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G = C4×F8  order 224 = 25·7

Direct product of C4 and F8

direct product, metabelian, soluble, monomial, A-group

Aliases: C4×F8, C23⋊C28, C24.C14, (C23×C4)⋊C7, C2.(C2×F8), (C2×F8).C2, SmallGroup(224,173)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C23 — C4×F8
Chief series
C1 — C23 — C24 — C2×F8 — C4×F8
Lower central
C23 — C4×F8
Upper central
C1 — C4

Generators and relations for C4×F8
 G = < a,b,c,d,e | a4=b2=c2=d2=e7=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=dc=cd, ece-1=b, ede-1=c >

C1
C2
7C2
7C2
8C7
C4
7C22
7C22
7C4
7C22
7C22
7C22
8C14
C23
7C23
7C2×C4
7C23
7C2×C4
7C2×C4
7C2×C4
8C28
C24
7C22×C4
7C22×C4
F8
C23×C4
C2×F8
C4×F8

Permutation representations of C4×F8
►On 28 points - transitive group 28T37
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 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), (1,19)(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), (1,19)(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)], [(1,19),(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(28,37);
 

32 conjugacy classes

class 1 2A2B2C4A4B4C4D7A···7F14A···14F28A···28L
order122244447···714···1428···28
size117711778···88···88···8

32 irreducible representations

dim111111777
type++++
imageC1C2C4C7C14C28F8C2×F8C4×F8
kernelC4×F8C2×F8F8C23×C4C24C23C4C2C1
# reps1126612112

Matrix representation of C4×F8 ►in GL7(𝔽29)

17000000
01700000
00170000
00017000
00001700
00000170
00000017
,
28000000
02800000
00280000
0001000
00002800
0000010
0000001
,
28000000
02800000
0010000
00028000
0000100
0000010
00000028
,
28000000
0100000
00280000
0001000
0000100
00000280
00000028
,
0100000
0010000
0001000
0000100
0000010
0000001
1000000

G:=sub<GL(7,GF(29))| [17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17,0,0,0,0,0,0,0,17],[28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[28,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28],[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] >;
 

C4×F8 in GAP, Magma, Sage, TeX 

C_4\times F_8
 
% in TeX
 
G:=Group("C4xF8");
 
// GroupNames label
 
G:=SmallGroup(224,173);
 
// by ID
 
G=gap.SmallGroup(224,173);
 
# by ID
 
G:=PCGroup([6,-2,-7,-2,-2,2,2,84,681,1690,2531]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^2=c^2=d^2=e^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=d*c=c*d,e*c*e^-1=b,e*d*e^-1=c>;
 
// generators/relations
 

Export

Subgroup lattice of C4×F8 in TeX

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