C2^3:F8 - GroupNames

G = C₂³⋊F₈ order 448 = 2⁶·7

2^nd semidirect product of C₂³ and F₈ acting via F₈/C₂³=C₇

metabelian, soluble, monomial, A-group

Aliases: C₂⁶⋊₃C₇, C₂³⋊₂F₈, SmallGroup(448,1394)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂⁶ — C₂³⋊F₈

Chief series

C₁ — C₂³ — C₂⁶ — C₂³⋊F₈

Lower central

C₂⁶ — C₂³⋊F₈

Upper central

C₁

Generators and relations for C₂³⋊F₈
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=g⁷=1, ab=ba, gbg^-1=ac=ca, ad=da, ae=ea, af=fa, gag^-1=c, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, gcg^-1=abc, de=ed, df=fd, gdg^-1=fe=ef, geg^-1=d, gfg^-1=e >

Subgroups: 2906 in 411 conjugacy classes, 5 normal (3 characteristic)
C₁, C₂, C₂², C₇, C₂³, C₂³, C₂⁴, C₂⁵, F₈, C₂⁶, C₂³⋊F₈
Quotients: C₁, C₇, F₈, C₂³⋊F₈

Character table of C₂³⋊F₈

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	7A	7B	7C	7D	7E	7F
size	1	7	7	7	7	7	7	7	7	7	64	64	64	64	64	64
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	1	1	1	1	ζ₇³	ζ₇⁶	ζ₇²	ζ₇⁵	ζ₇	ζ₇⁴	linear of order 7
ρ₃	1	1	1	1	1	1	1	1	1	1	ζ₇⁵	ζ₇³	ζ₇	ζ₇⁶	ζ₇⁴	ζ₇²	linear of order 7
ρ₄	1	1	1	1	1	1	1	1	1	1	ζ₇²	ζ₇⁴	ζ₇⁶	ζ₇	ζ₇³	ζ₇⁵	linear of order 7
ρ₅	1	1	1	1	1	1	1	1	1	1	ζ₇⁴	ζ₇	ζ₇⁵	ζ₇²	ζ₇⁶	ζ₇³	linear of order 7
ρ₆	1	1	1	1	1	1	1	1	1	1	ζ₇	ζ₇²	ζ₇³	ζ₇⁴	ζ₇⁵	ζ₇⁶	linear of order 7
ρ₇	1	1	1	1	1	1	1	1	1	1	ζ₇⁶	ζ₇⁵	ζ₇⁴	ζ₇³	ζ₇²	ζ₇	linear of order 7
ρ₈	7	3	-1	-1	-1	-1	-5	3	-1	3	0	0	0	0	0	0	orthogonal faithful
ρ₉	7	3	-1	-1	3	3	-1	-1	-1	-5	0	0	0	0	0	0	orthogonal faithful
ρ₁₀	7	-1	-1	3	-1	3	-1	-5	-1	3	0	0	0	0	0	0	orthogonal faithful
ρ₁₁	7	-5	-1	3	3	-1	-1	3	-1	-1	0	0	0	0	0	0	orthogonal faithful
ρ₁₂	7	-1	-1	-1	-5	3	3	3	-1	-1	0	0	0	0	0	0	orthogonal faithful
ρ₁₃	7	-1	-1	-5	3	-1	3	-1	-1	3	0	0	0	0	0	0	orthogonal faithful
ρ₁₄	7	-1	7	-1	-1	-1	-1	-1	-1	-1	0	0	0	0	0	0	orthogonal lifted from F₈
ρ₁₅	7	-1	-1	-1	-1	-1	-1	-1	7	-1	0	0	0	0	0	0	orthogonal lifted from F₈
ρ₁₆	7	3	-1	3	-1	-5	3	-1	-1	-1	0	0	0	0	0	0	orthogonal faithful

Permutation representations of C₂³⋊F₈
►On 14 points - transitive group 14T21

Generators in S₁₄

(2 9)(3 10)(4 11)(6 13)

(2 9)(5 12)(6 13)(7 14)

(1 8)(2 9)(3 10)(5 12)

