ES+(2,2):2D7 - GroupNames

G = 2₊¹⁺⁴⋊₂D₇ order 448 = 2⁶·7

2^nd semidirect product of 2₊¹⁺⁴ and D₇ acting via D₇/C₇=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₂D₇, C₇⋊₃C₂≀C₂², (C₂×C₂₈)⋊₃D₄, (C₂²×C₁₄)⋊₄D₄, (C₂×D₄).₈₅D₁₄, C₂³⋊₂(C₇⋊D₄), C₂³⋊D₁₄⋊₁₉C₂, C₁₄.₈₃C₂²≀C₂, C₂³⋊Dic₇⋊₁₀C₂, (C₂³×D₇)⋊₂C₂², C₂³.D₇⋊₇C₂², C₂³.₆(C₂²×D₇), (C₂²×C₁₄).₆C₂³, (C₇×2₊¹⁺⁴)⋊₆C₂, (D₄×C₁₄).₁₇₉C₂², C₂.₁₇(C₂⁴⋊D₇), (C₂×C₄)⋊₂(C₇⋊D₄), (C₂×C₁₄).₄₅(C₂×D₄), C₂².₁₇(C₂×C₇⋊D₄), SmallGroup(448,778)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₄ — 2₊¹⁺⁴⋊₂D₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×C₁₄ — C₂³×D₇ — C₂³⋊D₁₄ — 2₊¹⁺⁴⋊₂D₇

Lower central

C₇ — C₁₄ — C₂²×C₁₄ — 2₊¹⁺⁴⋊₂D₇

Upper central

C₁ — C₂ — C₂³ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴⋊₂D₇
G = < a,b,c,d,e,f | a⁴=b²=d²=e⁷=f²=1, c²=a², bab=a^-1, ac=ca, ad=da, ae=ea, faf=a^-1cd, fcf=bc=cb, fdf=bd=db, be=eb, bf=fb, dcd=a²c, ce=ec, de=ed, fef=e^-1 >

Subgroups: 1140 in 198 conjugacy classes, 43 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, Dic₇, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂≀C₂², D₁₄⋊C₄, C₂³.D₇, C₂×C₇⋊D₄, D₄×C₁₄, D₄×C₁₄, C₇×C₄○D₄, C₂³×D₇, C₂³⋊Dic₇, C₂³⋊D₁₄, C₇×2₊¹⁺⁴, 2₊¹⁺⁴⋊₂D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, D₁₄, C₂²≀C₂, C₇⋊D₄, C₂²×D₇, C₂≀C₂², C₂×C₇⋊D₄, C₂⁴⋊D₇, 2₊¹⁺⁴⋊₂D₇

Smallest permutation representation of 2₊¹⁺⁴⋊₂D₇
►On 56 points

Generators in S₅₆

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

(29 36)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(43 50)(44 51)(45 52)(46 53)(47 54)(48 55)(49 56)

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

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	7A	7B	7C	14A	14B	14C	14D	···	14AD	28A	···	28R
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	7	7	7	14	14	14	14	···	14	28	···	28
size	1	1	2	2	2	4	4	4	28	28	4	4	4	56	56	56	2	2	2	2	2	2	4	···	4	4	···	4

67 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

C₇⋊D₄

C₂≀C₂²

2₊¹⁺⁴⋊₂D₇

kernel

2₊¹⁺⁴⋊₂D₇

C₂³⋊Dic₇

C₂³⋊D₁₄

C₇×2₊¹⁺⁴

C₂×C₂₈

C₂²×C₁₄

2₊¹⁺⁴

C₂×D₄

C₂×C₄

C₂³

C₇

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊₂D₇ ►in GL₆(𝔽₂₉)

1	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	16
0	0	0	0	28	1
0	0	18	1	0	28
0	0	18	0	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	18	0	28	0
0	0	18	0	0	28

1	0	0	0	0	0
0	28	0	0	0	0
0	0	1	13	0	0
0	0	11	28	0	0
0	0	18	1	0	28
0	0	0	1	1	0

28	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	18	1	0	0
0	0	11	0	1	0
0	0	0	0	0	28

25	0	0	0	0	0
0	7	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	16	0	0	0	0
20	0	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	0	28
0	0	0	0	28	0

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,18,18,0,0,0,0,1,0,0,0,0,28,0,0,0,0,16,1,28,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,18,18,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,11,18,0,0,0,13,28,1,1,0,0,0,0,0,1,0,0,0,0,28,0],[28,0,0,0,0,0,0,1,0,0,0,0,0,0,28,18,11,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[25,0,0,0,0,0,0,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,20,0,0,0,0,16,0,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,0,0,0,0,28,0] >;

2₊¹⁺⁴⋊₂D₇ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_2D_7

% in TeX

G:=Group("ES+(2,2):2D7");

// GroupNames label

G:=SmallGroup(448,778);

// by ID

G=gap.SmallGroup(448,778);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,570,438,18822]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^7=f^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1*c*d,f*c*f=b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊2D7 order 448 = 26·7

2nd semidirect product of 2+ 1+4 and D7 acting via D7/C7=C2

G = 2₊¹⁺⁴⋊₂D₇ order 448 = 2⁶·7

2^nd semidirect product of 2₊¹⁺⁴ and D₇ acting via D₇/C₇=C₂