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G = C8×F7order 336 = 24·3·7

Direct product of C8 and F7

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

Aliases: C8×F7, D7⋊C24, C563C6, D14.2C12, Dic7.2C12, (C8×D7)⋊C3, C7⋊C86C6, C7⋊C246C2, C71(C2×C24), C7⋊C12.2C4, C2.1(C4×F7), (C4×D7).3C6, (C4×F7).3C2, (C2×F7).2C4, C4.12(C2×F7), C14.1(C2×C12), C28.13(C2×C6), C7⋊C31(C2×C8), (C8×C7⋊C3)⋊3C2, (C4×C7⋊C3).13C22, (C2×C7⋊C3).1(C2×C4), SmallGroup(336,7)

Series: Derived Chief Lower central Upper central

C1C7 — C8×F7
C1C7C14C28C4×C7⋊C3C4×F7 — C8×F7
C7 — C8×F7
C1C8

Generators and relations for C8×F7
 G = < a,b,c | a8=b7=c6=1, ab=ba, ac=ca, cbc-1=b5 >

7C2
7C2
7C3
7C22
7C4
7C6
7C6
7C6
7C8
7C2×C4
7C12
7C2×C6
7C12
7C2×C8
7C24
7C24
7C2×C12
7C2×C24

Smallest permutation representation of C8×F7
On 56 points
Generators in S56
(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)
(1 31 13 44 33 17 49)(2 32 14 45 34 18 50)(3 25 15 46 35 19 51)(4 26 16 47 36 20 52)(5 27 9 48 37 21 53)(6 28 10 41 38 22 54)(7 29 11 42 39 23 55)(8 30 12 43 40 24 56)
(9 53 37 21 27 48)(10 54 38 22 28 41)(11 55 39 23 29 42)(12 56 40 24 30 43)(13 49 33 17 31 44)(14 50 34 18 32 45)(15 51 35 19 25 46)(16 52 36 20 26 47)

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

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

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

56 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D6A···6F 7 8A8B8C8D8E8F8G8H12A···12H 14 24A···24P28A28B56A56B56C56D
order12223344446···678888888812···121424···24282856565656
size11777711777···76111177777···767···7666666

56 irreducible representations

dim111111111111116666
type++++++
imageC1C2C2C2C3C4C4C6C6C6C8C12C12C24F7C2×F7C4×F7C8×F7
kernelC8×F7C7⋊C24C8×C7⋊C3C4×F7C8×D7C7⋊C12C2×F7C7⋊C8C56C4×D7F7Dic7D14D7C8C4C2C1
# reps1111222222844161124

Matrix representation of C8×F7 in GL6(𝔽337)

8500000
0850000
0085000
0008500
0000850
0000085
,
336336336336336336
100000
010000
001000
000100
000010
,
000010
001000
100000
000001
000100
010000

G:=sub<GL(6,GF(337))| [85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85,0,0,0,0,0,0,85],[336,1,0,0,0,0,336,0,1,0,0,0,336,0,0,1,0,0,336,0,0,0,1,0,336,0,0,0,0,1,336,0,0,0,0,0],[0,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,1,0,0] >;

C8×F7 in GAP, Magma, Sage, TeX

C_8\times F_7
% in TeX

G:=Group("C8xF7");
// GroupNames label

G:=SmallGroup(336,7);
// by ID

G=gap.SmallGroup(336,7);
# by ID

G:=PCGroup([6,-2,-2,-3,-2,-2,-7,79,69,10373,1745]);
// Polycyclic

G:=Group<a,b,c|a^8=b^7=c^6=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations

Export

Subgroup lattice of C8×F7 in TeX

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