C6xF7 - GroupNames

G = C₆×F₇ order 252 = 2²·3²·7

Direct product of C₆ and F₇

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C₆×F₇, C₇⋊C₆², C₄₂⋊₂C₆, D₁₄⋊C₃², C₁₄⋊(C₃×C₆), D₇⋊(C₃×C₆), (C₆×D₇)⋊C₃, C₂₁⋊₃(C₂×C₆), (C₃×D₇)⋊₂C₆, (C₂×C₇⋊C₃)⋊C₆, C₇⋊C₃⋊(C₂×C₆), (C₆×C₇⋊C₃)⋊₂C₂, (C₃×C₇⋊C₃)⋊₃C₂², SmallGroup(252,28)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₇ — C₆×F₇

Chief series

C₁ — C₇ — C₂₁ — C₃×C₇⋊C₃ — C₃×F₇ — C₆×F₇

Lower central

C₇ — C₆×F₇

Upper central

C₁ — C₆

Generators and relations for C₆×F₇
G = < a,b,c | a⁶=b⁷=c⁶=1, ab=ba, ac=ca, cbc^-1=b⁵ >

Subgroups: 216 in 60 conjugacy classes, 34 normal (13 characteristic)
C₁, C₂, C₂, C₃, C₃, C₂², C₆, C₆, C₇, C₃², C₂×C₆, D₇, C₁₄, C₃×C₆, C₇⋊C₃, C₂₁, D₁₄, C₆², F₇, C₂×C₇⋊C₃, C₃×D₇, C₄₂, C₃×C₇⋊C₃, C₂×F₇, C₆×D₇, C₃×F₇, C₆×C₇⋊C₃, C₆×F₇
Quotients: C₁, C₂, C₃, C₂², C₆, C₃², C₂×C₆, C₃×C₆, C₆², F₇, C₂×F₇, C₃×F₇, C₆×F₇

Smallest permutation representation of C₆×F₇
►On 42 points

Generators in S₄₂

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

(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)

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

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

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

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

42 conjugacy classes

class	1	2A	2B	2C	3A	3B	3C	···	3H	6A	6B	6C	···	6X	7	14	21A	21B	42A	42B
order	1	2	2	2	3	3	3	···	3	6	6	6	···	6	7	14	21	21	42	42
size	1	1	7	7	1	1	7	···	7	1	1	7	···	7	6	6	6	6	6	6

42 irreducible representations

dim	1	1	1	1	1	1	1	1	1	6	6	6	6
type	+	+	+							+	+
image	C₁	C₂	C₂	C₃	C₃	C₆	C₆	C₆	C₆	F₇	C₂×F₇	C₃×F₇	C₆×F₇
kernel	C₆×F₇	C₃×F₇	C₆×C₇⋊C₃	C₂×F₇	C₆×D₇	F₇	C₂×C₇⋊C₃	C₃×D₇	C₄₂	C₆	C₃	C₂	C₁
# reps	1	2	1	6	2	12	6	4	2	1	1	2	2

Matrix representation of C₆×F₇ ►in GL₆(𝔽₄₃)

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

42	42	42	42	42	42
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

42	0	0	0	0	0
0	0	0	0	0	42
0	0	0	42	0	0
0	42	0	0	0	0
1	1	1	1	1	1
0	0	0	0	42	0

G:=sub<GL(6,GF(43))| [7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[42,1,0,0,0,0,42,0,1,0,0,0,42,0,0,1,0,0,42,0,0,0,1,0,42,0,0,0,0,1,42,0,0,0,0,0],[42,0,0,0,1,0,0,0,0,42,1,0,0,0,0,0,1,0,0,0,42,0,1,0,0,0,0,0,1,42,0,42,0,0,1,0] >;

C₆×F₇ in GAP, Magma, Sage, TeX

C_6\times F_7

% in TeX

G:=Group("C6xF7");

// GroupNames label

G:=SmallGroup(252,28);

// by ID

G=gap.SmallGroup(252,28);

# by ID

G:=PCGroup([5,-2,-2,-3,-3,-7,5404,914]);

// Polycyclic

G:=Group<a,b,c|a^6=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;

// generators/relations

G = C6×F7 order 252 = 22·32·7

Direct product of C6 and F7

G = C₆×F₇ order 252 = 2²·3²·7

Direct product of C₆ and F₇