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G = C4×F7  order 168 = 23·3·7

Direct product of C4 and F7

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C4×F7, D7⋊C12, C28⋊2C6, D14.C6, Dic7⋊2C6, (C4×D7)⋊C3, C7⋊C12⋊2C2, C7⋊1(C2×C12), (C2×F7).C2, C2.1(C2×F7), C14.2(C2×C6), C7⋊C3⋊1(C2×C4), (C4×C7⋊C3)⋊2C2, (C2×C7⋊C3).2C22, SmallGroup(168,8)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C7 — C4×F7
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7
Lower central
C7 — C4×F7
Upper central
C1 — C4

Generators and relations for C4×F7
 G = < a,b,c | a4=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

C1
C2
7C2
7C2
7C3
C7
C4
7C4
7C22
7C6
7C6
7C6
D7
C14
D7
C7⋊C3
7C2×C4
7C12
7C2×C6
7C12
D14
C28
Dic7
C2×C7⋊C3
F7
F7
7C2×C12
C4×D7
C2×F7
C7⋊C12
C4×C7⋊C3
C4×F7

Character table of C4×F7

 class 12A2B2C3A3B4A4B4C4D6A6B6C6D6E6F712A12B12C12D12E12F12G12H1428A28B
 size 1177771177777777677777777666
ρ11111111111111111111111111111    trivial
ρ211-1-11111-1-1-1-1-1-1111-1111-1-1-11111    linear of order 2
ρ311-1-111-1-111-1-1-1-11111-1-1-1111-11-1-1    linear of order 2
ρ4111111-1-1-1-11111111-1-1-1-1-1-1-1-11-1-1    linear of order 2
ρ51111ζ32ζ31111ζ32ζ32ζ3ζ3ζ3ζ321ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32111    linear of order 3
ρ611-1-1ζ32ζ311-1-1ζ6ζ6ζ65ζ65ζ3ζ321ζ65ζ32ζ3ζ3ζ6ζ6ζ65ζ32111    linear of order 6
ρ711-1-1ζ3ζ3211-1-1ζ65ζ65ζ6ζ6ζ32ζ31ζ6ζ3ζ32ζ32ζ65ζ65ζ6ζ3111    linear of order 6
ρ81111ζ32ζ3-1-1-1-1ζ32ζ32ζ3ζ3ζ3ζ321ζ65ζ6ζ65ζ65ζ6ζ6ζ65ζ61-1-1    linear of order 6
ρ911-1-1ζ3ζ32-1-111ζ65ζ65ζ6ζ6ζ32ζ31ζ32ζ65ζ6ζ6ζ3ζ3ζ32ζ651-1-1    linear of order 6
ρ1011-1-1ζ32ζ3-1-111ζ6ζ6ζ65ζ65ζ3ζ321ζ3ζ6ζ65ζ65ζ32ζ32ζ3ζ61-1-1    linear of order 6
ρ111111ζ3ζ321111ζ3ζ3ζ32ζ32ζ32ζ31ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3111    linear of order 3
ρ121111ζ3ζ32-1-1-1-1ζ3ζ3ζ32ζ32ζ32ζ31ζ6ζ65ζ6ζ6ζ65ζ65ζ6ζ651-1-1    linear of order 6
ρ131-1-1111-ii-ii-111-1-1-11-i-ii-ii-iii-1-ii    linear of order 4
ρ141-11-111-iii-i1-1-11-1-11i-ii-i-ii-ii-1-ii    linear of order 4
ρ151-1-1111i-ii-i-111-1-1-11ii-ii-ii-i-i-1i-i    linear of order 4
ρ161-11-111i-i-ii1-1-11-1-11-ii-iii-ii-i-1i-i    linear of order 4
ρ171-1-11ζ32ζ3-ii-iiζ6ζ32ζ3ζ65ζ65ζ61ζ43ζ3ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ4ζ32-1-ii    linear of order 12
ρ181-11-1ζ32ζ3-iii-iζ32ζ6ζ65ζ3ζ65ζ61ζ4ζ3ζ43ζ32ζ4ζ3ζ43ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ32-1-ii    linear of order 12
ρ191-1-11ζ3ζ32-ii-iiζ65ζ3ζ32ζ6ζ6ζ651ζ43ζ32ζ43ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ4ζ3-1-ii    linear of order 12
ρ201-1-11ζ3ζ32i-ii-iζ65ζ3ζ32ζ6ζ6ζ651ζ4ζ32ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ43ζ3-1i-i    linear of order 12
ρ211-11-1ζ3ζ32i-i-iiζ3ζ65ζ6ζ32ζ6ζ651ζ43ζ32ζ4ζ3ζ43ζ32ζ4ζ32ζ4ζ3ζ43ζ3ζ4ζ32ζ43ζ3-1i-i    linear of order 12
ρ221-1-11ζ32ζ3i-ii-iζ6ζ32ζ3ζ65ζ65ζ61ζ4ζ3ζ4ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ32ζ43ζ3ζ43ζ32-1i-i    linear of order 12
ρ231-11-1ζ3ζ32-iii-iζ3ζ65ζ6ζ32ζ6ζ651ζ4ζ32ζ43ζ3ζ4ζ32ζ43ζ32ζ43ζ3ζ4ζ3ζ43ζ32ζ4ζ3-1-ii    linear of order 12
ρ241-11-1ζ32ζ3i-i-iiζ32ζ6ζ65ζ3ζ65ζ61ζ43ζ3ζ4ζ32ζ43ζ3ζ4ζ3ζ4ζ32ζ43ζ32ζ4ζ3ζ43ζ32-1i-i    linear of order 12
ρ25660000-6-600000000-100000000-111    orthogonal lifted from C2×F7
ρ266600006600000000-100000000-1-1-1    orthogonal lifted from F7
ρ276-600006i-6i00000000-1000000001-ii    complex faithful
ρ286-60000-6i6i00000000-1000000001i-i    complex faithful

