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## G = C22×F8order 224 = 25·7

### Direct product of C22 and F8

Aliases: C22×F8, C25⋊C7, C24⋊C14, C23⋊(C2×C14), SmallGroup(224,195)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C22×F8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e2=f7=1, ab=ba, 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: 419 in 72 conjugacy classes, 15 normal (6 characteristic)
C1, C2, C2, C22, C22, C7, C23, C23, C14, C24, C24, C2×C14, C25, F8, C2×F8, C22×F8
Quotients: C1, C2, C22, C7, C14, C2×C14, F8, C2×F8, C22×F8

Permutation representations of C22×F8
On 28 points - transitive group 28T38
Generators in S28
(1 24)(2 25)(3 26)(4 27)(5 28)(6 22)(7 23)(8 17)(9 18)(10 19)(11 20)(12 21)(13 15)(14 16)
(1 8)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)
(1 8)(2 25)(3 19)(4 20)(5 12)(7 23)(9 18)(10 26)(11 27)(14 16)(17 24)(21 28)
(1 24)(2 9)(3 26)(4 20)(5 21)(6 13)(8 17)(10 19)(11 27)(12 28)(15 22)(18 25)
(2 25)(3 10)(4 27)(5 21)(6 15)(7 14)(9 18)(11 20)(12 28)(13 22)(16 23)(19 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)| (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,25)(3,19)(4,20)(5,12)(7,23)(9,18)(10,26)(11,27)(14,16)(17,24)(21,28), (1,24)(2,9)(3,26)(4,20)(5,21)(6,13)(8,17)(10,19)(11,27)(12,28)(15,22)(18,25), (2,25)(3,10)(4,27)(5,21)(6,15)(7,14)(9,18)(11,20)(12,28)(13,22)(16,23)(19,26), (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( (1,24)(2,25)(3,26)(4,27)(5,28)(6,22)(7,23)(8,17)(9,18)(10,19)(11,20)(12,21)(13,15)(14,16), (1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28), (1,8)(2,25)(3,19)(4,20)(5,12)(7,23)(9,18)(10,26)(11,27)(14,16)(17,24)(21,28), (1,24)(2,9)(3,26)(4,20)(5,21)(6,13)(8,17)(10,19)(11,27)(12,28)(15,22)(18,25), (2,25)(3,10)(4,27)(5,21)(6,15)(7,14)(9,18)(11,20)(12,28)(13,22)(16,23)(19,26), (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([[(1,24),(2,25),(3,26),(4,27),(5,28),(6,22),(7,23),(8,17),(9,18),(10,19),(11,20),(12,21),(13,15),(14,16)], [(1,8),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28)], [(1,8),(2,25),(3,19),(4,20),(5,12),(7,23),(9,18),(10,26),(11,27),(14,16),(17,24),(21,28)], [(1,24),(2,9),(3,26),(4,20),(5,21),(6,13),(8,17),(10,19),(11,27),(12,28),(15,22),(18,25)], [(2,25),(3,10),(4,27),(5,21),(6,15),(7,14),(9,18),(11,20),(12,28),(13,22),(16,23),(19,26)], [(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(28,38);

On 28 points - transitive group 28T39
Generators in S28
(1 12)(2 13)(3 14)(4 8)(5 9)(6 10)(7 11)(15 27)(16 28)(17 22)(18 23)(19 24)(20 25)(21 26)
(1 19)(2 20)(3 21)(4 15)(5 16)(6 17)(7 18)(8 27)(9 28)(10 22)(11 23)(12 24)(13 25)(14 26)
(2 20)(3 21)(4 15)(7 18)(8 27)(11 23)(13 25)(14 26)
(1 19)(3 21)(4 15)(5 16)(8 27)(9 28)(12 24)(14 26)
(2 20)(4 15)(5 16)(6 17)(8 27)(9 28)(10 22)(13 25)
(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)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26), (2,20)(3,21)(4,15)(7,18)(8,27)(11,23)(13,25)(14,26), (1,19)(3,21)(4,15)(5,16)(8,27)(9,28)(12,24)(14,26), (2,20)(4,15)(5,16)(6,17)(8,27)(9,28)(10,22)(13,25), (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)(2,13)(3,14)(4,8)(5,9)(6,10)(7,11)(15,27)(16,28)(17,22)(18,23)(19,24)(20,25)(21,26), (1,19)(2,20)(3,21)(4,15)(5,16)(6,17)(7,18)(8,27)(9,28)(10,22)(11,23)(12,24)(13,25)(14,26), (2,20)(3,21)(4,15)(7,18)(8,27)(11,23)(13,25)(14,26), (1,19)(3,21)(4,15)(5,16)(8,27)(9,28)(12,24)(14,26), (2,20)(4,15)(5,16)(6,17)(8,27)(9,28)(10,22)(13,25), (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),(2,13),(3,14),(4,8),(5,9),(6,10),(7,11),(15,27),(16,28),(17,22),(18,23),(19,24),(20,25),(21,26)], [(1,19),(2,20),(3,21),(4,15),(5,16),(6,17),(7,18),(8,27),(9,28),(10,22),(11,23),(12,24),(13,25),(14,26)], [(2,20),(3,21),(4,15),(7,18),(8,27),(11,23),(13,25),(14,26)], [(1,19),(3,21),(4,15),(5,16),(8,27),(9,28),(12,24),(14,26)], [(2,20),(4,15),(5,16),(6,17),(8,27),(9,28),(10,22),(13,25)], [(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(28,39);

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 7 7 type + + + + image C1 C2 C7 C14 F8 C2×F8 kernel C22×F8 C2×F8 C25 C24 C22 C2 # reps 1 3 6 18 1 3

Matrix representation of C22×F8 in GL8(𝔽29)

 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 28 0 0 0 0 6 23 0 0 1 0 0 0 20 24 0 0 0 1 0 0 0 0 24 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 6 0 28 1 0 0 0 0 6 0 16 0 1 0 0 0 0 5 0 0 0 28 0 0 0 5 0 0 0 0 28
,
 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 28 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 17 0 1 0 0 0 0 23 0 13 0 28 0 0 0 9 0 19 0 0 28 0 0 2 0 24 0 0 0 28
,
 23 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 23 12 1 27 0 0 0 0 0 0 0 28 1 0 0 0 0 0 0 16 0 1 0 0 0 0 0 10 0 0 1 0 0 0 0 5 0 0 0

G:=sub<GL(8,GF(29))| [28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,0,6,20,0,0,0,28,0,0,23,24,0,0,0,0,1,1,0,0,24,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,28,0,0,6,6,0,0,0,0,1,0,0,0,5,5,0,0,0,28,28,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,23,9,2,0,0,28,0,17,0,0,0,0,0,0,1,0,13,19,24,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[23,0,0,0,0,0,0,0,0,0,0,23,0,0,0,0,0,1,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,27,28,16,10,5,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

C22×F8 in GAP, Magma, Sage, TeX

C_2^2\times F_8
% in TeX

G:=Group("C2^2xF8");
// GroupNames label

G:=SmallGroup(224,195);
// by ID

G=gap.SmallGroup(224,195);
# by ID

G:=PCGroup([6,-2,-2,-7,-2,2,2,351,856,1277]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^2=f^7=1,a*b=b*a,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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