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G = D4xF8  order 448 = 26·7

Direct product of D4 and F8

direct product, metabelian, soluble, monomial

Aliases: D4xF8, C25:C14, C4:(C2xF8), (C4xF8):C2, (D4xC23):C7, C22:(C2xF8), (C23xC4):C14, (C22xF8):C2, C23:2(C7xD4), C2.2(C22xF8), C24.2(C2xC14), (C2xF8).2C22, SmallGroup(448,1363)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C24 — D4xF8
Chief series
C1 — C23 — C24 — C2xF8 — C22xF8 — D4xF8
Lower central
C23 — C24 — D4xF8
Upper central
C1 — C2 — D4

Generators and relations for D4xF8
 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, C2xC4, D4, D4, C23, C23, C14, C22xC4, C2xD4, C24, C24, C28, C2xC14, C23xC4, C22xD4, C25, C7xD4, F8, D4xC23, C2xF8, C2xF8, C4xF8, C22xF8, D4xF8
Quotients: C1, C2, C22, C7, D4, C14, C2xC14, C7xD4, F8, C2xF8, C22xF8, D4xF8

Permutation representations of D4xF8
►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++++++++
imageC1C2C2C7C14C14D4xF8D4C7xD4F8C2xF8C2xF8
kernelD4xF8C4xF8C22xF8D4xC23C23xC4C25C1F8C23D4C4C22
# reps1126612116112

Matrix representation of D4xF8 ►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] >;
 

D4xF8 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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