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## G = C22×F7order 168 = 23·3·7

### Direct product of C22 and F7

Aliases: C22×F7, D143C6, C7⋊C3⋊C23, C14⋊(C2×C6), D7⋊(C2×C6), C7⋊(C22×C6), (C2×C14)⋊4C6, (C22×D7)⋊3C3, (C2×C7⋊C3)⋊C22, (C22×C7⋊C3)⋊2C2, SmallGroup(168,47)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C7 — C22×F7
 Chief series C1 — C7 — C7⋊C3 — F7 — C2×F7 — C22×F7
 Lower central C7 — C22×F7
 Upper central C1 — C22

Generators and relations for C22×F7
G = < a,b,c,d | a2=b2=c7=d6=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 226 in 64 conjugacy classes, 37 normal (8 characteristic)
C1, C2 [×3], C2 [×4], C3, C22, C22 [×6], C6 [×7], C7, C23, C2×C6 [×7], D7 [×4], C14 [×3], C7⋊C3, C22×C6, D14 [×6], C2×C14, F7 [×4], C2×C7⋊C3 [×3], C22×D7, C2×F7 [×6], C22×C7⋊C3, C22×F7
Quotients: C1, C2 [×7], C3, C22 [×7], C6 [×7], C23, C2×C6 [×7], C22×C6, F7, C2×F7 [×3], C22×F7

Character table of C22×F7

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 6A 6B 6C 6D 6E 6F 6G 6H 6I 6J 6K 6L 6M 6N 7 14A 14B 14C size 1 1 1 1 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 6 6 6 6 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 -1 1 -1 1 1 1 -1 -1 -1 1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 linear of order 2 ρ3 1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 -1 linear of order 2 ρ4 1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 linear of order 2 ρ6 1 1 1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 -1 1 -1 -1 1 -1 1 1 1 1 1 linear of order 2 ρ7 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 1 -1 linear of order 2 ρ8 1 1 -1 -1 -1 -1 1 1 1 1 -1 1 -1 -1 -1 -1 1 1 -1 1 1 -1 -1 1 1 1 -1 -1 linear of order 2 ρ9 1 -1 1 -1 -1 1 -1 1 ζ3 ζ32 ζ6 ζ6 ζ65 ζ3 ζ32 ζ3 ζ65 ζ6 ζ32 ζ3 ζ32 ζ6 ζ65 ζ65 1 -1 -1 1 linear of order 6 ρ10 1 1 1 1 -1 -1 -1 -1 ζ32 ζ3 ζ65 ζ3 ζ32 ζ6 ζ65 ζ32 ζ6 ζ65 ζ3 ζ6 ζ65 ζ3 ζ6 ζ32 1 1 1 1 linear of order 6 ρ11 1 1 1 1 -1 -1 -1 -1 ζ3 ζ32 ζ6 ζ32 ζ3 ζ65 ζ6 ζ3 ζ65 ζ6 ζ32 ζ65 ζ6 ζ32 ζ65 ζ3 1 1 1 1 linear of order 6 ρ12 1 -1 -1 1 -1 1 1 -1 ζ32 ζ3 ζ65 ζ65 ζ32 ζ32 ζ3 ζ6 ζ32 ζ3 ζ65 ζ6 ζ65 ζ3 ζ6 ζ6 1 -1 1 -1 linear of order 6 ρ13 1 1 -1 -1 -1 -1 1 1 ζ3 ζ32 ζ6 ζ32 ζ65 ζ65 ζ6 ζ65 ζ3 ζ32 ζ6 ζ3 ζ32 ζ6 ζ65 ζ3 1 1 -1 -1 linear of order 6 ρ14 1 -1 -1 1 1 -1 -1 1 ζ32 ζ3 ζ3 ζ65 ζ32 ζ6 ζ65 ζ6 ζ6 ζ65 ζ65 ζ32 ζ3 ζ3 ζ32 ζ6 1 -1 1 -1 linear of order 6 ρ15 1 1 1 1 1 1 1 1 ζ32 ζ3 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 1 1 1 1 linear of order 3 ρ16 1 -1 -1 1 -1 1 1 -1 ζ3 ζ32 ζ6 ζ6 ζ3 ζ3 ζ32 ζ65 ζ3 ζ32 ζ6 ζ65 ζ6 ζ32 ζ65 ζ65 1 -1 1 -1 linear of order 6 ρ17 1 1 -1 -1 -1 -1 1 1 ζ32 ζ3 ζ65 ζ3 ζ6 ζ6 ζ65 ζ6 ζ32 ζ3 ζ65 ζ32 ζ3 ζ65 ζ6 ζ32 1 1 -1 -1 linear of order 6 ρ18 1 1 1 1 1 1 1 1 ζ3 ζ32 ζ32 ζ32 ζ3 ζ3 ζ32 ζ3 ζ3 ζ32 ζ32 ζ3 ζ32 ζ32 ζ3 ζ3 1 1 1 1 linear of order 3 ρ19 1 -1 1 -1 -1 1 -1 1 ζ32 ζ3 ζ65 ζ65 ζ6 ζ32 ζ3 ζ32 ζ6 ζ65 ζ3 ζ32 ζ3 ζ65 ζ6 ζ6 1 -1 -1 1 linear of order 6 ρ20 1 -1 1 -1 1 -1 1 -1 ζ3 ζ32 ζ32 ζ6 ζ65 ζ65 ζ6 ζ3 ζ3 ζ32 ζ32 ζ65 ζ6 ζ6 ζ3 ζ65 1 -1 -1 1 linear of order 6 ρ21 1 -1 1 -1 1 -1 1 -1 ζ32 ζ3 ζ3 ζ65 ζ6 ζ6 ζ65 ζ32 ζ32 ζ3 ζ3 ζ6 ζ65 ζ65 ζ32 ζ6 1 -1 -1 1 linear of order 6 ρ22 1 -1 -1 1 1 -1 -1 1 ζ3 ζ32 ζ32 ζ6 ζ3 ζ65 ζ6 ζ65 ζ65 ζ6 ζ6 ζ3 ζ32 ζ32 ζ3 ζ65 1 -1 1 -1 linear of order 6 ρ23 1 1 -1 -1 1 1 -1 -1 ζ32 ζ3 ζ3 ζ3 ζ6 ζ32 ζ3 ζ6 ζ6 ζ65 ζ65 ζ6 ζ65 ζ65 ζ32 ζ32 1 1 -1 -1 linear of order 6 ρ24 1 1 -1 -1 1 1 -1 -1 ζ3 ζ32 ζ32 ζ32 ζ65 ζ3 ζ32 ζ65 ζ65 ζ6 ζ6 ζ65 ζ6 ζ6 ζ3 ζ3 1 1 -1 -1 linear of order 6 ρ25 6 -6 6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 1 -1 orthogonal lifted from C2×F7 ρ26 6 -6 -6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 -1 1 orthogonal lifted from C2×F7 ρ27 6 6 6 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 -1 -1 orthogonal lifted from F7 ρ28 6 6 -6 -6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 -1 1 1 orthogonal lifted from C2×F7

