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G = D14⋊C12  order 336 = 24·3·7

The semidirect product of D14 and C12 acting via C12/C2=C6

metabelian, supersoluble, monomial

Aliases: D14⋊C12, D14⋊C4⋊C3, (C2×F7)⋊C4, (C2×C4)⋊1F7, (C2×C28)⋊6C6, C2.5(C4×F7), C14.6(C3×D4), C2.2(C4⋊F7), (C22×D7).C6, (C22×F7).C2, C14.5(C2×C12), (C2×Dic7)⋊1C6, C22.6(C2×F7), C2.2(Dic7⋊C6), (C2×C7⋊C12)⋊1C2, C7⋊1(C3×C22⋊C4), (C2×C7⋊C3).6D4, C7⋊C3⋊1(C22⋊C4), (C2×C14).5(C2×C6), (C22×C7⋊C3).5C22, (C2×C4×C7⋊C3)⋊6C2, (C2×C7⋊C3).4(C2×C4), SmallGroup(336,17)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C14 — D14⋊C12
Chief series
C1 — C7 — C14 — C2×C14 — C22×C7⋊C3 — C22×F7 — D14⋊C12
Lower central
C7 — C14 — D14⋊C12
Upper central
C1 — C22 — C2×C4

Generators and relations for D14⋊C12
 G = < a,b,c | a14=b2=c12=1, bab=a-1, cac-1=a9, cbc-1=ab >

Subgroups: 320 in 68 conjugacy classes, 28 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C7, C2×C4, C2×C4, C23, C12, C2×C6, D7, C14, C22⋊C4, C7⋊C3, C2×C12, C22×C6, Dic7, C28, D14, D14, C2×C14, F7, C2×C7⋊C3, C3×C22⋊C4, C2×Dic7, C2×C28, C22×D7, C7⋊C12, C4×C7⋊C3, C2×F7, C2×F7, C22×C7⋊C3, D14⋊C4, C2×C7⋊C12, C2×C4×C7⋊C3, C22×F7, D14⋊C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, F7, C3×C22⋊C4, C2×F7, C4×F7, C4⋊F7, Dic7⋊C6, D14⋊C12

Smallest permutation representation of D14⋊C12
►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 14)(2 13)(3 12)(4 11)(5 10)(6 9)(7 8)(15 28)(16 27)(17 26)(18 25)(19 24)(20 23)(21 22)(29 33)(30 32)(34 42)(35 41)(36 40)(37 39)(43 47)(44 46)(48 56)(49 55)(50 54)(51 53)

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

38 conjugacy classes

class 1 2A2B2C2D2E3A3B4A4B4C4D6A···6F6G6H6I6J 7 12A···12H14A14B14C28A28B28C28D
order1222223344446···66666712···1214141428282828
size11111414772214147···714141414614···146666666

38 irreducible representations

dim11111111112266666
type++++++++
imageC1C2C2C2C3C4C6C6C6C12D4C3×D4F7C2×F7C4×F7C4⋊F7Dic7⋊C6
kernelD14⋊C12C2×C7⋊C12C2×C4×C7⋊C3C22×F7D14⋊C4C2×F7C2×Dic7C2×C28C22×D7D14C2×C7⋊C3C14C2×C4C22C2C2C2
# reps11112422282411222

Matrix representation of D14⋊C12 ►in GL8(𝔽337)

3360000000
0336000000
00000010
00000001
00336336336336336336
00100000
00010000
00001000
,
3360000000
2991000000
00000010
00000100
00001000
00010000
00100000
00336336336336336336
,
2803000000
30857000000
0018900000
0000189000
0000001890
00148148148148148148
0001890000
0000018900

G:=sub<GL(8,GF(337))| [336,0,0,0,0,0,0,0,0,336,0,0,0,0,0,0,0,0,0,0,336,1,0,0,0,0,0,0,336,0,1,0,0,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,0,0,1,0,336,0,0,0,0,0,0,1,336,0,0,0],[336,299,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,336,0,0,0,0,0,1,0,336,0,0,0,0,1,0,0,336,0,0,0,1,0,0,0,336,0,0,1,0,0,0,0,336,0,0,0,0,0,0,0,336],[280,308,0,0,0,0,0,0,3,57,0,0,0,0,0,0,0,0,189,0,0,148,0,0,0,0,0,0,0,148,189,0,0,0,0,189,0,148,0,0,0,0,0,0,0,148,0,189,0,0,0,0,189,148,0,0,0,0,0,0,0,148,0,0] >;
 

D14⋊C12 in GAP, Magma, Sage, TeX 

D_{14}\rtimes C_{12}
 
% in TeX
 
G:=Group("D14:C12");
 
// GroupNames label
 
G:=SmallGroup(336,17);
 
// by ID
 
G=gap.SmallGroup(336,17);
 
# by ID
 
G:=PCGroup([6,-2,-2,-3,-2,-2,-7,313,79,10373,1745]);
 
// Polycyclic
 
G:=Group<a,b,c|a^14=b^2=c^12=1,b*a*b=a^-1,c*a*c^-1=a^9,c*b*c^-1=a*b>;
 
// generators/relations
 

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