(1 8)(2 9)(3 10)(6 13)

(2 9)(3 10)(4 11)(7 14)

(1 8)(3 10)(4 11)(5 12)

(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)

G:=sub<Sym(14)| (2,9)(3,10)(4,11)(6,13), (2,9)(5,12)(6,13)(7,14), (1,8)(2,9)(3,10)(5,12), (1,8)(2,9)(3,10)(6,13), (2,9)(3,10)(4,11)(7,14), (1,8)(3,10)(4,11)(5,12), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)>;

G:=Group( (2,9)(3,10)(4,11)(6,13), (2,9)(5,12)(6,13)(7,14), (1,8)(2,9)(3,10)(5,12), (1,8)(2,9)(3,10)(6,13), (2,9)(3,10)(4,11)(7,14), (1,8)(3,10)(4,11)(5,12), (1,2,3,4,5,6,7)(8,9,10,11,12,13,14) );

G=PermutationGroup([[(2,9),(3,10),(4,11),(6,13)], [(2,9),(5,12),(6,13),(7,14)], [(1,8),(2,9),(3,10),(5,12)], [(1,8),(2,9),(3,10),(6,13)], [(2,9),(3,10),(4,11),(7,14)], [(1,8),(3,10),(4,11),(5,12)], [(1,2,3,4,5,6,7),(8,9,10,11,12,13,14)]])

G:=TransitiveGroup(14,21);

►On 28 points - transitive group 28T62

Generators in S₂₈

(2 16)(3 8)(4 9)(5 26)(6 20)(7 28)(10 19)(11 27)(12 21)(14 23)(17 24)(18 25)

(1 22)(2 16)(3 24)(5 19)(6 11)(7 12)(8 17)(10 26)(13 15)(14 23)(20 27)(21 28)

(1 15)(2 14)(3 8)(4 25)(5 19)(6 27)(9 18)(10 26)(11 20)(13 22)(16 23)(17 24)

(1 22)(2 14)(3 24)(4 18)(5 19)(6 11)(8 17)(9 25)(10 26)(13 15)(16 23)(20 27)

(2 23)(3 8)(4 25)(5 19)(6 20)(7 12)(9 18)(10 26)(11 27)(14 16)(17 24)(21 28)

(1 13)(3 24)(4 9)(5 26)(6 20)(7 21)(8 17)(10 19)(11 27)(12 28)(15 22)(18 25)

(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)

G:=sub<Sym(28)| (2,16)(3,8)(4,9)(5,26)(6,20)(7,28)(10,19)(11,27)(12,21)(14,23)(17,24)(18,25), (1,22)(2,16)(3,24)(5,19)(6,11)(7,12)(8,17)(10,26)(13,15)(14,23)(20,27)(21,28), (1,15)(2,14)(3,8)(4,25)(5,19)(6,27)(9,18)(10,26)(11,20)(13,22)(16,23)(17,24), (1,22)(2,14)(3,24)(4,18)(5,19)(6,11)(8,17)(9,25)(10,26)(13,15)(16,23)(20,27), (2,23)(3,8)(4,25)(5,19)(6,20)(7,12)(9,18)(10,26)(11,27)(14,16)(17,24)(21,28), (1,13)(3,24)(4,9)(5,26)(6,20)(7,21)(8,17)(10,19)(11,27)(12,28)(15,22)(18,25), (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)>;

G:=Group( (2,16)(3,8)(4,9)(5,26)(6,20)(7,28)(10,19)(11,27)(12,21)(14,23)(17,24)(18,25), (1,22)(2,16)(3,24)(5,19)(6,11)(7,12)(8,17)(10,26)(13,15)(14,23)(20,27)(21,28), (1,15)(2,14)(3,8)(4,25)(5,19)(6,27)(9,18)(10,26)(11,20)(13,22)(16,23)(17,24), (1,22)(2,14)(3,24)(4,18)(5,19)(6,11)(8,17)(9,25)(10,26)(13,15)(16,23)(20,27), (2,23)(3,8)(4,25)(5,19)(6,20)(7,12)(9,18)(10,26)(11,27)(14,16)(17,24)(21,28), (1,13)(3,24)(4,9)(5,26)(6,20)(7,21)(8,17)(10,19)(11,27)(12,28)(15,22)(18,25), (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) );

