Copied to
clipboard

G = D14⋊C4order 112 = 24·7

The semidirect product of D14 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D14⋊C4, C14.6D4, C2.2D28, C22.6D14, (C2×C4)⋊1D7, (C2×C28)⋊1C2, C2.5(C4×D7), C71(C22⋊C4), C14.5(C2×C4), (C22×D7).C2, (C2×Dic7)⋊1C2, C2.2(C7⋊D4), (C2×C14).6C22, SmallGroup(112,13)

Series: Derived Chief Lower central Upper central

C1C14 — D14⋊C4
C1C7C14C2×C14C22×D7 — D14⋊C4
C7C14 — D14⋊C4
C1C22C2×C4

Generators and relations for D14⋊C4
 G = < a,b,c | a14=b2=c4=1, bab=a-1, ac=ca, cbc-1=a7b >

14C2
14C2
2C4
7C22
7C22
14C4
14C22
14C22
2D7
2D7
7C2×C4
7C23
2D14
2D14
2Dic7
2C28
7C22⋊C4

Smallest permutation representation of D14⋊C4
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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 28)(8 27)(9 26)(10 25)(11 24)(12 23)(13 22)(14 21)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 56)(41 55)(42 54)
(1 47 21 29)(2 48 22 30)(3 49 23 31)(4 50 24 32)(5 51 25 33)(6 52 26 34)(7 53 27 35)(8 54 28 36)(9 55 15 37)(10 56 16 38)(11 43 17 39)(12 44 18 40)(13 45 19 41)(14 46 20 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), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,56)(41,55)(42,54), (1,47,21,29)(2,48,22,30)(3,49,23,31)(4,50,24,32)(5,51,25,33)(6,52,26,34)(7,53,27,35)(8,54,28,36)(9,55,15,37)(10,56,16,38)(11,43,17,39)(12,44,18,40)(13,45,19,41)(14,46,20,42)>;

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

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

D14⋊C4 is a maximal subgroup of
C42⋊D7  C4×D28  C4.D28  C422D7  D7×C22⋊C4  Dic74D4  C22⋊D28  D14.D4  D14⋊D4  Dic7.D4  C22.D28  C4⋊C47D7  D28⋊C4  D14.5D4  C4⋊D28  D14⋊Q8  D142Q8  C4⋊C4⋊D7  C4×C7⋊D4  C23.23D14  C287D4  C23⋊D14  Dic7⋊D4  D143Q8  C28.23D4  D14⋊C12  D14⋊Dic3  D42⋊C4  C2.D84
D14⋊C4 is a maximal quotient of
Dic14⋊C4  C23.1D14  C14.D8  C14.Q16  C28.44D4  D14⋊C8  C2.D56  C28.46D4  C4.12D28  D284C4  C14.C42  D14⋊Dic3  D42⋊C4  C2.D84

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D7A7B7C14A···14I28A···28L
order122222444477714···1428···28
size111114142214142222···22···2

34 irreducible representations

dim11111222222
type++++++++
imageC1C2C2C2C4D4D7D14C4×D7D28C7⋊D4
kernelD14⋊C4C2×Dic7C2×C28C22×D7D14C14C2×C4C22C2C2C2
# reps11114233666

Matrix representation of D14⋊C4 in GL4(𝔽29) generated by

252500
41100
00118
00180
,
4400
182500
0010
002428
,
17000
01700
001618
002613
G:=sub<GL(4,GF(29))| [25,4,0,0,25,11,0,0,0,0,11,18,0,0,8,0],[4,18,0,0,4,25,0,0,0,0,1,24,0,0,0,28],[17,0,0,0,0,17,0,0,0,0,16,26,0,0,18,13] >;

D14⋊C4 in GAP, Magma, Sage, TeX

D_{14}\rtimes C_4
% in TeX

G:=Group("D14:C4");
// GroupNames label

G:=SmallGroup(112,13);
// by ID

G=gap.SmallGroup(112,13);
# by ID

G:=PCGroup([5,-2,-2,-2,-2,-7,101,26,2404]);
// Polycyclic

G:=Group<a,b,c|a^14=b^2=c^4=1,b*a*b=a^-1,a*c=c*a,c*b*c^-1=a^7*b>;
// generators/relations

Export

Subgroup lattice of D14⋊C4 in TeX

׿
×
𝔽