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## G = D4×F8order 448 = 26·7

### Direct product of D4 and F8

Aliases: D4×F8, C25⋊C14, C4⋊(C2×F8), (C4×F8)⋊C2, (D4×C23)⋊C7, C22⋊(C2×F8), (C23×C4)⋊C14, (C22×F8)⋊C2, C232(C7×D4), C2.2(C22×F8), C24.2(C2×C14), (C2×F8).2C22, SmallGroup(448,1363)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — D4×F8
 Chief series C1 — C23 — C24 — C2×F8 — C22×F8 — D4×F8
 Lower central C23 — C24 — D4×F8
 Upper central C1 — C2 — D4

Generators and relations for D4×F8
G = < a,b,c,d,e,f | a4=b2=c2=d2=e2=f7=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, fcf-1=ed=de, fdf-1=c, fef-1=d >

Subgroups: 1027 in 127 conjugacy classes, 18 normal (12 characteristic)
C1, C2, C2, C4, C4, C22, C22, C7, C2×C4, D4, D4, C23, C23, C14, C22×C4, C2×D4, C24, C24, C28, C2×C14, C23×C4, C22×D4, C25, C7×D4, F8, D4×C23, C2×F8, C2×F8, C4×F8, C22×F8, D4×F8
Quotients: C1, C2, C22, C7, D4, C14, C2×C14, C7×D4, F8, C2×F8, C22×F8, D4×F8

Permutation representations of D4×F8
On 28 points - transitive group 28T59
Generators in S28
(1 12 19 24)(2 13 20 25)(3 14 21 26)(4 8 15 27)(5 9 16 28)(6 10 17 22)(7 11 18 23)
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 15)(9 16)(10 17)(11 18)(12 19)(13 20)(14 21)
(1 19)(4 15)(6 17)(7 18)(8 27)(10 22)(11 23)(12 24)
(1 19)(2 20)(5 16)(7 18)(9 28)(11 23)(12 24)(13 25)
(1 19)(2 20)(3 21)(6 17)(10 22)(12 24)(13 25)(14 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)| (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,19)(4,15)(6,17)(7,18)(8,27)(10,22)(11,23)(12,24), (1,19)(2,20)(5,16)(7,18)(9,28)(11,23)(12,24)(13,25), (1,19)(2,20)(3,21)(6,17)(10,22)(12,24)(13,25)(14,26), (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( (1,12,19,24)(2,13,20,25)(3,14,21,26)(4,8,15,27)(5,9,16,28)(6,10,17,22)(7,11,18,23), (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,15)(9,16)(10,17)(11,18)(12,19)(13,20)(14,21), (1,19)(4,15)(6,17)(7,18)(8,27)(10,22)(11,23)(12,24), (1,19)(2,20)(5,16)(7,18)(9,28)(11,23)(12,24)(13,25), (1,19)(2,20)(3,21)(6,17)(10,22)(12,24)(13,25)(14,26), (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([[(1,12,19,24),(2,13,20,25),(3,14,21,26),(4,8,15,27),(5,9,16,28),(6,10,17,22),(7,11,18,23)], [(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,15),(9,16),(10,17),(11,18),(12,19),(13,20),(14,21)], [(1,19),(4,15),(6,17),(7,18),(8,27),(10,22),(11,23),(12,24)], [(1,19),(2,20),(5,16),(7,18),(9,28),(11,23),(12,24),(13,25)], [(1,19),(2,20),(3,21),(6,17),(10,22),(12,24),(13,25),(14,26)], [(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(28,59);

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 7A ··· 7F 14A ··· 14F 14G ··· 14R 28A ··· 28F order 1 2 2 2 2 2 2 2 4 4 7 ··· 7 14 ··· 14 14 ··· 14 28 ··· 28 size 1 1 2 2 7 7 14 14 2 14 8 ··· 8 8 ··· 8 16 ··· 16 16 ··· 16

40 irreducible representations

 dim 1 1 1 1 1 1 14 2 2 7 7 7 type + + + + + + + + image C1 C2 C2 C7 C14 C14 D4×F8 D4 C7×D4 F8 C2×F8 C2×F8 kernel D4×F8 C4×F8 C22×F8 D4×C23 C23×C4 C25 C1 F8 C23 D4 C4 C22 # reps 1 1 2 6 6 12 1 1 6 1 1 2

Matrix representation of D4×F8 in GL9(ℤ)

 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 -1
,
 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1 0 0 0 1 0 0 1 0 -1 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 0 0 -1 1 0 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 -1 0 1 0 0 1 0 0 0 -1 0 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 -1 0 0 0 0 0 1 0 0 -1 0 0
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -1 0 0 0 0 0 0 1 0 -1 0 0 0 0 0 1 0 0 -1 0 0 0 0 1 0 0 0 -1 0 0 0 1 0 0 0 0 -1 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 -1
,
 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0

G:=sub<GL(9,Integers())| [0,-1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1],[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0] >;

D4×F8 in GAP, Magma, Sage, TeX

D_4\times F_8
% in TeX

G:=Group("D4xF8");
// GroupNames label

G:=SmallGroup(448,1363);
// by ID

G=gap.SmallGroup(448,1363);
# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,2,2,421,998,2371,3450]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=c^2=d^2=e^2=f^7=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,f*c*f^-1=e*d=d*e,f*d*f^-1=c,f*e*f^-1=d>;
// generators/relations

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