G=PermutationGroup([[(2,16),(3,8),(4,9),(5,26),(6,20),(7,28),(10,19),(11,27),(12,21),(14,23),(17,24),(18,25)], [(1,22),(2,16),(3,24),(5,19),(6,11),(7,12),(8,17),(10,26),(13,15),(14,23),(20,27),(21,28)], [(1,15),(2,14),(3,8),(4,25),(5,19),(6,27),(9,18),(10,26),(11,20),(13,22),(16,23),(17,24)], [(1,22),(2,14),(3,24),(4,18),(5,19),(6,11),(8,17),(9,25),(10,26),(13,15),(16,23),(20,27)], [(2,23),(3,8),(4,25),(5,19),(6,20),(7,12),(9,18),(10,26),(11,27),(14,16),(17,24),(21,28)], [(1,13),(3,24),(4,9),(5,26),(6,20),(7,21),(8,17),(10,19),(11,27),(12,28),(15,22),(18,25)], [(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)]])

G:=TransitiveGroup(28,62);

►On 28 points - transitive group 28T63

Generators in S₂₈

(2 20)(3 8)(4 9)(5 23)(6 17)(7 25)(10 16)(11 24)(12 18)(14 27)(15 22)(21 28)

(1 26)(2 20)(3 28)(5 16)(6 11)(7 12)(8 21)(10 23)(13 19)(14 27)(17 24)(18 25)

(1 19)(2 14)(3 8)(4 22)(5 16)(6 24)(9 15)(10 23)(11 17)(13 26)(20 27)(21 28)

(1 26)(2 14)(4 15)(5 10)(6 17)(7 25)(9 22)(11 24)(12 18)(13 19)(16 23)(20 27)

(1 26)(2 27)(3 8)(5 16)(6 11)(7 18)(10 23)(12 25)(13 19)(14 20)(17 24)(21 28)

(1 19)(2 27)(3 28)(4 9)(6 17)(7 12)(8 21)(11 24)(13 26)(14 20)(15 22)(18 25)

(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)

G:=sub<Sym(28)| (2,20)(3,8)(4,9)(5,23)(6,17)(7,25)(10,16)(11,24)(12,18)(14,27)(15,22)(21,28), (1,26)(2,20)(3,28)(5,16)(6,11)(7,12)(8,21)(10,23)(13,19)(14,27)(17,24)(18,25), (1,19)(2,14)(3,8)(4,22)(5,16)(6,24)(9,15)(10,23)(11,17)(13,26)(20,27)(21,28), (1,26)(2,14)(4,15)(5,10)(6,17)(7,25)(9,22)(11,24)(12,18)(13,19)(16,23)(20,27), (1,26)(2,27)(3,8)(5,16)(6,11)(7,18)(10,23)(12,25)(13,19)(14,20)(17,24)(21,28), (1,19)(2,27)(3,28)(4,9)(6,17)(7,12)(8,21)(11,24)(13,26)(14,20)(15,22)(18,25), (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)>;

G:=Group( (2,20)(3,8)(4,9)(5,23)(6,17)(7,25)(10,16)(11,24)(12,18)(14,27)(15,22)(21,28), (1,26)(2,20)(3,28)(5,16)(6,11)(7,12)(8,21)(10,23)(13,19)(14,27)(17,24)(18,25), (1,19)(2,14)(3,8)(4,22)(5,16)(6,24)(9,15)(10,23)(11,17)(13,26)(20,27)(21,28), (1,26)(2,14)(4,15)(5,10)(6,17)(7,25)(9,22)(11,24)(12,18)(13,19)(16,23)(20,27), (1,26)(2,27)(3,8)(5,16)(6,11)(7,18)(10,23)(12,25)(13,19)(14,20)(17,24)(21,28), (1,19)(2,27)(3,28)(4,9)(6,17)(7,12)(8,21)(11,24)(13,26)(14,20)(15,22)(18,25), (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) );

