Copied to
clipboard

G = D4×F7  order 336 = 24·3·7

Direct product of D4 and F7

direct product, metabelian, supersoluble, monomial

Aliases: D4×F7, D28⋊2C6, (D4×D7)⋊C3, C28⋊(C2×C6), D7⋊(C3×D4), C7⋊2(C6×D4), C4⋊F7⋊2C2, C4⋊1(C2×F7), (C7×D4)⋊2C6, (C4×D7)⋊1C6, C7⋊D4⋊1C6, (C4×F7)⋊1C2, D14⋊2(C2×C6), C7⋊C12⋊1C22, C22⋊2(C2×F7), Dic7⋊C6⋊1C2, Dic7⋊1(C2×C6), (C22×D7)⋊2C6, (C22×F7)⋊2C2, (C2×F7)⋊2C22, C2.6(C22×F7), C14.5(C22×C6), C7⋊C3⋊2(C2×D4), (C2×C14)⋊(C2×C6), (D4×C7⋊C3)⋊2C2, (C4×C7⋊C3)⋊C22, (C22×C7⋊C3)⋊C22, (C2×C7⋊C3).5C23, Aut(D28), Hol(C28), SmallGroup(336,125)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D4×F7
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C22×F7 — D4×F7
Lower central
C7 — C14 — D4×F7
Upper central
C1 — C2 — D4

Generators and relations for D4×F7
 G = < a,b,c,d | a4=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 500 in 108 conjugacy classes, 44 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C7, C2×C4, D4, D4, C23, C12, C2×C6, D7, D7, C14, C14, C2×D4, C7⋊C3, C2×C12, C3×D4, C22×C6, Dic7, C28, D14, D14, D14, C2×C14, F7, F7, C2×C7⋊C3, C2×C7⋊C3, C6×D4, C4×D7, D28, C7⋊D4, C7×D4, C22×D7, C7⋊C12, C4×C7⋊C3, C2×F7, C2×F7, C2×F7, C22×C7⋊C3, D4×D7, C4×F7, C4⋊F7, Dic7⋊C6, D4×C7⋊C3, C22×F7, D4×F7
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, F7, C6×D4, C2×F7, C22×F7, D4×F7

Permutation representations of D4×F7
►On 28 points - transitive group 28T41
Generators in S28
(1 15 8 22)(2 16 9 23)(3 17 10 24)(4 18 11 25)(5 19 12 26)(6 20 13 27)(7 21 14 28)

(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 8)(2 11 3 14 5 13)(4 10 7 12 6 9)(15 22)(16 25 17 28 19 27)(18 24 21 26 20 23)
 
G:=sub<Sym(28)| (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28), (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), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23)>;
 
G:=Group( (1,15,8,22)(2,16,9,23)(3,17,10,24)(4,18,11,25)(5,19,12,26)(6,20,13,27)(7,21,14,28), (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), (1,8)(2,11,3,14,5,13)(4,10,7,12,6,9)(15,22)(16,25,17,28,19,27)(18,24,21,26,20,23) );
 
G=PermutationGroup([[(1,15,8,22),(2,16,9,23),(3,17,10,24),(4,18,11,25),(5,19,12,26),(6,20,13,27),(7,21,14,28)], [(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)], [(1,8),(2,11,3,14,5,13),(4,10,7,12,6,9),(15,22),(16,25,17,28,19,27),(18,24,21,26,20,23)]])
 
G:=TransitiveGroup(28,41);
 

35 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B4A4B6A···6F6G···6N 7 12A12B12C12D14A14B14C 28 
order1222222233446···66···671212121214141428
size1122771414772147···714···146141414146121212

35 irreducible representations

dim1111111111111222666
type+++++++++++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4×F7D4C3×D4F7C2×F7C2×F7
kernelD4×F7C4×F7C4⋊F7Dic7⋊C6D4×C7⋊C3C22×F7D4×D7C4×D7D28C7⋊D4C7×D4C22×D7C1F7D7D4C4C22
# reps111212222424124112

Matrix representation of D4×F7 ►in GL8(𝔽337)

1330000000
241336000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
3367000000
01000000
00100000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00336336336336336336
00100000
00010000
00001000
00000100
00000010
,
1280000000
0128000000
00100000
00000001
00000100
00010000
00336336336336336336
00000010

G:=sub<GL(8,GF(337))| [1,241,0,0,0,0,0,0,330,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[336,0,0,0,0,0,0,0,7,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,1,0,0,0,336,0,0,0,0,1,0,0,336,0,0,0,0,0],[128,0,0,0,0,0,0,0,0,128,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;
 

D4×F7 in GAP, Magma, Sage, TeX 

D_4\times F_7
 
% in TeX
 
G:=Group("D4xF7");
 
// GroupNames label
 
G:=SmallGroup(336,125);
 
// by ID
 
G=gap.SmallGroup(336,125);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,260,10373,887]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^7=d^6=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