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G = D4⋊2F7  order 336 = 24·3·7

The semidirect product of D4 and F7 acting through Inn(D4)

metabelian, supersoluble, monomial

Aliases: D4⋊2F7, Dic14⋊2C6, D4⋊2D7⋊C3, (C7×D4)⋊3C6, (C4×D7)⋊2C6, C7⋊D4⋊2C6, (C4×F7)⋊2C2, C4.F7⋊2C2, C4.5(C2×F7), C28.5(C2×C6), Dic7⋊C6⋊2C2, (C2×Dic7)⋊3C6, D14.2(C2×C6), C7⋊C12.3C22, C22.1(C2×F7), C2.7(C22×F7), C14.6(C22×C6), Dic7.3(C2×C6), (C2×F7).2C22, (D4×C7⋊C3)⋊3C2, C7⋊2(C3×C4○D4), (C2×C7⋊C12)⋊3C2, C7⋊C3⋊2(C4○D4), (C2×C14).(C2×C6), (C4×C7⋊C3).5C22, (C2×C7⋊C3).6C23, (C22×C7⋊C3).C22, SmallGroup(336,126)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D4⋊2F7
Chief series
C1 — C7 — C14 — C2×C7⋊C3 — C2×F7 — C4×F7 — D4⋊2F7
Lower central
C7 — C14 — D4⋊2F7
Upper central
C1 — C2 — D4

Generators and relations for D4⋊2F7
 G = < a,b,c,d | a4=b2=c7=d6=1, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a2b, dcd-1=c5 >

Subgroups: 308 in 80 conjugacy classes, 40 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C7, C2×C4, D4, D4, Q8, C12, C2×C6, D7, C14, C14, C4○D4, C7⋊C3, C2×C12, C3×D4, C3×Q8, Dic7, Dic7, C28, D14, C2×C14, F7, C2×C7⋊C3, C2×C7⋊C3, C3×C4○D4, Dic14, C4×D7, C2×Dic7, C7⋊D4, C7×D4, C7⋊C12, C7⋊C12, C4×C7⋊C3, C2×F7, C22×C7⋊C3, D4⋊2D7, C4.F7, C4×F7, C2×C7⋊C12, Dic7⋊C6, D4×C7⋊C3, D4⋊2F7
Quotients: C1, C2, C3, C22, C6, C23, C2×C6, C4○D4, C22×C6, F7, C3×C4○D4, C2×F7, C22×F7, D4⋊2F7

Smallest permutation representation of D4⋊2F7
►On 56 points
Generators in S56
(1 36 8 29)(2 37 9 30)(3 38 10 31)(4 39 11 32)(5 40 12 33)(6 41 13 34)(7 42 14 35)(15 50 22 43)(16 51 23 44)(17 52 24 45)(18 53 25 46)(19 54 26 47)(20 55 27 48)(21 56 28 49)

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

(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)

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

35 conjugacy classes

class 1 2A2B2C2D3A3B4A4B4C4D4E6A6B6C···6H 7 12A12B12C12D12E···12J14A14B14C 28 
order122223344444666···671212121212···1214141428
size1122147727714147714···146777714···146121212

35 irreducible representations

dim1111111111111222666
type++++++-+++
imageC1C2C2C2C2C2C3C6C6C6C6C6D4⋊2F7C4○D4C3×C4○D4F7C2×F7C2×F7
kernelD4⋊2F7C4.F7C4×F7C2×C7⋊C12Dic7⋊C6D4×C7⋊C3D4⋊2D7Dic14C4×D7C2×Dic7C7⋊D4C7×D4C1C7⋊C3C7D4C4C22
# reps111221222442124112

Matrix representation of D4⋊2F7 ►in GL8(𝔽337)

1890000000
0148000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
0189000000
1480000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
10000000
01000000
00336336336336336336
00100000
00010000
00001000
00000100
00000010
,
1280000000
0209000000
00100000
00000001
00000100
00010000
00336336336336336336
00000010

G:=sub<GL(8,GF(337))| [189,0,0,0,0,0,0,0,0,148,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[0,148,0,0,0,0,0,0,189,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[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],[128,0,0,0,0,0,0,0,0,209,0,0,0,0,0,0,0,0,1,0,0,0,336,0,0,0,0,0,0,1,336,0,0,0,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,0,336,1,0,0,0,1,0,0,336,0] >;
 

D4⋊2F7 in GAP, Magma, Sage, TeX 

D_4\rtimes_2F_7
 
% in TeX
 
G:=Group("D4:2F7");
 
// GroupNames label
 
G:=SmallGroup(336,126);
 
// by ID
 
G=gap.SmallGroup(336,126);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-3,-2,-7,151,506,260,10373,887]);
 
// 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,d*b*d^-1=a^2*b,d*c*d^-1=c^5>;
 
// generators/relations
 

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