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G = C26⋊C7  order 448 = 26·7

2nd semidirect product of C26 and C7 acting faithfully

metabelian, soluble, monomial, A-group

Aliases: C26⋊2C7, C23⋊1F8, SmallGroup(448,1393)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C26 — C26⋊C7
Chief series
C1 — C23 — C26 — C26⋊C7
Lower central
C26 — C26⋊C7
Upper central
C1

Generators and relations for C26⋊C7
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g7=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, gag-1=cb=bc, bd=db, be=eb, bf=fb, gbg-1=a, cd=dc, ce=ec, cf=fc, gcg-1=b, de=ed, df=fd, gdg-1=fe=ef, geg-1=d, gfg-1=e >

Subgroups: 2962 in 424 conjugacy classes, 12 normal (3 characteristic)
C1, C2, C22, C7, C23, C23, C24, C25, F8, C26, C26⋊C7
Quotients: C1, C7, F8, C26⋊C7

Character table of C26⋊C7

 class 12A2B2C2D2E2F2G2H2I7A7B7C7D7E7F
 size 1777777777646464646464
ρ11111111111111111    trivial
ρ21111111111ζ73ζ76ζ72ζ75ζ7ζ74    linear of order 7
ρ31111111111ζ75ζ73ζ7ζ76ζ74ζ72    linear of order 7
ρ41111111111ζ72ζ74ζ76ζ7ζ73ζ75    linear of order 7
ρ51111111111ζ74ζ7ζ75ζ72ζ76ζ73    linear of order 7
ρ61111111111ζ7ζ72ζ73ζ74ζ75ζ76    linear of order 7
ρ71111111111ζ76ζ75ζ74ζ73ζ72ζ7    linear of order 7
ρ87-1-1-17-1-1-1-1-1000000    orthogonal lifted from F8
ρ97-1-1-1-1-1-17-1-1000000    orthogonal lifted from F8
ρ107-1-1-1-1-17-1-1-1000000    orthogonal lifted from F8
ρ117-1-17-1-1-1-1-1-1000000    orthogonal lifted from F8
ρ127-1-1-1-17-1-1-1-1000000    orthogonal lifted from F8
ρ137-1-1-1-1-1-1-1-17000000    orthogonal lifted from F8
ρ147-17-1-1-1-1-1-1-1000000    orthogonal lifted from F8
ρ157-1-1-1-1-1-1-17-1000000    orthogonal lifted from F8
ρ1677-1-1-1-1-1-1-1-1000000    orthogonal lifted from F8

Permutation representations of C26⋊C7
►On 28 points - transitive group 28T60
Generators in S28
(1 20)(2 25)(4 11)(5 28)(6 13)(7 19)(8 24)(9 21)(12 17)(14 23)(16 27)(18 22)

(1 20)(2 21)(3 26)(5 12)(6 22)(7 14)(8 24)(9 25)(10 15)(13 18)(17 28)(19 23)

(1 8)(2 21)(3 15)(4 27)(6 13)(7 23)(9 25)(10 26)(11 16)(14 19)(18 22)(20 24)

(1 20)(4 16)(6 18)(7 19)(8 24)(11 27)(13 22)(14 23)

(1 20)(2 21)(5 17)(7 19)(8 24)(9 25)(12 28)(14 23)

(1 20)(2 21)(3 15)(6 18)(8 24)(9 25)(10 26)(13 22)

(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)| (1,20)(2,25)(4,11)(5,28)(6,13)(7,19)(8,24)(9,21)(12,17)(14,23)(16,27)(18,22), (1,20)(2,21)(3,26)(5,12)(6,22)(7,14)(8,24)(9,25)(10,15)(13,18)(17,28)(19,23), (1,8)(2,21)(3,15)(4,27)(6,13)(7,23)(9,25)(10,26)(11,16)(14,19)(18,22)(20,24), (1,20)(4,16)(6,18)(7,19)(8,24)(11,27)(13,22)(14,23), (1,20)(2,21)(5,17)(7,19)(8,24)(9,25)(12,28)(14,23), (1,20)(2,21)(3,15)(6,18)(8,24)(9,25)(10,26)(13,22), (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( (1,20)(2,25)(4,11)(5,28)(6,13)(7,19)(8,24)(9,21)(12,17)(14,23)(16,27)(18,22), (1,20)(2,21)(3,26)(5,12)(6,22)(7,14)(8,24)(9,25)(10,15)(13,18)(17,28)(19,23), (1,8)(2,21)(3,15)(4,27)(6,13)(7,23)(9,25)(10,26)(11,16)(14,19)(18,22)(20,24), (1,20)(4,16)(6,18)(7,19)(8,24)(11,27)(13,22)(14,23), (1,20)(2,21)(5,17)(7,19)(8,24)(9,25)(12,28)(14,23), (1,20)(2,21)(3,15)(6,18)(8,24)(9,25)(10,26)(13,22), (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([[(1,20),(2,25),(4,11),(5,28),(6,13),(7,19),(8,24),(9,21),(12,17),(14,23),(16,27),(18,22)], [(1,20),(2,21),(3,26),(5,12),(6,22),(7,14),(8,24),(9,25),(10,15),(13,18),(17,28),(19,23)], [(1,8),(2,21),(3,15),(4,27),(6,13),(7,23),(9,25),(10,26),(11,16),(14,19),(18,22),(20,24)], [(1,20),(4,16),(6,18),(7,19),(8,24),(11,27),(13,22),(14,23)], [(1,20),(2,21),(5,17),(7,19),(8,24),(9,25),(12,28),(14,23)], [(1,20),(2,21),(3,15),(6,18),(8,24),(9,25),(10,26),(13,22)], [(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(28,60);
 

