C6xF8 - GroupNames

G = C₆xF₈ order 336 = 2⁴·3·7

Direct product of C₆ and F₈

direct product, metabelian, soluble, monomial, A-group

Aliases: C₆xF₈, C₂⁴:C₂₁, C₂³:C₄₂, (C₂³xC₆):C₇, (C₂²xC₆):₂C₁₄, SmallGroup(336,213)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₆xF₈

Chief series

C₁ — C₂³ — F₈ — C₃xF₈ — C₆xF₈

Lower central

C₂³ — C₆xF₈

Upper central

C₁ — C₆

Generators and relations for C₆xF₈
G = < a,b,c,d,e | a⁶=b²=c²=d²=e⁷=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=dc=cd, ece^-1=b, ede^-1=c >

Subgroups: 170 in 34 conjugacy classes, 12 normal (all characteristic)
Quotients: C₁, C₂, C₃, C₆, C₇, C₁₄, C₂₁, C₄₂, F₈, C₂xF₈, C₃xF₈, C₆xF₈

C₁

C₂

₇C₂

C₃

₈C₇

₇C₂²

C₆

₇C₆

₈C₁₄

₈C₂₁

C₂³

₇C₂³

₇C₂xC₆

₈C₄₂

C₂⁴

C₂²xC₆

₇C₂²xC₆

F₈

C₂³xC₆

C₂xF₈

C₃xF₈

C₆xF₈

Smallest permutation representation of C₆xF₈
►On 42 points

Generators in S₄₂

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

(2 20)(4 15)(5 16)(6 17)(8 41)(9 42)(10 36)(13 39)(22 31)(25 34)(27 29)(28 30)

(3 21)(5 16)(6 17)(7 18)(9 42)(10 36)(11 37)(14 40)(22 31)(23 32)(26 35)(28 30)

(1 19)(4 15)(6 17)(7 18)(8 41)(10 36)(11 37)(12 38)(22 31)(23 32)(24 33)(27 29)

(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)

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

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

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

48 conjugacy classes

class	1	2A	2B	2C	3A	3B	6A	6B	6C	6D	6E	6F	7A	···	7F	14A	···	14F	21A	···	21L	42A	···	42L
order	1	2	2	2	3	3	6	6	6	6	6	6	7	···	7	14	···	14	21	···	21	42	···	42
size	1	1	7	7	1	1	1	1	7	7	7	7	8	···	8	8	···	8	8	···	8	8	···	8

48 irreducible representations

dim	1	1	1	1	1	1	1	1	7	7	7	7
type	+	+							+	+
image	C₁	C₂	C₃	C₆	C₇	C₁₄	C₂₁	C₄₂	F₈	C₂xF₈	C₃xF₈	C₆xF₈
kernel	C₆xF₈	C₃xF₈	C₂xF₈	F₈	C₂³xC₆	C₂²xC₆	C₂⁴	C₂³	C₆	C₃	C₂	C₁
# reps	1	1	2	2	6	6	12	12	1	1	2	2

Matrix representation of C₆xF₈ ►in GL₇(F₄₃)

7	0	0	0	0	0	0
0	7	0	0	0	0	0
0	0	7	0	0	0	0
0	0	0	7	0	0	0
0	0	0	0	7	0	0
0	0	0	0	0	7	0
0	0	0	0	0	0	7

1	0	0	0	0	0	0
0	1	0	0	0	0	0
16	0	42	0	0	0	0
21	0	0	42	0	0	0
41	0	0	0	42	0	0
0	0	0	0	0	1	0
11	0	0	0	0	0	42

1	0	0	0	0	0	0
4	42	0	0	0	0	0
16	0	42	0	0	0	0
21	0	0	42	0	0	0
0	0	0	0	1	0	0
35	0	0	0	0	42	0
0	0	0	0	0	0	1

42	0	0	0	0	0	0
0	42	0	0	0	0	0
0	0	42	0	0	0	0
22	0	0	1	0	0	0
0	0	0	0	42	0	0
8	0	0	0	0	1	0
32	0	0	0	0	0	1

4	41	0	0	0	0	0
0	39	1	0	0	0	0
0	27	0	1	0	0	0
0	22	0	0	1	0	0
0	2	0	0	0	1	0
0	8	0	0	0	0	1
0	32	0	0	0	0	0

G:=sub<GL(7,GF(43))| [7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7,0,0,0,0,0,0,0,7],[1,0,16,21,41,0,11,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42],[1,4,16,21,0,35,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1],[42,0,0,22,0,8,32,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,41,39,27,22,2,8,32,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₆xF₈ in GAP, Magma, Sage, TeX

C_6\times F_8

% in TeX

G:=Group("C6xF8");

// GroupNames label

G:=SmallGroup(336,213);

// by ID

G=gap.SmallGroup(336,213);

# by ID

G:=PCGroup([6,-2,-3,-7,-2,2,2,351,856,1277]);

// Polycyclic

G:=Group<a,b,c,d,e|a^6=b^2=c^2=d^2=e^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=d*c=c*d,e*c*e^-1=b,e*d*e^-1=c>;

// generators/relations

Export

Subgroup lattice of C₆xF₈ in TeX

G = C6xF8 order 336 = 24·3·7

Direct product of C6 and F8

G = C₆xF₈ order 336 = 2⁴·3·7

Direct product of C₆ and F₈