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G = D4.F7  order 336 = 24·3·7

The non-split extension by D4 of F7 acting via F7/C7⋊C3=C2

metabelian, supersoluble, monomial

Aliases: D4.F7, Dic14⋊1C6, C7⋊C8⋊2C6, D4.D7⋊C3, C7⋊C24⋊2C2, C7⋊C3⋊2SD16, C7⋊2(C3×SD16), C4.F7⋊1C2, C4.2(C2×F7), C28.2(C2×C6), (C7×D4).1C6, C14.8(C3×D4), C2.5(Dic7⋊C6), (D4×C7⋊C3).1C2, (C2×C7⋊C3).8D4, (C4×C7⋊C3).2C22, SmallGroup(336,19)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C28 — D4.F7
Chief series
C1 — C7 — C14 — C28 — C4×C7⋊C3 — C4.F7 — D4.F7
Lower central
C7 — C14 — C28 — D4.F7
Upper central
C1 — C2 — C4 — D4

Generators and relations for D4.F7
 G = < a,b,c,d | a4=b2=c7=1, d6=a2, bab=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c5 >

C1
C2
4C2
7C3
C7
C4
2C22
14C4
7C6
28C6
C14
4C14
C7⋊C3
D4
7Q8
7C8
7C12
14C2×C6
14C12
C28
2C2×C14
2Dic7
C2×C7⋊C3
4C2×C7⋊C3
7SD16
7C3×Q8
7C3×D4
7C24
Dic14
C7×D4
C7⋊C8
C4×C7⋊C3
2C22×C7⋊C3
2C7⋊C12
7C3×SD16
D4.D7
C4.F7
C7⋊C24
D4×C7⋊C3
D4.F7

Character table of D4.F7

 class 12A2B3A3B4A4B6A6B6C6D78A8B12A12B12C12D14A14B14C24A24B24C24D28
 size 114772287728286141414142828612121414141412
ρ111111111111111111111111111    trivial
ρ211-1111-111-1-111111-1-11-1-111111    linear of order 2
ρ311-1111111-1-11-1-111111-1-1-1-1-1-11    linear of order 2
ρ4111111-111111-1-111-1-1111-1-1-1-11    linear of order 2
ρ5111ζ32ζ31-1ζ3ζ32ζ32ζ31-1-1ζ32ζ3ζ65ζ6111ζ6ζ6ζ65ζ651    linear of order 6
ρ611-1ζ32ζ31-1ζ3ζ32ζ6ζ65111ζ32ζ3ζ65ζ61-1-1ζ32ζ32ζ3ζ31    linear of order 6
ρ711-1ζ3ζ321-1ζ32ζ3ζ65ζ6111ζ3ζ32ζ6ζ651-1-1ζ3ζ3ζ32ζ321    linear of order 6
ρ8111ζ3ζ321-1ζ32ζ3ζ3ζ321-1-1ζ3ζ32ζ6ζ65111ζ65ζ65ζ6ζ61    linear of order 6
ρ911-1ζ32ζ311ζ3ζ32ζ6ζ651-1-1ζ32ζ3ζ3ζ321-1-1ζ6ζ6ζ65ζ651    linear of order 6
ρ10111ζ32ζ311ζ3ζ32ζ32ζ3111ζ32ζ3ζ3ζ32111ζ32ζ32ζ3ζ31    linear of order 3
ρ1111-1ζ3ζ3211ζ32ζ3ζ65ζ61-1-1ζ3ζ32ζ32ζ31-1-1ζ65ζ65ζ6ζ61    linear of order 6
ρ12111ζ3ζ3211ζ32ζ3ζ3ζ32111ζ3ζ32ζ32ζ3111ζ3ζ3ζ32ζ321    linear of order 3
ρ1322022-202200200-2-2002000000-2    orthogonal lifted from D4
ρ14220-1+√-3-1-√-3-20-1-√-3-1+√-3002001-√-31+√-3002000000-2    complex lifted from C3×D4
ρ15220-1-√-3-1+√-3-20-1+√-3-1-√-3002001+√-31-√-3002000000-2    complex lifted from C3×D4
ρ162-202200-2-2002√-2-√-20000-200-√-2√-2-√-2√-20    complex lifted from SD16
ρ172-202200-2-2002-√-2√-20000-200√-2-√-2√-2-√-20    complex lifted from SD16
ρ182-20-1+√-3-1-√-3001+√-31-√-3002√-2-√-20000-200ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ3ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ320    complex lifted from C3×SD16
ρ192-20-1-√-3-1+√-3001-√-31+√-3002-√-2√-20000-200ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ32ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ30    complex lifted from C3×SD16
ρ202-20-1+√-3-1-√-3001+√-31-√-3002-√-2√-20000-200ζ83ζ3+ζ8ζ3ζ87ζ3+ζ85ζ3ζ83ζ32+ζ8ζ32ζ87ζ32+ζ85ζ320    complex lifted from C3×SD16
ρ212-20-1-√-3-1+√-3001-√-31+√-3002√-2-√-20000-200ζ87ζ32+ζ85ζ32ζ83ζ32+ζ8ζ32ζ87ζ3+ζ85ζ3ζ83ζ3+ζ8ζ30    complex lifted from C3×SD16
ρ2266600600000-1000000-1-1-10000-1    orthogonal lifted from F7
ρ2366-600600000-1000000-1110000-1    orthogonal lifted from C2×F7
ρ2466000-600000-1000000-1-√-7√-700001    complex lifted from Dic7⋊C6
ρ2566000-600000-1000000-1√-7-√-700001    complex lifted from Dic7⋊C6
ρ2612-12000000000-200000020000000    symplectic faithful, Schur index 2

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

(1 7)(2 4)(3 5)(9 55)(11 45)(13 47)(15 49)(17 51)(19 53)(21 27)(22 37)(23 29)(24 39)(25 31)(26 41)(28 43)(30 33)(32 35)(46 52)(48 54)(50 56)

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

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

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

01000000
3360000000
0033600000
0003360000
0000336000
0000033600
0000003360
0000000336
,
01000000
10000000
0033600000
0003360000
0000336000
00000100
00000010
00000001
,
10000000
01000000
003361241000
0001241000
003361251000
0000033610
0000033601
000001241213
,
26275000000
7575000000
00000010
000001251213
000001213336
00010000
001251213000
001213336000

G:=sub<GL(8,GF(337))| [0,336,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,336,0,336,0,0,0,0,0,124,124,125,0,0,0,0,0,1,1,1,0,0,0,0,0,0,0,0,336,336,124,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,213],[262,75,0,0,0,0,0,0,75,75,0,0,0,0,0,0,0,0,0,0,0,0,125,1,0,0,0,0,0,1,1,213,0,0,0,0,0,0,213,336,0,0,0,125,1,0,0,0,0,0,1,1,213,0,0,0,0,0,0,213,336,0,0,0] >;
 

D4.F7 in GAP, Magma, Sage, TeX 

D_4.F_7
 
% in TeX
 
G:=Group("D4.F7");
 
// GroupNames label
 
G:=SmallGroup(336,19);
 
// by ID
 
G=gap.SmallGroup(336,19);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,144,169,867,441,69,10373,1745]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^7=1,d^6=a^2,b*a*b=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^5>;
 
// generators/relations
 

Export

Subgroup lattice of D4.F7 in TeX
Character table of D4.F7 in TeX

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