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G = C7⋊C24order 168 = 23·3·7

The semidirect product of C7 and C24 acting via C24/C4=C6

metacyclic, supersoluble, monomial, Z-group

Aliases: C7⋊C24, C14.C12, C4.2F7, C28.2C6, C7⋊C8⋊C3, C7⋊C3⋊C8, C2.(C7⋊C12), (C2×C7⋊C3).C4, (C4×C7⋊C3).2C2, SmallGroup(168,1)

Series: Derived Chief Lower central Upper central

C1C7 — C7⋊C24
C1C7C14C28C4×C7⋊C3 — C7⋊C24
C7 — C7⋊C24
C1C4

Generators and relations for C7⋊C24
 G = < a,b | a7=b24=1, bab-1=a3 >

7C3
7C6
7C8
7C12
7C24

Character table of C7⋊C24

 class 123A3B4A4B6A6B78A8B8C8D12A12B12C12D1424A24B24C24D24E24F24G24H28A28B
 size 1177117767777777767777777766
ρ11111111111111111111111111111    trivial
ρ2111111111-1-1-1-111111-1-1-1-1-1-1-1-111    linear of order 2
ρ311ζ3ζ3211ζ32ζ311111ζ3ζ32ζ3ζ321ζ3ζ32ζ32ζ32ζ32ζ3ζ3ζ311    linear of order 3
ρ411ζ32ζ311ζ3ζ321-1-1-1-1ζ32ζ3ζ32ζ31ζ6ζ65ζ65ζ65ζ65ζ6ζ6ζ611    linear of order 6
ρ511ζ3ζ3211ζ32ζ31-1-1-1-1ζ3ζ32ζ3ζ321ζ65ζ6ζ6ζ6ζ6ζ65ζ65ζ6511    linear of order 6
ρ611ζ32ζ311ζ3ζ3211111ζ32ζ3ζ32ζ31ζ32ζ3ζ3ζ3ζ3ζ32ζ32ζ3211    linear of order 3
ρ71111-1-1111-iii-i-1-1-1-11-iii-i-iii-i-1-1    linear of order 4
ρ81111-1-1111i-i-ii-1-1-1-11i-i-iii-i-ii-1-1    linear of order 4
ρ91-111i-i-1-11ζ8ζ87ζ83ζ85-i-iii-1ζ8ζ87ζ83ζ85ζ8ζ87ζ83ζ85i-i    linear of order 8
ρ101-111-ii-1-11ζ83ζ85ζ8ζ87ii-i-i-1ζ83ζ85ζ8ζ87ζ83ζ85ζ8ζ87-ii    linear of order 8
ρ111-111-ii-1-11ζ87ζ8ζ85ζ83ii-i-i-1ζ87ζ8ζ85ζ83ζ87ζ8ζ85ζ83-ii    linear of order 8
ρ121-111i-i-1-11ζ85ζ83ζ87ζ8-i-iii-1ζ85ζ83ζ87ζ8ζ85ζ83ζ87ζ8i-i    linear of order 8
ρ1311ζ32ζ3-1-1ζ3ζ321-iii-iζ6ζ65ζ6ζ651ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32-1-1    linear of order 12
ρ1411ζ32ζ3-1-1ζ3ζ321i-i-iiζ6ζ65ζ6ζ651ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32-1-1    linear of order 12
ρ1511ζ3ζ32-1-1ζ32ζ31-iii-iζ65ζ6ζ65ζ61ζ43ζ3ζ4ζ32ζ4ζ32ζ43ζ32ζ43ζ32ζ4ζ3ζ4ζ3ζ43ζ3-1-1    linear of order 12
ρ1611ζ3ζ32-1-1ζ32ζ31i-i-iiζ65ζ6ζ65ζ61ζ4ζ3ζ43ζ32ζ43ζ32ζ4ζ32ζ4ζ32ζ43ζ3ζ43ζ3ζ4ζ3-1-1    linear of order 12
ρ171-1ζ3ζ32-iiζ6ζ651ζ87ζ8ζ85ζ83ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32-1ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3-ii    linear of order 24
ρ181-1ζ32ζ3-iiζ65ζ61ζ83ζ85ζ8ζ87ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3-1ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32-ii    linear of order 24
ρ191-1ζ3ζ32i-iζ6ζ651ζ8ζ87ζ83ζ85ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32-1ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3i-i    linear of order 24
ρ201-1ζ32ζ3-iiζ65ζ61ζ87ζ8ζ85ζ83ζ82ζ32ζ82ζ3ζ86ζ32ζ86ζ3-1ζ87ζ32ζ8ζ3ζ85ζ3ζ83ζ3ζ87ζ3ζ8ζ32ζ85ζ32ζ83ζ32-ii    linear of order 24
ρ211-1ζ32ζ3i-iζ65ζ61ζ85ζ83ζ87ζ8ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3-1ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32i-i    linear of order 24
ρ221-1ζ3ζ32-iiζ6ζ651ζ83ζ85ζ8ζ87ζ82ζ3ζ82ζ32ζ86ζ3ζ86ζ32-1ζ83ζ3ζ85ζ32ζ8ζ32ζ87ζ32ζ83ζ32ζ85ζ3ζ8ζ3ζ87ζ3-ii    linear of order 24
ρ231-1ζ32ζ3i-iζ65ζ61ζ8ζ87ζ83ζ85ζ86ζ32ζ86ζ3ζ82ζ32ζ82ζ3-1ζ8ζ32ζ87ζ3ζ83ζ3ζ85ζ3ζ8ζ3ζ87ζ32ζ83ζ32ζ85ζ32i-i    linear of order 24
ρ241-1ζ3ζ32i-iζ6ζ651ζ85ζ83ζ87ζ8ζ86ζ3ζ86ζ32ζ82ζ3ζ82ζ32-1ζ85ζ3ζ83ζ32ζ87ζ32ζ8ζ32ζ85ζ32ζ83ζ3ζ87ζ3ζ8ζ3i-i    linear of order 24
ρ2566006600-100000000-100000000-1-1    orthogonal lifted from F7
ρ266600-6-600-100000000-10000000011    symplectic lifted from C7⋊C12, Schur index 2
ρ276-6006i-6i00-100000000100000000-ii    complex faithful
ρ286-600-6i6i00-100000000100000000i-i    complex faithful

