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G = C22×F8  order 224 = 25·7

Direct product of C22 and F8

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

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 2A2B2C2D2E2F2G7A···7F14A···14R
order122222227···714···14
size111177778···88···8

32 irreducible representations

dim111177
type++++
imageC1C2C7C14F8C2×F8
kernelC22×F8C2×F8C25C24C22C2
# reps1361813

Matrix representation of C22×F8 ►in GL8(𝔽29)

280000000
028000000
002800000
000280000
000028000
000002800
000000280
000000028
,
10000000
028000000
002800000
000280000
000028000
000002800
000000280
000000028
,
10000000
028000000
002800000
00010000
000128000
062300100
0202400010
0002400028
,
10000000
028000000
00100000
000280000
060281000
060160100
005000280
005000028
,
10000000
01000000
002800000
00010000
001701000
02301302800
0901900280
0202400028
,
230000000
00100000
00010000
02312127000
000028100
000016010
000010001
00005000

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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