G=PermutationGroup([[(2,20),(3,8),(4,9),(5,23),(6,17),(7,25),(10,16),(11,24),(12,18),(14,27),(15,22),(21,28)], [(1,26),(2,20),(3,28),(5,16),(6,11),(7,12),(8,21),(10,23),(13,19),(14,27),(17,24),(18,25)], [(1,19),(2,14),(3,8),(4,22),(5,16),(6,24),(9,15),(10,23),(11,17),(13,26),(20,27),(21,28)], [(1,26),(2,14),(4,15),(5,10),(6,17),(7,25),(9,22),(11,24),(12,18),(13,19),(16,23),(20,27)], [(1,26),(2,27),(3,8),(5,16),(6,11),(7,18),(10,23),(12,25),(13,19),(14,20),(17,24),(21,28)], [(1,19),(2,27),(3,28),(4,9),(6,17),(7,12),(8,21),(11,24),(13,26),(14,20),(15,22),(18,25)], [(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)]])

G:=TransitiveGroup(28,63);

►On 28 points - transitive group 28T64

Generators in S₂₈

(1 18)(2 19)(3 20)(5 15)(8 22)(10 24)(13 27)(14 28)

(1 18)(4 21)(5 15)(6 16)(9 23)(10 24)(11 25)(13 27)

(1 18)(2 19)(4 21)(7 17)(9 23)(12 26)(13 27)(14 28)

(2 28)(3 8)(4 23)(5 15)(6 16)(7 12)(9 21)(10 24)(11 25)(14 19)(17 26)(20 22)

(1 13)(3 22)(4 9)(5 24)(6 16)(7 17)(8 20)(10 15)(11 25)(12 26)(18 27)(21 23)

(1 18)(2 14)(4 23)(5 10)(6 25)(7 17)(9 21)(11 16)(12 26)(13 27)(15 24)(19 28)

(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)

G:=sub<Sym(28)| (1,18)(2,19)(3,20)(5,15)(8,22)(10,24)(13,27)(14,28), (1,18)(4,21)(5,15)(6,16)(9,23)(10,24)(11,25)(13,27), (1,18)(2,19)(4,21)(7,17)(9,23)(12,26)(13,27)(14,28), (2,28)(3,8)(4,23)(5,15)(6,16)(7,12)(9,21)(10,24)(11,25)(14,19)(17,26)(20,22), (1,13)(3,22)(4,9)(5,24)(6,16)(7,17)(8,20)(10,15)(11,25)(12,26)(18,27)(21,23), (1,18)(2,14)(4,23)(5,10)(6,25)(7,17)(9,21)(11,16)(12,26)(13,27)(15,24)(19,28), (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)>;

G:=Group( (1,18)(2,19)(3,20)(5,15)(8,22)(10,24)(13,27)(14,28), (1,18)(4,21)(5,15)(6,16)(9,23)(10,24)(11,25)(13,27), (1,18)(2,19)(4,21)(7,17)(9,23)(12,26)(13,27)(14,28), (2,28)(3,8)(4,23)(5,15)(6,16)(7,12)(9,21)(10,24)(11,25)(14,19)(17,26)(20,22), (1,13)(3,22)(4,9)(5,24)(6,16)(7,17)(8,20)(10,15)(11,25)(12,26)(18,27)(21,23), (1,18)(2,14)(4,23)(5,10)(6,25)(7,17)(9,21)(11,16)(12,26)(13,27)(15,24)(19,28), (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) );

G=PermutationGroup([[(1,18),(2,19),(3,20),(5,15),(8,22),(10,24),(13,27),(14,28)], [(1,18),(4,21),(5,15),(6,16),(9,23),(10,24),(11,25),(13,27)], [(1,18),(2,19),(4,21),(7,17),(9,23),(12,26),(13,27),(14,28)], [(2,28),(3,8),(4,23),(5,15),(6,16),(7,12),(9,21),(10,24),(11,25),(14,19),(17,26),(20,22)], [(1,13),(3,22),(4,9),(5,24),(6,16),(7,17),(8,20),(10,15),(11,25),(12,26),(18,27),(21,23)], [(1,18),(2,14),(4,23),(5,10),(6,25),(7,17),(9,21),(11,16),(12,26),(13,27),(15,24),(19,28)], [(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)]])

