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G = C22×F7order 168 = 23·3·7

Direct product of C22 and F7

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

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

C1C7 — C22×F7
C1C7C7⋊C3F7C2×F7 — C22×F7
C7 — C22×F7
C1C22

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 12A2B2C2D2E2F2G3A3B6A6B6C6D6E6F6G6H6I6J6K6L6M6N714A14B14C
 size 1111777777777777777777776666
ρ11111111111111111111111111111    trivial
ρ21-11-1-11-1111-1-1-1111-1-1111-1-1-11-1-11    linear of order 2
ρ31-1-11-111-111-1-1111-111-1-1-11-1-11-11-1    linear of order 2
ρ411-1-111-1-11111-111-1-1-1-1-1-1-11111-1-1    linear of order 2
ρ51-11-11-11-1111-1-1-1-11111-1-1-11-11-1-11    linear of order 2
ρ61111-1-1-1-111-111-1-11-1-11-1-11-111111    linear of order 2
ρ71-1-111-1-11111-11-1-1-1-1-1-11111-11-11-1    linear of order 2
ρ811-1-1-1-11111-11-1-1-1-111-111-1-1111-1-1    linear of order 2
ρ91-11-1-11-11ζ3ζ32ζ6ζ6ζ65ζ3ζ32ζ3ζ65ζ6ζ32ζ3ζ32ζ6ζ65ζ651-1-11    linear of order 6
ρ101111-1-1-1-1ζ32ζ3ζ65ζ3ζ32ζ6ζ65ζ32ζ6ζ65ζ3ζ6ζ65ζ3ζ6ζ321111    linear of order 6
ρ111111-1-1-1-1ζ3ζ32ζ6ζ32ζ3ζ65ζ6ζ3ζ65ζ6ζ32ζ65ζ6ζ32ζ65ζ31111    linear of order 6
ρ121-1-11-111-1ζ32ζ3ζ65ζ65ζ32ζ32ζ3ζ6ζ32ζ3ζ65ζ6ζ65ζ3ζ6ζ61-11-1    linear of order 6
ρ1311-1-1-1-111ζ3ζ32ζ6ζ32ζ65ζ65ζ6ζ65ζ3ζ32ζ6ζ3ζ32ζ6ζ65ζ311-1-1    linear of order 6
ρ141-1-111-1-11ζ32ζ3ζ3ζ65ζ32ζ6ζ65ζ6ζ6ζ65ζ65ζ32ζ3ζ3ζ32ζ61-11-1    linear of order 6
ρ1511111111ζ32ζ3ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ321111    linear of order 3
ρ161-1-11-111-1ζ3ζ32ζ6ζ6ζ3ζ3ζ32ζ65ζ3ζ32ζ6ζ65ζ6ζ32ζ65ζ651-11-1    linear of order 6
ρ1711-1-1-1-111ζ32ζ3ζ65ζ3ζ6ζ6ζ65ζ6ζ32ζ3ζ65ζ32ζ3ζ65ζ6ζ3211-1-1    linear of order 6
ρ1811111111ζ3ζ32ζ32ζ32ζ3ζ3ζ32ζ3ζ3ζ32ζ32ζ3ζ32ζ32ζ3ζ31111    linear of order 3
ρ191-11-1-11-11ζ32ζ3ζ65ζ65ζ6ζ32ζ3ζ32ζ6ζ65ζ3ζ32ζ3ζ65ζ6ζ61-1-11    linear of order 6
ρ201-11-11-11-1ζ3ζ32ζ32ζ6ζ65ζ65ζ6ζ3ζ3ζ32ζ32ζ65ζ6ζ6ζ3ζ651-1-11    linear of order 6
ρ211-11-11-11-1ζ32ζ3ζ3ζ65ζ6ζ6ζ65ζ32ζ32ζ3ζ3ζ6ζ65ζ65ζ32ζ61-1-11    linear of order 6
ρ221-1-111-1-11ζ3ζ32ζ32ζ6ζ3ζ65ζ6ζ65ζ65ζ6ζ6ζ3ζ32ζ32ζ3ζ651-11-1    linear of order 6
ρ2311-1-111-1-1ζ32ζ3ζ3ζ3ζ6ζ32ζ3ζ6ζ6ζ65ζ65ζ6ζ65ζ65ζ32ζ3211-1-1    linear of order 6
ρ2411-1-111-1-1ζ3ζ32ζ32ζ32ζ65ζ3ζ32ζ65ζ65ζ6ζ6ζ65ζ6ζ6ζ3ζ311-1-1    linear of order 6
ρ256-66-600000000000000000000-111-1    orthogonal lifted from C2×F7
ρ266-6-6600000000000000000000-11-11    orthogonal lifted from C2×F7
ρ27666600000000000000000000-1-1-1-1    orthogonal lifted from F7
ρ2866-6-600000000000000000000-1-111    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)

42000000
04200000
00420000
00042000
00004200
00000420
00000042
,
1000000
04200000
00420000
00042000
00004200
00000420
00000042
,
1000000
00000042
01000042
00100042
00010042
00001042
00000142
,
6000000
00000420
00042000
04200000
00000042
00004200
00420000

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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Character table of C22×F7 in TeX

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