Permutation representations of C22×F7
On 28 points - transitive group 28T24
Generators in S28
(1 15)(2 16)(3 17)(4 18)(5 19)(6 20)(7 21)(8 22)(9 23)(10 24)(11 25)(12 26)(13 27)(14 28)
(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 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 22)(2 25 3 28 5 27)(4 24 7 26 6 23)(8 15)(9 18 10 21 12 20)(11 17 14 19 13 16)

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

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

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

G:=TransitiveGroup(28,24);

C22×F7 is a maximal subgroup of   D14⋊C12
C22×F7 is a maximal quotient of   D286C6  D42F7  Q83F7

Matrix representation of C22×F7 in GL7(𝔽43)

 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42
,
 1 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42 0 0 0 0 0 0 0 42
,
 1 0 0 0 0 0 0 0 0 0 0 0 0 42 0 1 0 0 0 0 42 0 0 1 0 0 0 42 0 0 0 1 0 0 42 0 0 0 0 1 0 42 0 0 0 0 0 1 42
,
 6 0 0 0 0 0 0 0 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42 0 0 0 0 0 0 0 0 0 0 0 42 0 0 0 0 42 0 0 0 0 42 0 0 0 0

G:=sub<GL(7,GF(43))| [42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42],[1,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,42],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,42,42,42,42,42,42],[6,0,0,0,0,0,0,0,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0,42,0,0,0,0,0,0,0,0,0,42,0,0] >;

C22×F7 in GAP, Magma, Sage, TeX

C_2^2\times F_7
% in TeX

G:=Group("C2^2xF7");
// GroupNames label

G:=SmallGroup(168,47);
// by ID

G=gap.SmallGroup(168,47);
# by ID

G:=PCGroup([5,-2,-2,-2,-3,-7,3604,319]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^7=d^6=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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