Permutation representations of C4×F7
►On 28 points - transitive group 28T26
Generators in S28
(1 22 8 15)(2 23 9 16)(3 24 10 17)(4 25 11 18)(5 26 12 19)(6 27 13 20)(7 28 14 21)

(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)

(2 4 3 7 5 6)(9 11 10 14 12 13)(16 18 17 21 19 20)(23 25 24 28 26 27)
 
G:=sub<Sym(28)| (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27)>;
 
G:=Group( (1,22,8,15)(2,23,9,16)(3,24,10,17)(4,25,11,18)(5,26,12,19)(6,27,13,20)(7,28,14,21), (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), (2,4,3,7,5,6)(9,11,10,14,12,13)(16,18,17,21,19,20)(23,25,24,28,26,27) );
 
G=PermutationGroup([[(1,22,8,15),(2,23,9,16),(3,24,10,17),(4,25,11,18),(5,26,12,19),(6,27,13,20),(7,28,14,21)], [(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)], [(2,4,3,7,5,6),(9,11,10,14,12,13),(16,18,17,21,19,20),(23,25,24,28,26,27)]])
 
G:=TransitiveGroup(28,26);
 

C4×F7 is a maximal subgroup of   C8⋊F7  D28⋊6C6  D4⋊2F7  Q8⋊3F7
C4×F7 is a maximal quotient of   C8⋊F7  Dic7⋊C12  D14⋊C12

Matrix representation of C4×F7 ►in GL6(𝔽337)

18900000
01890000
00189000
00018900
00001890
00000189
,
336336336336336336
100000
010000
001000
000100
000010
,
33600000
00000336
00033600
03360000
111111
00003360

G:=sub<GL(6,GF(337))| [189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189,0,0,0,0,0,0,189],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[336,0,0,0,1,0,0,0,0,336,1,0,0,0,0,0,1,0,0,0,336,0,1,0,0,0,0,0,1,336,0,336,0,0,1,0] >;
 

C4×F7 in GAP, Magma, Sage, TeX 

C_4\times F_7
 
% in TeX
 
G:=Group("C4xF7");
 
// GroupNames label
 
G:=SmallGroup(168,8);
 
// by ID
 
G=gap.SmallGroup(168,8);
 
# by ID
 
G:=PCGroup([5,-2,-2,-3,-2,-7,66,3604,614]);
 
// Polycyclic
 
G:=Group<a,b,c|a^4=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
 
// generators/relations
 

Export

Subgroup lattice of C4×F7 in TeX
Character table of C4×F7 in TeX

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