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## G = C4×F8order 224 = 25·7

### Direct product of C4 and F8

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 >

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 2A 2B 2C 4A 4B 4C 4D 7A ··· 7F 14A ··· 14F 28A ··· 28L order 1 2 2 2 4 4 4 4 7 ··· 7 14 ··· 14 28 ··· 28 size 1 1 7 7 1 1 7 7 8 ··· 8 8 ··· 8 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 7 7 7 type + + + + image C1 C2 C4 C7 C14 C28 F8 C2×F8 C4×F8 kernel C4×F8 C2×F8 F8 C23×C4 C24 C23 C4 C2 C1 # reps 1 1 2 6 6 12 1 1 2

Matrix representation of C4×F8 in GL7(𝔽29)

 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 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

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

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