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G = D4×F8  order 448 = 26·7

Direct product of D4 and F8

direct product, metabelian, soluble, monomial

Aliases: D4×F8, C25⋊C14, C4⋊(C2×F8), (C4×F8)⋊C2, (D4×C23)⋊C7, C22⋊(C2×F8), (C23×C4)⋊C14, (C22×F8)⋊C2, C23⋊2(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 2A2B2C2D2E2F2G4A4B7A···7F14A···14F14G···14R28A···28F
order12222222447···714···1414···1428···28
size11227714142148···88···816···1616···16

40 irreducible representations

dim1111111422777
type++++++++
imageC1C2C2C7C14C14D4×F8D4C7×D4F8C2×F8C2×F8
kernelD4×F8C4×F8C22×F8D4×C23C23×C4C25C1F8C23D4C4C22
# reps1126612116112

Matrix representation of D4×F8 ►in GL9(ℤ)

010000000
-100000000
00-1000000
000-100000
0000-10000
00000-1000
000000-100
0000000-10
00000000-1
,
010000000
100000000
001000000
000100000
000010000
000001000
000000100
000000010
000000001
,
100000000
010000000
0000-10100
0000-10010
0000-10000
0000-10001
0010-10000
0001-10000
0000-11000
,
100000000
010000000
000010-100
000000-101
001000-100
000000-110
000000-100
000001-100
000100-100
,
100000000
010000000
00000001-1
00000010-1
00000100-1
00001000-1
00010000-1
00100000-1
00000000-1
,
100000000
010000000
000000010
001000000
000100000
000000001
000010000
000001000
000000100

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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