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## G = D4×F7order 336 = 24·3·7

### Direct product of D4 and F7

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 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 6A ··· 6F 6G ··· 6N 7 12A 12B 12C 12D 14A 14B 14C 28 order 1 2 2 2 2 2 2 2 3 3 4 4 6 ··· 6 6 ··· 6 7 12 12 12 12 14 14 14 28 size 1 1 2 2 7 7 14 14 7 7 2 14 7 ··· 7 14 ··· 14 6 14 14 14 14 6 12 12 12

35 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 12 2 2 6 6 6 type + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C3 C6 C6 C6 C6 C6 D4×F7 D4 C3×D4 F7 C2×F7 C2×F7 kernel D4×F7 C4×F7 C4⋊F7 Dic7⋊C6 D4×C7⋊C3 C22×F7 D4×D7 C4×D7 D28 C7⋊D4 C7×D4 C22×D7 C1 F7 D7 D4 C4 C22 # reps 1 1 1 2 1 2 2 2 2 4 2 4 1 2 4 1 1 2

Matrix representation of D4×F7 in GL8(𝔽337)

 1 330 0 0 0 0 0 0 241 336 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 7 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 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 0 336 336 336 336 336 336 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
,
 128 0 0 0 0 0 0 0 0 128 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 336 336 336 336 336 336 0 0 0 0 0 0 1 0

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

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