G:=TransitiveGroup(28,64);

►On 28 points - transitive group 28T65

Generators in S₂₈

(2 16)(3 8)(4 9)(5 26)(6 20)(7 28)(10 19)(11 27)(12 21)(14 23)(17 24)(18 25)

(1 22)(2 16)(3 24)(5 19)(6 11)(7 12)(8 17)(10 26)(13 15)(14 23)(20 27)(21 28)

(1 15)(2 14)(3 8)(4 25)(5 19)(6 27)(9 18)(10 26)(11 20)(13 22)(16 23)(17 24)

(1 22)(2 14)(3 8)(4 18)(6 27)(7 21)(9 25)(11 20)(12 28)(13 15)(16 23)(17 24)

(1 15)(2 23)(3 8)(4 9)(5 19)(7 28)(10 26)(12 21)(13 22)(14 16)(17 24)(18 25)

(1 22)(2 16)(3 24)(4 9)(5 10)(6 20)(8 17)(11 27)(13 15)(14 23)(18 25)(19 26)

(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)

G:=sub<Sym(28)| (2,16)(3,8)(4,9)(5,26)(6,20)(7,28)(10,19)(11,27)(12,21)(14,23)(17,24)(18,25), (1,22)(2,16)(3,24)(5,19)(6,11)(7,12)(8,17)(10,26)(13,15)(14,23)(20,27)(21,28), (1,15)(2,14)(3,8)(4,25)(5,19)(6,27)(9,18)(10,26)(11,20)(13,22)(16,23)(17,24), (1,22)(2,14)(3,8)(4,18)(6,27)(7,21)(9,25)(11,20)(12,28)(13,15)(16,23)(17,24), (1,15)(2,23)(3,8)(4,9)(5,19)(7,28)(10,26)(12,21)(13,22)(14,16)(17,24)(18,25), (1,22)(2,16)(3,24)(4,9)(5,10)(6,20)(8,17)(11,27)(13,15)(14,23)(18,25)(19,26), (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)>;

G:=Group( (2,16)(3,8)(4,9)(5,26)(6,20)(7,28)(10,19)(11,27)(12,21)(14,23)(17,24)(18,25), (1,22)(2,16)(3,24)(5,19)(6,11)(7,12)(8,17)(10,26)(13,15)(14,23)(20,27)(21,28), (1,15)(2,14)(3,8)(4,25)(5,19)(6,27)(9,18)(10,26)(11,20)(13,22)(16,23)(17,24), (1,22)(2,14)(3,8)(4,18)(6,27)(7,21)(9,25)(11,20)(12,28)(13,15)(16,23)(17,24), (1,15)(2,23)(3,8)(4,9)(5,19)(7,28)(10,26)(12,21)(13,22)(14,16)(17,24)(18,25), (1,22)(2,16)(3,24)(4,9)(5,10)(6,20)(8,17)(11,27)(13,15)(14,23)(18,25)(19,26), (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) );

G=PermutationGroup([[(2,16),(3,8),(4,9),(5,26),(6,20),(7,28),(10,19),(11,27),(12,21),(14,23),(17,24),(18,25)], [(1,22),(2,16),(3,24),(5,19),(6,11),(7,12),(8,17),(10,26),(13,15),(14,23),(20,27),(21,28)], [(1,15),(2,14),(3,8),(4,25),(5,19),(6,27),(9,18),(10,26),(11,20),(13,22),(16,23),(17,24)], [(1,22),(2,14),(3,8),(4,18),(6,27),(7,21),(9,25),(11,20),(12,28),(13,15),(16,23),(17,24)], [(1,15),(2,23),(3,8),(4,9),(5,19),(7,28),(10,26),(12,21),(13,22),(14,16),(17,24),(18,25)], [(1,22),(2,16),(3,24),(4,9),(5,10),(6,20),(8,17),(11,27),(13,15),(14,23),(18,25),(19,26)], [(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)]])

