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G = C4.F7order 168 = 23·3·7

The non-split extension by C4 of F7 acting via F7/C7⋊C3=C2

metacyclic, supersoluble, monomial

Aliases: C4.F7, Dic14⋊C3, C28.1C6, Dic7.C6, C7⋊C3⋊Q8, C7⋊(C3×Q8), C7⋊C12.C2, C2.3(C2×F7), C14.1(C2×C6), (C4×C7⋊C3).1C2, (C2×C7⋊C3).1C22, SmallGroup(168,7)

Series: Derived Chief Lower central Upper central

C1C14 — C4.F7
C1C7C14C2×C7⋊C3C7⋊C12 — C4.F7
C7C14 — C4.F7
C1C2C4

Generators and relations for C4.F7
 G = < a,b,c | a4=b7=1, c6=a2, ab=ba, cac-1=a-1, cbc-1=b5 >

7C3
7C4
7C4
7C6
7Q8
7C12
7C12
7C12
7C3×Q8

Character table of C4.F7

 class 123A3B4A4B4C6A6B712A12B12C12D12E12F1428A28B
 size 117721414776141414141414666
ρ11111111111111111111    trivial
ρ21111-1-11111-111-1-1-11-1-1    linear of order 2
ρ311111-1-1111-1-1-11-11111    linear of order 2
ρ41111-11-11111-1-1-11-11-1-1    linear of order 2
ρ511ζ32ζ3-11-1ζ32ζ31ζ32ζ65ζ6ζ6ζ3ζ651-1-1    linear of order 6
ρ611ζ3ζ32-11-1ζ3ζ321ζ3ζ6ζ65ζ65ζ32ζ61-1-1    linear of order 6
ρ711ζ32ζ3111ζ32ζ31ζ32ζ3ζ32ζ32ζ3ζ3111    linear of order 3
ρ811ζ3ζ32111ζ3ζ321ζ3ζ32ζ3ζ3ζ32ζ32111    linear of order 3
ρ911ζ3ζ32-1-11ζ3ζ321ζ65ζ32ζ3ζ65ζ6ζ61-1-1    linear of order 6
ρ1011ζ32ζ3-1-11ζ32ζ31ζ6ζ3ζ32ζ6ζ65ζ651-1-1    linear of order 6
ρ1111ζ32ζ31-1-1ζ32ζ31ζ6ζ65ζ6ζ32ζ65ζ3111    linear of order 6
ρ1211ζ3ζ321-1-1ζ3ζ321ζ65ζ6ζ65ζ3ζ6ζ32111    linear of order 6
ρ132-222000-2-22000000-200    symplectic lifted from Q8, Schur index 2
ρ142-2-1+-3-1--30001--31+-32000000-200    complex lifted from C3×Q8
ρ152-2-1--3-1+-30001+-31--32000000-200    complex lifted from C3×Q8
ρ166600-60000-1000000-111    orthogonal lifted from C2×F7
ρ17660060000-1000000-1-1-1    orthogonal lifted from F7
ρ186-60000000-10000001-77    symplectic faithful, Schur index 2
ρ196-60000000-100000017-7    symplectic faithful, Schur index 2

Smallest permutation representation of C4.F7
On 56 points
Generators in S56
(1 7 3 5)(2 6 4 8)(9 35 15 41)(10 42 16 36)(11 37 17 43)(12 44 18 38)(13 39 19 33)(14 34 20 40)(21 51 27 45)(22 46 28 52)(23 53 29 47)(24 48 30 54)(25 55 31 49)(26 50 32 56)
(1 22 30 41 26 33 37)(2 34 42 23 38 27 31)(3 28 24 35 32 39 43)(4 40 36 29 44 21 25)(5 52 48 15 56 19 11)(6 20 16 53 12 45 49)(7 46 54 9 50 13 17)(8 14 10 47 18 51 55)
(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 29 30 31 32)(33 34 35 36 37 38 39 40 41 42 43 44)(45 46 47 48 49 50 51 52 53 54 55 56)

G:=sub<Sym(56)| (1,7,3,5)(2,6,4,8)(9,35,15,41)(10,42,16,36)(11,37,17,43)(12,44,18,38)(13,39,19,33)(14,34,20,40)(21,51,27,45)(22,46,28,52)(23,53,29,47)(24,48,30,54)(25,55,31,49)(26,50,32,56), (1,22,30,41,26,33,37)(2,34,42,23,38,27,31)(3,28,24,35,32,39,43)(4,40,36,29,44,21,25)(5,52,48,15,56,19,11)(6,20,16,53,12,45,49)(7,46,54,9,50,13,17)(8,14,10,47,18,51,55), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56)>;

G:=Group( (1,7,3,5)(2,6,4,8)(9,35,15,41)(10,42,16,36)(11,37,17,43)(12,44,18,38)(13,39,19,33)(14,34,20,40)(21,51,27,45)(22,46,28,52)(23,53,29,47)(24,48,30,54)(25,55,31,49)(26,50,32,56), (1,22,30,41,26,33,37)(2,34,42,23,38,27,31)(3,28,24,35,32,39,43)(4,40,36,29,44,21,25)(5,52,48,15,56,19,11)(6,20,16,53,12,45,49)(7,46,54,9,50,13,17)(8,14,10,47,18,51,55), (1,2,3,4)(5,6,7,8)(9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40,41,42,43,44)(45,46,47,48,49,50,51,52,53,54,55,56) );

G=PermutationGroup([(1,7,3,5),(2,6,4,8),(9,35,15,41),(10,42,16,36),(11,37,17,43),(12,44,18,38),(13,39,19,33),(14,34,20,40),(21,51,27,45),(22,46,28,52),(23,53,29,47),(24,48,30,54),(25,55,31,49),(26,50,32,56)], [(1,22,30,41,26,33,37),(2,34,42,23,38,27,31),(3,28,24,35,32,39,43),(4,40,36,29,44,21,25),(5,52,48,15,56,19,11),(6,20,16,53,12,45,49),(7,46,54,9,50,13,17),(8,14,10,47,18,51,55)], [(1,2,3,4),(5,6,7,8),(9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40,41,42,43,44),(45,46,47,48,49,50,51,52,53,54,55,56)])

C4.F7 is a maximal subgroup of   C56⋊C6  C8.F7  D4.F7  Q8.2F7  D286C6  D42F7  Q8×F7
C4.F7 is a maximal quotient of   Dic7⋊C12  C28⋊C12

Matrix representation of C4.F7 in GL6(𝔽3)

200001
012002
222001
200122
200220
100001
,
020010
000200
022121
022210
022201
211201
,
010120
020120
010101
120121
100011
001222

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

C4.F7 in GAP, Magma, Sage, TeX

C_4.F_7
% in TeX

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

G:=SmallGroup(168,7);
// by ID

G=gap.SmallGroup(168,7);
# by ID

G:=PCGroup([5,-2,-2,-3,-2,-7,60,141,66,3604,614]);
// Polycyclic

G:=Group<a,b,c|a^4=b^7=1,c^6=a^2,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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