D4:8D14 - GroupNames

G = D₄:₈D₁₄ order 224 = 2⁵·7

4^th semidirect product of D₄ and D₁₄ acting through Inn(D₄)

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₄:₈D₁₄, Q₈:₇D₁₄, C₇:₂2₊¹⁺⁴, D₂₈:₁₁C₂², C₂₈.₂₆C₂³, C₁₄.₁₂C₂⁴, D₁₄.₇C₂³, Dic₇.₇C₂³, Dic₁₄:₁₂C₂², C₄oD₄:₃D₇, (D₄xD₇):₅C₂, (C₂xC₄):₄D₁₄, C₄oD₂₈:₈C₂, (C₂xD₂₈):₁₃C₂, (C₂xC₂₈):₅C₂², Q₈:₂D₇:₅C₂, (C₇xD₄):₉C₂², (C₄xD₇):₂C₂², C₇:D₄:₅C₂², (C₇xQ₈):₈C₂², (C₂xC₁₄).₄C₂³, C₂.₁₃(C₂³xD₇), C₄.₃₃(C₂²xD₇), (C₂²xD₇):₄C₂², C₂².₃(C₂²xD₇), (C₇xC₄oD₄):₄C₂, SmallGroup(224,185)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — D₄:₈D₁₄

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²xD₇ — D₄xD₇ — D₄:₈D₁₄

Lower central

C₇ — C₁₄ — D₄:₈D₁₄

Upper central

C₁ — C₂ — C₄oD₄

Generators and relations for D₄:₈D₁₄
G = < a,b,c,d | a⁴=b²=c¹⁴=d²=1, bab=dad=a^-1, ac=ca, cbc^-1=a²b, bd=db, dcd=c^-1 >

Subgroups: 742 in 166 conjugacy classes, 85 normal (12 characteristic)
C₁, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₇, C₂xC₄, C₂xC₄, D₄, D₄, Q₈, Q₈, C₂³, D₇, C₁₄, C₁₄, C₂xD₄, C₄oD₄, C₄oD₄, Dic₇, C₂₈, C₂₈, D₁₄, D₁₄, C₂xC₁₄, 2₊¹⁺⁴, Dic₁₄, C₄xD₇, D₂₈, C₇:D₄, C₂xC₂₈, C₇xD₄, C₇xQ₈, C₂²xD₇, C₂xD₂₈, C₄oD₂₈, D₄xD₇, Q₈:₂D₇, C₇xC₄oD₄, D₄:₈D₁₄
Quotients: C₁, C₂, C₂², C₂³, D₇, C₂⁴, D₁₄, 2₊¹⁺⁴, C₂²xD₇, C₂³xD₇, D₄:₈D₁₄

Smallest permutation representation of D₄:₈D₁₄
►On 56 points

Generators in S₅₆

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

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

(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 52)(30 51)(31 50)(32 49)(33 48)(34 47)(35 46)(36 45)(37 44)(38 43)(39 56)(40 55)(41 54)(42 53)

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

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

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

47 conjugacy classes

class	1	2A	2B	2C	2D	2E	···	2J	4A	4B	4C	4D	4E	4F	7A	7B	7C	14A	14B	14C	14D	···	14L	28A	···	28F	28G	···	28O
order	1	2	2	2	2	2	···	2	4	4	4	4	4	4	7	7	7	14	14	14	14	···	14	28	···	28	28	···	28
size	1	1	2	2	2	14	···	14	2	2	2	2	14	14	2	2	2	2	2	2	4	···	4	2	···	2	4	···	4

47 irreducible representations

dim

type

image

C₁

C₂

D₇

D₁₄

2₊¹⁺⁴

D₄:₈D₁₄

kernel

D₄:₈D₁₄

C₂xD₂₈

C₄oD₂₈

D₄xD₇

Q₈:₂D₇

C₇xC₄oD₄

C₄oD₄

C₂xC₄

D₄

Q₈

C₇

C₁

# reps

Matrix representation of D₄:₈D₁₄ ►in GL₄(F₂₉) generated by

27	6	6	23
4	2	0	25
0	0	8	23
0	0	6	21

27	6	6	23
4	2	0	25
10	14	8	23
0	14	6	21

26	19	1	21
3	0	7	7
0	0	22	10
0	0	19	10

2	23	23	6
15	27	25	0
0	0	23	8
0	0	21	6

G:=sub<GL(4,GF(29))| [27,4,0,0,6,2,0,0,6,0,8,6,23,25,23,21],[27,4,10,0,6,2,14,14,6,0,8,6,23,25,23,21],[26,3,0,0,19,0,0,0,1,7,22,19,21,7,10,10],[2,15,0,0,23,27,0,0,23,25,23,21,6,0,8,6] >;

D₄:₈D₁₄ in GAP, Magma, Sage, TeX

D_4\rtimes_8D_{14}

% in TeX

G:=Group("D4:8D14");

// GroupNames label

G:=SmallGroup(224,185);

// by ID

G=gap.SmallGroup(224,185);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-7,188,579,69,6917]);

// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^14=d^2=1,b*a*b=d*a*d=a^-1,a*c=c*a,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^-1>;

// generators/relations

G = D4:8D14 order 224 = 25·7

4th semidirect product of D4 and D14 acting through Inn(D4)

G = D₄:₈D₁₄ order 224 = 2⁵·7

4^th semidirect product of D₄ and D₁₄ acting through Inn(D₄)