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G = C4:F7order 168 = 23·3·7

The semidirect product of C4 and F7 acting via F7/C7:C3=C2

metacyclic, supersoluble, monomial

Aliases: C4:F7, D28:C3, C28:1C6, D14:1C6, C7:C3:1D4, C7:1(C3xD4), (C2xF7):1C2, C2.4(C2xF7), C14.3(C2xC6), (C4xC7:C3):1C2, (C2xC7:C3).3C22, SmallGroup(168,9)

Series: Derived Chief Lower central Upper central

C1C14 — C4:F7
C1C7C14C2xC7:C3C2xF7 — C4:F7
C7C14 — C4:F7
C1C2C4

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

Subgroups: 142 in 32 conjugacy classes, 15 normal (11 characteristic)
Quotients: C1, C2, C3, C22, C6, D4, C2xC6, C3xD4, F7, C2xF7, C4:F7
14C2
14C2
7C3
7C22
7C22
7C6
14C6
14C6
2D7
2D7
7D4
7C2xC6
7C2xC6
7C12
2F7
2F7
7C3xD4

Character table of C4:F7

 class 12A2B2C3A3B46A6B6C6D6E6F712A12B1428A28B
 size 111414772771414141461414666
ρ11111111111111111111    trivial
ρ211-1111-111-11-111-1-11-1-1    linear of order 2
ρ3111-111-1111-11-11-1-11-1-1    linear of order 2
ρ411-1-111111-1-1-1-1111111    linear of order 2
ρ51111ζ3ζ321ζ3ζ32ζ3ζ32ζ32ζ31ζ3ζ32111    linear of order 3
ρ611-1-1ζ3ζ321ζ3ζ32ζ65ζ6ζ6ζ651ζ3ζ32111    linear of order 6
ρ7111-1ζ3ζ32-1ζ3ζ32ζ3ζ6ζ32ζ651ζ65ζ61-1-1    linear of order 6
ρ8111-1ζ32ζ3-1ζ32ζ3ζ32ζ65ζ3ζ61ζ6ζ651-1-1    linear of order 6
ρ911-11ζ3ζ32-1ζ3ζ32ζ65ζ32ζ6ζ31ζ65ζ61-1-1    linear of order 6
ρ1011-1-1ζ32ζ31ζ32ζ3ζ6ζ65ζ65ζ61ζ32ζ3111    linear of order 6
ρ1111-11ζ32ζ3-1ζ32ζ3ζ6ζ3ζ65ζ321ζ6ζ651-1-1    linear of order 6
ρ121111ζ32ζ31ζ32ζ3ζ32ζ3ζ3ζ321ζ32ζ3111    linear of order 3
ρ132-200220-2-20000200-200    orthogonal lifted from D4
ρ142-200-1--3-1+-301+-31--30000200-200    complex lifted from C3xD4
ρ152-200-1+-3-1--301--31+-30000200-200    complex lifted from C3xD4
ρ166600006000000-100-1-1-1    orthogonal lifted from F7
ρ17660000-6000000-100-111    orthogonal lifted from C2xF7
ρ186-600000000000-10017-7    orthogonal faithful
ρ196-600000000000-1001-77    orthogonal faithful

Permutation representations of C4:F7
On 28 points - transitive group 28T23
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)
(1 15)(2 18 3 21 5 20)(4 17 7 19 6 16)(8 22)(9 25 10 28 12 27)(11 24 14 26 13 23)

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

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

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

G:=TransitiveGroup(28,23);

C4:F7 is a maximal subgroup of   C56:C6  D56:C3  D4:F7  Q8:2F7  D28:6C6  D4xF7  Q8:3F7
C4:F7 is a maximal quotient of   C56:C6  D56:C3  C8.F7  C28:C12  D14:C12

Matrix representation of C4:F7 in GL6(F3)

202110
102212
101010
201210
002210
212100
,
020222
010110
110100
120010
102210
220020
,
110200
021100
000110
020001
020200
010000

G:=sub<GL(6,GF(3))| [2,1,1,2,0,2,0,0,0,0,0,1,2,2,1,1,2,2,1,2,0,2,2,1,1,1,1,1,1,0,0,2,0,0,0,0],[0,0,1,1,1,2,2,1,1,2,0,2,0,0,0,0,2,0,2,1,1,0,2,0,2,1,0,1,1,2,2,0,0,0,0,0],[1,0,0,0,0,0,1,2,0,2,2,1,0,1,0,0,0,0,2,1,1,0,2,0,0,0,1,0,0,0,0,0,0,1,0,0] >;

C4:F7 in GAP, Magma, Sage, TeX

C_4\rtimes F_7
% in TeX

G:=Group("C4:F7");
// GroupNames label

G:=SmallGroup(168,9);
// by ID

G=gap.SmallGroup(168,9);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,141,66,3604,614]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=c^6=1,a*b=b*a,c*a*c^-1=a^-1,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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