Smallest permutation representation of C7⋊C24
On 56 points
Generators in S56
(1 33 49 12 41 20 28)(2 13 29 50 21 34 42)(3 51 43 30 35 14 22)(4 31 23 44 15 52 36)(5 45 37 24 53 32 16)(6 25 17 38 9 46 54)(7 39 55 18 47 26 10)(8 19 11 56 27 40 48)
(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)| (1,33,49,12,41,20,28)(2,13,29,50,21,34,42)(3,51,43,30,35,14,22)(4,31,23,44,15,52,36)(5,45,37,24,53,32,16)(6,25,17,38,9,46,54)(7,39,55,18,47,26,10)(8,19,11,56,27,40,48), (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( (1,33,49,12,41,20,28)(2,13,29,50,21,34,42)(3,51,43,30,35,14,22)(4,31,23,44,15,52,36)(5,45,37,24,53,32,16)(6,25,17,38,9,46,54)(7,39,55,18,47,26,10)(8,19,11,56,27,40,48), (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([(1,33,49,12,41,20,28),(2,13,29,50,21,34,42),(3,51,43,30,35,14,22),(4,31,23,44,15,52,36),(5,45,37,24,53,32,16),(6,25,17,38,9,46,54),(7,39,55,18,47,26,10),(8,19,11,56,27,40,48)], [(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)])

C7⋊C24 is a maximal subgroup of   C8×F7  C8⋊F7  C28.C12  D4⋊F7  D4.F7  Q82F7  Q8.2F7
C7⋊C24 is a maximal quotient of   C7⋊C48

Matrix representation of C7⋊C24 in GL7(𝔽337)

1000000
033610000
033601000
033600100
033600010
033600001
033600000
,
241000000
013413425402030
001342031340254
01345120300203
013400134254203
05102031342030
00134051203203

G:=sub<GL(7,GF(337))| [1,0,0,0,0,0,0,0,336,336,336,336,336,336,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0],[241,0,0,0,0,0,0,0,134,0,134,134,51,0,0,134,134,51,0,0,134,0,254,203,203,0,203,0,0,0,134,0,134,134,51,0,203,0,0,254,203,203,0,0,254,203,203,0,203] >;

C7⋊C24 in GAP, Magma, Sage, TeX

C_7\rtimes C_{24}
% in TeX

G:=Group("C7:C24");
// GroupNames label

G:=SmallGroup(168,1);
// by ID

G=gap.SmallGroup(168,1);
# by ID

G:=PCGroup([5,-2,-3,-2,-2,-7,30,42,3604,1209]);
// Polycyclic

G:=Group<a,b|a^7=b^24=1,b*a*b^-1=a^3>;
// generators/relations

Export

Subgroup lattice of C7⋊C24 in TeX
Character table of C7⋊C24 in TeX

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