G:=TransitiveGroup(28,65);

►On 28 points - transitive group 28T66

Generators in S₂₈

(2 19)(3 20)(4 21)(6 16)(8 26)(9 27)(11 22)(14 25)

(2 19)(5 15)(6 16)(7 17)(10 28)(11 22)(12 23)(14 25)

(1 18)(2 19)(3 20)(5 15)(8 26)(10 28)(13 24)(14 25)

(2 25)(3 26)(4 27)(7 23)(8 20)(9 21)(12 17)(14 19)

(1 24)(3 26)(4 27)(5 28)(8 20)(9 21)(10 15)(13 18)

(2 25)(4 27)(5 28)(6 22)(9 21)(10 15)(11 16)(14 19)

(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)

G:=sub<Sym(28)| (2,19)(3,20)(4,21)(6,16)(8,26)(9,27)(11,22)(14,25), (2,19)(5,15)(6,16)(7,17)(10,28)(11,22)(12,23)(14,25), (1,18)(2,19)(3,20)(5,15)(8,26)(10,28)(13,24)(14,25), (2,25)(3,26)(4,27)(7,23)(8,20)(9,21)(12,17)(14,19), (1,24)(3,26)(4,27)(5,28)(8,20)(9,21)(10,15)(13,18), (2,25)(4,27)(5,28)(6,22)(9,21)(10,15)(11,16)(14,19), (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)>;

G:=Group( (2,19)(3,20)(4,21)(6,16)(8,26)(9,27)(11,22)(14,25), (2,19)(5,15)(6,16)(7,17)(10,28)(11,22)(12,23)(14,25), (1,18)(2,19)(3,20)(5,15)(8,26)(10,28)(13,24)(14,25), (2,25)(3,26)(4,27)(7,23)(8,20)(9,21)(12,17)(14,19), (1,24)(3,26)(4,27)(5,28)(8,20)(9,21)(10,15)(13,18), (2,25)(4,27)(5,28)(6,22)(9,21)(10,15)(11,16)(14,19), (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) );

G=PermutationGroup([[(2,19),(3,20),(4,21),(6,16),(8,26),(9,27),(11,22),(14,25)], [(2,19),(5,15),(6,16),(7,17),(10,28),(11,22),(12,23),(14,25)], [(1,18),(2,19),(3,20),(5,15),(8,26),(10,28),(13,24),(14,25)], [(2,25),(3,26),(4,27),(7,23),(8,20),(9,21),(12,17),(14,19)], [(1,24),(3,26),(4,27),(5,28),(8,20),(9,21),(10,15),(13,18)], [(2,25),(4,27),(5,28),(6,22),(9,21),(10,15),(11,16),(14,19)], [(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)]])

G:=TransitiveGroup(28,66);

Polynomial with Galois group C₂³⋊F₈ over ℚ

action	f(x)	Disc(f)
14T21	x¹⁴-8x¹²+9x¹⁰+75x⁸-227x⁶+136x⁴+112x²-49	2¹⁴·7¹⁰·43¹²

Matrix representation of C₂³⋊F₈ ►in GL₇(ℤ)

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
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
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
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
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
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
0	0	0	0	0	0	-1

0	0	0	0	0	1	0
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

G:=sub<GL(7,Integers())| [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,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,-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,-1,0,0,0,0,0,0,0,1],[-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,-1,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0] >;

C₂³⋊F₈ in GAP, Magma, Sage, TeX

C_2^3\rtimes F_8

% in TeX

G:=Group("C2^3:F8");

// GroupNames label

G:=SmallGroup(448,1394);

// by ID

G=gap.SmallGroup(448,1394);

# by ID

G:=PCGroup([7,-7,-2,2,2,-2,2,2,491,1031,591,3924,9413,13726]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂³⋊F₈ in TeX

G = C23⋊F8 order 448 = 26·7

2nd semidirect product of C23 and F8 acting via F8/C23=C7

G = C₂³⋊F₈ order 448 = 2⁶·7

2^nd semidirect product of C₂³ and F₈ acting via F₈/C₂³=C₇