:₁₈C₂², (C₂²xD₁₂):₁₇C₂, (C₂xD₁₂):₄₉C₂², (C₂²xC₄).₂₈₁D₆, C₂².₇₃(C₂xD₁₂), (C₂²xC₆).₁₁₈D₄, C₂.₂₇(C₂²xD₁₂), (C₂xC₁₂).₅₁₁C₂³, (C₂xDic₆):₅₇C₂², (C₃xM₄(2)):₁₉C₂², (C₂²xC₁₂).₂₆₆C₂², C₃:₁(C₂xC₈:C₂²), (C₂xC₂₄:C₂):₄C₂, (C₂xC₆).₆₂(C₂xD₄), (C₂xC₄oD₁₂):₂₆C₂, (C₂xC₄).₂₂₃(C₂²xS₃), SmallGroup(192,1305)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₂ — C₂xC₈:D₆

Chief series

C₁ — C₃ — C₆ — C₁₂ — D₁₂ — C₂xD₁₂ — C₂²xD₁₂ — C₂xC₈:D₆

Lower central

C₃ — C₆ — C₁₂ — C₂xC₈:D₆

Upper central

C₁ — C₂² — C₂²xC₄ — C₂xM₄(2)

Generators and relations for C₂xC₈:D₆
G = < a,b,c,d | a²=b⁸=c⁶=d²=1, ab=ba, ac=ca, ad=da, cbc^-1=b⁵, dbd=b^-1, dcd=c^-1 >

Subgroups: 984 in 298 conjugacy classes, 111 normal (25 characteristic)
C₁, C₂, C₂, C₂, C₃, C₄, C₄, C₄, C₂², C₂², C₂², S₃, C₆, C₆, C₆, C₈, C₂xC₄, C₂xC₄, C₂xC₄, D₄, Q₈, C₂³, C₂³, Dic₃, C₁₂, C₁₂, D₆, C₂xC₆, C₂xC₆, C₂xC₆, C₂xC₈, M₄(2), D₈, SD₁₆, C₂²xC₄, C₂²xC₄, C₂xD₄, C₂xQ₈, C₄oD₄, C₂⁴, C₂₄, Dic₆, Dic₆, C₄xS₃, D₁₂, D₁₂, C₂xDic₃, C₃:D₄, C₂xC₁₂, C₂xC₁₂, C₂²xS₃, C₂²xC₆, C₂xM₄(2), C₂xD₈, C₂xSD₁₆, C₈:C₂², C₂²xD₄, C₂xC₄oD₄, C₂₄:C₂, D₂₄, C₂xC₂₄, C₃xM₄(2), C₂xDic₆, S₃xC₂xC₄, C₂xD₁₂, C₂xD₁₂, C₂xD₁₂, C₄oD₁₂, C₄oD₁₂, C₂xC₃:D₄, C₂²xC₁₂, S₃xC₂³, C₂xC₈:C₂², C₂xC₂₄:C₂, C₂xD₂₄, C₈:D₆, C₆xM₄(2), C₂²xD₁₂, C₂xC₄oD₁₂, C₂xC₈:D₆
Quotients: C₁, C₂, C₂², S₃, D₄, C₂³, D₆, C₂xD₄, C₂⁴, D₁₂, C₂²xS₃, C₈:C₂², C₂²xD₄, C₂xD₁₂, S₃xC₂³, C₂xC₈:C₂², C₈:D₆, C₂²xD₁₂, C₂xC₈:D₆

Smallest permutation representation of C₂xC₈:D₆
►On 48 points

Generators in S₄₈

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

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

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2K	3	4A	4B	4C	4D	4E	4F	6A	6B	6C	6D	6E	8A	8B	8C	8D	12A	12B	12C	12D	12E	12F	24A	···	24H
order	1	2	2	2	2	2	2	···	2	3	4	4	4	4	4	4	6	6	6	6	6	8	8	8	8	12	12	12	12	12	12	24	···	24
size	1	1	1	1	2	2	12	···	12	2	2	2	2	2	12	12	2	2	2	4	4	4	4	4	4	2	2	2	2	4	4	4	···	4

42 irreducible representations

dim

type

image

C₁

C₂

S₃

D₄

D₆

D₁₂

C₈:C₂²

C₈:D₆

kernel

C₂xC₈:D₆

C₂xC₂₄:C₂

C₂xD₂₄

C₈:D₆

C₆xM₄(2)

C₂²xD₁₂

C₂xC₄oD₁₂

C₂xM₄(2)

C₂xC₁₂

C₂²xC₆

C₂xC₈

M₄(2)

C₂²xC₄

C₂xC₄

C₂³

C₆

C₂

# reps

Matrix representation of C₂xC₈:D₆ ►in GL₆(F₇₃)

72	0	0	0	0	0
0	72	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

72	0	0	0	0	0
0	72	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	66	14	0	0
0	0	59	7	0	0

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

0	72	0	0	0	0
72	0	0	0	0	0
0	0	1	0	0	0
0	0	1	72	0	0
0	0	0	0	7	59
0	0	0	0	66	66

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,66,59,0,0,0,0,14,7,0,0,1,0,0,0,0,0,0,1,0,0],[0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,72,0],[0,72,0,0,0,0,72,0,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,7,66,0,0,0,0,59,66] >;

C₂xC₈:D₆ in GAP, Magma, Sage, TeX

C_2\times C_8\rtimes D_6

% in TeX

G:=Group("C2xC8:D6");

// GroupNames label

G:=SmallGroup(192,1305);

// by ID

G=gap.SmallGroup(192,1305);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,675,297,80,1684,102,6278]);

// Polycyclic

G:=Group<a,b,c,d|a^2=b^8=c^6=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^5,d*b*d=b^-1,d*c*d=c^-1>;

// generators/relations

G = C2xC8:D6 order 192 = 26·3

Direct product of C2 and C8:D6

G = C₂xC₈:D₆ order 192 = 2⁶·3

Direct product of C₂ and C₈:D₆