C8xF8 - GroupNames

G = C₈xF₈ order 448 = 2⁶·7

Direct product of C₈ and F₈

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

Aliases: C₈xF₈, C₂³:C₅₆, C₂⁴.C₂₈, (C₂³xC₈):C₇, C₂.(C₄xF₈), (C₂xF₈).C₄, C₄.₂(C₂xF₈), (C₄xF₈).₂C₂, (C₂³xC₄).₂C₁₄, SmallGroup(448,919)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₈xF₈

Chief series

C₁ — C₂³ — C₂⁴ — C₂³xC₄ — C₄xF₈ — C₈xF₈

Lower central

C₂³ — C₈xF₈

Upper central

C₁ — C₈

Generators and relations for C₈xF₈
G = < a,b,c,d,e | a⁸=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: 205 in 39 conjugacy classes, 12 normal (all characteristic)
Quotients: C₁, C₂, C₄, C₇, C₈, C₁₄, C₂₈, C₅₆, F₈, C₂xF₈, C₄xF₈, C₈xF₈

C₁

C₂

₇C₂

₈C₇

C₄

₇C₂²

₇C₄

₇C₂²

₈C₁₄

C₈

C₂³

₇C₂³

₇C₂xC₄

₇C₈

₈C₂₈

C₂⁴

₇C₂xC₈

₇C₂²xC₄

₇C₂xC₈

F₈

₈C₅₆

C₂³xC₄

₇C₂²xC₈

C₂xF₈

C₂³xC₈

C₄xF₈

C₈xF₈

Smallest permutation representation of C₈xF₈
►On 56 points

Generators in S₅₆

(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)(49 50 51 52 53 54 55 56)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)

(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)

(1 5)(2 6)(3 7)(4 8)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)

(1 31 38 55 23 47 11)(2 32 39 56 24 48 12)(3 25 40 49 17 41 13)(4 26 33 50 18 42 14)(5 27 34 51 19 43 15)(6 28 35 52 20 44 16)(7 29 36 53 21 45 9)(8 30 37 54 22 46 10)

G:=sub<Sym(56)| (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)(49,50,51,52,53,54,55,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,5)(2,6)(3,7)(4,8)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,31,38,55,23,47,11)(2,32,39,56,24,48,12)(3,25,40,49,17,41,13)(4,26,33,50,18,42,14)(5,27,34,51,19,43,15)(6,28,35,52,20,44,16)(7,29,36,53,21,45,9)(8,30,37,54,22,46,10)>;

G:=Group( (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)(49,50,51,52,53,54,55,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(41,45)(42,46)(43,47)(44,48)(49,53)(50,54)(51,55)(52,56), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,21)(18,22)(19,23)(20,24)(25,29)(26,30)(27,31)(28,32), (1,5)(2,6)(3,7)(4,8)(25,29)(26,30)(27,31)(28,32)(33,37)(34,38)(35,39)(36,40)(41,45)(42,46)(43,47)(44,48), (1,31,38,55,23,47,11)(2,32,39,56,24,48,12)(3,25,40,49,17,41,13)(4,26,33,50,18,42,14)(5,27,34,51,19,43,15)(6,28,35,52,20,44,16)(7,29,36,53,21,45,9)(8,30,37,54,22,46,10) );

G=PermutationGroup([[(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),(49,50,51,52,53,54,55,56)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(41,45),(42,46),(43,47),(44,48),(49,53),(50,54),(51,55),(52,56)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,21),(18,22),(19,23),(20,24),(25,29),(26,30),(27,31),(28,32)], [(1,5),(2,6),(3,7),(4,8),(25,29),(26,30),(27,31),(28,32),(33,37),(34,38),(35,39),(36,40),(41,45),(42,46),(43,47),(44,48)], [(1,31,38,55,23,47,11),(2,32,39,56,24,48,12),(3,25,40,49,17,41,13),(4,26,33,50,18,42,14),(5,27,34,51,19,43,15),(6,28,35,52,20,44,16),(7,29,36,53,21,45,9),(8,30,37,54,22,46,10)]])

64 conjugacy classes

class	1	2A	2B	2C	4A	4B	4C	4D	7A	···	7F	8A	8B	8C	8D	8E	8F	8G	8H	14A	···	14F	28A	···	28L	56A	···	56X
order	1	2	2	2	4	4	4	4	7	···	7	8	8	8	8	8	8	8	8	14	···	14	28	···	28	56	···	56
size	1	1	7	7	1	1	7	7	8	···	8	1	1	1	1	7	7	7	7	8	···	8	8	···	8	8	···	8

64 irreducible representations

dim	1	1	1	1	1	1	1	1	7	7	7	7
type	+	+							+	+
image	C₁	C₂	C₄	C₇	C₈	C₁₄	C₂₈	C₅₆	F₈	C₂xF₈	C₄xF₈	C₈xF₈
kernel	C₈xF₈	C₄xF₈	C₂xF₈	C₂³xC₈	F₈	C₂³xC₄	C₂⁴	C₂³	C₈	C₄	C₂	C₁
# reps	1	1	2	6	4	6	12	24	1	1	2	4

Matrix representation of C₈xF₈ ►in GL₇(F₁₁₃)

18	0	0	0	0	0	0
0	18	0	0	0	0	0
0	0	18	0	0	0	0
0	0	0	18	0	0	0
0	0	0	0	18	0	0
0	0	0	0	0	18	0
0	0	0	0	0	0	18

112	0	0	0	0	0	0
0	112	0	0	0	0	0
0	0	112	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	112	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	1

112	0	0	0	0	0	0
0	112	0	0	0	0	0
0	0	1	0	0	0	0
0	0	0	112	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	1	0
0	0	0	0	0	0	112

112	0	0	0	0	0	0
0	1	0	0	0	0	0
0	0	112	0	0	0	0
0	0	0	1	0	0	0
0	0	0	0	1	0	0
0	0	0	0	0	112	0
0	0	0	0	0	0	112

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
0	0	0	0	0	0	1
1	0	0	0	0	0	0

G:=sub<GL(7,GF(113))| [18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18,0,0,0,0,0,0,0,18],[112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[112,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112],[112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,112,0,0,0,0,0,0,0,112],[0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0] >;

C₈xF₈ in GAP, Magma, Sage, TeX

C_8\times F_8

% in TeX

G:=Group("C8xF8");

// GroupNames label

G:=SmallGroup(448,919);

// by ID

G=gap.SmallGroup(448,919);

# by ID

G:=PCGroup([7,-2,-7,-2,-2,-2,2,2,98,58,1971,4716,6873]);

// Polycyclic

G:=Group<a,b,c,d,e|a^8=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 C₈xF₈ in TeX

G = C8xF8 order 448 = 26·7

Direct product of C8 and F8

G = C₈xF₈ order 448 = 2⁶·7

Direct product of C₈ and F₈