Matrix representation of C26⋊C7 ►in GL14(𝔽29)

280000000000000
028000000000000
002800000000000
00010000000000
000828000000000
8282800100000000
7282100010000000
00000001000000
00000000100000
000000000280000
000000000028000
00000002715002800
0000000002715010
0000000152000028
,
280000000000000
028000000000000
00100000000000
000280000000000
170211000000000
8280220100000000
008000280000000
00000001000000
000000002800000
000000000280000
000000000028000
00000000142713100
0000000130000280
0000000027132001
,
280000000000000
01000000000000
002800000000000
00010000000000
102801000000000
010702800000000
0102800280000000
000000028000000
000000002800000
000000000280000
00000000001000
0000000000162800
00000001627270010
00000001427130001
,
280000000000000
028000000000000
002800000000000
00010000000000
000828000000000
8282800100000000
7282100010000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
280000000000000
028000000000000
00100000000000
000280000000000
170211000000000
8280220100000000
008000280000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
280000000000000
01000000000000
002800000000000
00010000000000
102801000000000
010702800000000
0102800280000000
00000001000000
00000000100000
00000000010000
00000000001000
00000000000100
00000000000010
00000000000001
,
01000000000000
00100000000000
00010000000000
28221827000000000
000021100000000
000022010000000
00001000000000
00000000100000
00000000010000
00000000001000
00000002822182800
000000000002110
000000000002201
00000000000100

G:=sub<GL(14,GF(29))| [28,0,0,0,0,8,7,0,0,0,0,0,0,0,0,28,0,0,0,28,28,0,0,0,0,0,0,0,0,0,28,0,0,28,21,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,27,0,15,0,0,0,0,0,0,0,0,1,0,0,15,0,2,0,0,0,0,0,0,0,0,0,28,0,0,27,0,0,0,0,0,0,0,0,0,0,0,28,0,15,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28],[28,0,0,0,1,8,0,0,0,0,0,0,0,0,0,28,0,0,7,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,28,21,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,13,0,0,0,0,0,0,0,0,0,28,0,0,14,0,27,0,0,0,0,0,0,0,0,0,28,0,27,0,13,0,0,0,0,0,0,0,0,0,0,28,13,0,2,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,28,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,16,14,0,0,0,0,0,0,0,0,28,0,0,0,27,27,0,0,0,0,0,0,0,0,0,28,0,0,27,13,0,0,0,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,0,8,7,0,0,0,0,0,0,0,0,28,0,0,0,28,28,0,0,0,0,0,0,0,0,0,28,0,0,28,21,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,8,0,0,0,0,0,0,0,0,0,28,0,0,7,28,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,0,0,0,0,0,0,0,0,28,21,22,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[28,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,1,0,0,0,0,0,0,0,0,0,28,0,28,0,0,0,0,0,0,0,0,0,0,0,0,1,0,7,28,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1],[0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,27,21,22,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,0,0,1,0,0,22,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,8,0,0,0,0,0,0,0,0,0,0,0,0,0,28,21,22,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0] >;
 

C26⋊C7 in GAP, Magma, Sage, TeX 

C_2^6\rtimes C_7
 
% in TeX
 
G:=Group("C2^6:C7");
 
// GroupNames label
 
G:=SmallGroup(448,1393);
 
// by ID
 
G=gap.SmallGroup(448,1393);
 
# by ID
 
G:=PCGroup([7,-7,-2,2,2,-2,2,2,197,590,983,3924,9413,13726]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=g^7=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,g*a*g^-1=c*b=b*c,b*d=d*b,b*e=e*b,b*f=f*b,g*b*g^-1=a,c*d=d*c,c*e=e*c,c*f=f*c,g*c*g^-1=b,d*e=e*d,d*f=f*d,g*d*g^-1=f*e=e*f,g*e*g^-1=d,g*f*g^-1=e>;
 
// generators/relations
 

Export

Character table of C26⋊C7 in TeX

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