ES+(2,2):D7 - GroupNames

G = 2₊¹⁺⁴⋊D₇ order 448 = 2⁶·7

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

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₁D₇, (C₇×D₄)⋊₁₅D₄, (C₇×Q₈)⋊₁₅D₄, C₂₈⋊D₄⋊₈C₂, C₇⋊₅(D₄⋊₄D₄), D₄⋊₇(C₇⋊D₄), C₄○D₄.₉D₁₄, Q₈⋊₇(C₇⋊D₄), D₄⋊D₁₄⋊₅C₂, (C₂×D₄).₈₂D₁₄, C₂₈.₂₁₇(C₂×D₄), C₁₄.₈₀C₂²≀C₂, C₂₈.D₄⋊₁₁C₂, (C₂×D₂₈)⋊₁₅C₂², (C₂×C₂₈).₂₁C₂³, (C₄×Dic₇)⋊₈C₂², (C₂²×C₁₄).₂₄D₄, D₄⋊₂Dic₇⋊₁₁C₂, C₂³.₁₂(C₇⋊D₄), C₄.Dic₇⋊₁₀C₂², (C₇×2₊¹⁺⁴)⋊₁C₂, (D₄×C₁₄).₁₀₇C₂², C₂.₁₄(C₂⁴⋊D₇), C₄.₆₄(C₂×C₇⋊D₄), (C₂×C₁₄).₄₂(C₂×D₄), (C₂×C₄).₂₁(C₂²×D₇), C₂².₁₄(C₂×C₇⋊D₄), (C₇×C₄○D₄).₁₉C₂², SmallGroup(448,775)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₈ — 2₊¹⁺⁴⋊D₇

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂×C₂₈ — C₂×D₂₈ — D₄⋊D₁₄ — 2₊¹⁺⁴⋊D₇

Lower central

C₇ — C₁₄ — C₂×C₂₈ — 2₊¹⁺⁴⋊D₇

Upper central

C₁ — C₂ — C₂×C₄ — 2₊¹⁺⁴

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

Subgroups: 820 in 168 conjugacy classes, 43 normal (17 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, D₇, C₁₄, C₁₄, C₄², M₄(2), D₈, SD₁₆, C₂×D₄, C₂×D₄, C₄○D₄, C₄○D₄, Dic₇, C₂₈, C₂₈, D₁₄, C₂×C₁₄, C₂×C₁₄, C₄.D₄, C₄≀C₂, C₄⋊₁D₄, C₈⋊C₂², 2₊¹⁺⁴, C₇⋊C₈, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, D₄⋊₄D₄, C₄.Dic₇, C₄×Dic₇, D₄⋊D₇, Q₈⋊D₇, C₂×D₂₈, C₂×C₇⋊D₄, D₄×C₁₄, D₄×C₁₄, C₇×C₄○D₄, C₇×C₄○D₄, C₂₈.D₄, D₄⋊₂Dic₇, C₂₈⋊D₄, D₄⋊D₁₄, C₇×2₊¹⁺⁴, 2₊¹⁺⁴⋊D₇
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, D₁₄, C₂²≀C₂, C₇⋊D₄, C₂²×D₇, D₄⋊₄D₄, C₂×C₇⋊D₄, C₂⁴⋊D₇, 2₊¹⁺⁴⋊D₇

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

Generators in S₅₆

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

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

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

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

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

67 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	7A	7B	7C	8A	8B	14A	14B	14C	14D	···	14AD	28A	···	28R
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	7	7	7	8	8	14	14	14	14	···	14	28	···	28
size	1	1	2	4	4	4	4	56	2	2	4	4	28	28	2	2	2	56	56	2	2	2	4	···	4	4	···	4

67 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

C₇⋊D₄

D₄⋊₄D₄

2₊¹⁺⁴⋊D₇

kernel

2₊¹⁺⁴⋊D₇

C₂₈.D₄

D₄⋊₂Dic₇

C₂₈⋊D₄

D₄⋊D₁₄

C₇×2₊¹⁺⁴

C₇×D₄

C₇×Q₈

C₂²×C₁₄

2₊¹⁺⁴

C₂×D₄

C₄○D₄

D₄

Q₈

C₂³

C₇

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊D₇ ►in GL₆(𝔽₁₁₃)

112	0	0	0	0	0
0	112	0	0	0	0
0	0	112	41	0	0
0	0	22	1	0	0
0	0	0	41	112	2
0	0	1	0	112	1

1	112	0	0	0	0
0	112	0	0	0	0
0	0	0	0	112	0
0	0	0	1	22	91
0	0	112	0	0	0
0	0	112	0	1	112

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	72	0	0
0	0	91	112	0	0
0	0	0	41	112	2
0	0	112	41	112	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	41	112	2
0	0	0	1	22	91
0	0	1	72	0	0
0	0	1	72	1	112

49	47	0	0	0	0
0	30	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

83	47	0	0	0	0
53	30	0	0	0	0
0	0	112	41	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	72	1	112

G:=sub<GL(6,GF(113))| [112,0,0,0,0,0,0,112,0,0,0,0,0,0,112,22,0,1,0,0,41,1,41,0,0,0,0,0,112,112,0,0,0,0,2,1],[1,0,0,0,0,0,112,112,0,0,0,0,0,0,0,0,112,112,0,0,0,1,0,0,0,0,112,22,0,1,0,0,0,91,0,112],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,91,0,112,0,0,72,112,41,41,0,0,0,0,112,112,0,0,0,0,2,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,41,1,72,72,0,0,112,22,0,1,0,0,2,91,0,112],[49,0,0,0,0,0,47,30,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[83,53,0,0,0,0,47,30,0,0,0,0,0,0,112,0,0,0,0,0,41,1,0,72,0,0,0,0,1,1,0,0,0,0,0,112] >;

2₊¹⁺⁴⋊D₇ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes D_7

% in TeX

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

// GroupNames label

G:=SmallGroup(448,775);

// by ID

G=gap.SmallGroup(448,775);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,253,254,570,1684,851,438,102,18822]);

// 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=f*a*f=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f=a*b,d*c*d=f*c*f=a^2*c,c*e=e*c,d*e=e*d,f*d*f=c*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊D7 order 448 = 26·7

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

G = 2₊¹⁺⁴⋊D₇ order 448 = 2⁶·7

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