D7xES+(2,2) - GroupNames

G = D₇×2₊¹⁺⁴ order 448 = 2⁶·7

Direct product of D₇ and 2₊¹⁺⁴

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D₇×2₊¹⁺⁴, D₂₈⋊₁₃C₂³, C₁₄.₁₅C₂⁵, C₂₈.₅₀C₂⁴, D₁₄.₁₄C₂⁴, Dic₁₄⋊₁₂C₂³, Dic₇.₁₀C₂⁴, C₄○D₄⋊₁₀D₁₄, (C₂×D₄)⋊₃₁D₁₄, (C₂×C₂₈)⋊₂C₂³, (C₄×D₇)⋊₃C₂³, C₇⋊D₄⋊₆C₂³, D₄⋊₆D₁₄⋊₉C₂, Q₈⋊₉(C₂²×D₇), (C₇×D₄)⋊₁₁C₂³, (D₄×D₇)⋊₁₇C₂², D₄⋊₁₀(C₂²×D₇), (C₂×C₁₄).₆C₂⁴, D₄⋊₈D₁₄⋊₁₁C₂, (Q₈×D₇)⋊₂₀C₂², (C₇×Q₈)⋊₁₀C₂³, C₄.₄₇(C₂³×D₇), C₂.₁₆(D₇×C₂⁴), C₂³⋊₂(C₂²×D₇), C₇⋊₃(C₂×2₊¹⁺⁴), C₄○D₂₈⋊₁₂C₂², (D₄×C₁₄)⋊₂₅C₂², (C₂×D₂₈)⋊₄₀C₂², (C₂²×C₁₄)⋊₂C₂³, (C₂×Dic₇)⋊₆C₂³, (C₂²×D₇)⋊₆C₂³, D₄⋊₂D₇⋊₁₅C₂², C₂².₃(C₂³×D₇), Q₈⋊₂D₇⋊₁₅C₂², (C₂³×D₇)⋊₁₉C₂², (C₇×2₊¹⁺⁴)⋊₄C₂, (C₂×D₄×D₇)⋊₂₉C₂, (D₇×C₄○D₄)⋊₇C₂, (C₂×C₄×D₇)⋊₃₆C₂², (C₂×C₄)⋊₂(C₂²×D₇), (C₇×C₄○D₄)⋊₁₀C₂², (C₂×C₇⋊D₄)⋊₃₂C₂², SmallGroup(448,1379)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₄ — D₇×2₊¹⁺⁴

Chief series

C₁ — C₇ — C₁₄ — D₁₄ — C₂²×D₇ — C₂³×D₇ — C₂×D₄×D₇ — D₇×2₊¹⁺⁴

Lower central

C₇ — C₁₄ — D₇×2₊¹⁺⁴

Upper central

C₁ — C₂ — 2₊¹⁺⁴

Generators and relations for D₇×2₊¹⁺⁴
G = < a,b,c,d,e,f | a⁷=b²=c⁴=d²=f²=1, e²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=c²e >

Subgroups: 4084 in 898 conjugacy classes, 445 normal (8 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, D₇, D₇, C₁₄, C₁₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, Dic₁₄, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₇×Q₈, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂×2₊¹⁺⁴, C₂×C₄×D₇, C₂×D₂₈, C₄○D₂₈, D₄×D₇, D₄⋊₂D₇, Q₈×D₇, Q₈⋊₂D₇, C₂×C₇⋊D₄, D₄×C₁₄, C₇×C₄○D₄, C₂³×D₇, C₂×D₄×D₇, D₄⋊₆D₁₄, D₇×C₄○D₄, D₄⋊₈D₁₄, C₇×2₊¹⁺⁴, D₇×2₊¹⁺⁴
Quotients: C₁, C₂, C₂², C₂³, D₇, C₂⁴, D₁₄, 2₊¹⁺⁴, C₂⁵, C₂²×D₇, C₂×2₊¹⁺⁴, C₂³×D₇, D₇×C₂⁴, D₇×2₊¹⁺⁴

Smallest permutation representation of D₇×2₊¹⁺⁴
►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 12)(2 11)(3 10)(4 9)(5 8)(6 14)(7 13)(15 24)(16 23)(17 22)(18 28)(19 27)(20 26)(21 25)(29 38)(30 37)(31 36)(32 42)(33 41)(34 40)(35 39)(43 52)(44 51)(45 50)(46 56)(47 55)(48 54)(49 53)

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

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

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

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

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

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

85 conjugacy classes

class	1	2A	2B	···	2J	2K	2L	2M	···	2U	4A	···	4F	4G	···	4L	7A	7B	7C	14A	14B	14C	14D	···	14AD	28A	···	28R
order	1	2	2	···	2	2	2	2	···	2	4	···	4	4	···	4	7	7	7	14	14	14	14	···	14	28	···	28
size	1	1	2	···	2	7	7	14	···	14	2	···	2	14	···	14	2	2	2	2	2	2	4	···	4	4	···	4

85 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₇	D₁₄	D₁₄	2₊¹⁺⁴	D₇×2₊¹⁺⁴
kernel	D₇×2₊¹⁺⁴	C₂×D₄×D₇	D₄⋊₆D₁₄	D₇×C₄○D₄	D₄⋊₈D₁₄	C₇×2₊¹⁺⁴	2₊¹⁺⁴	C₂×D₄	C₄○D₄	D₇	C₁
# reps	1	9	9	6	6	1	3	27	18	2	3

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

0	1	0	0	0	0
28	3	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	28	0	0	0	0
28	0	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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	28	0
0	0	1	1	1	2
0	0	1	0	0	0
0	0	28	28	0	28

1	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	28	0
0	0	28	28	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	0	1	0	0
0	0	28	0	0	0
0	0	28	28	28	27
0	0	1	0	1	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	28	28	28	27
0	0	0	0	0	1

G:=sub<GL(6,GF(29))| [0,28,0,0,0,0,1,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,28,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,1,28,0,0,0,1,0,28,0,0,28,1,0,0,0,0,0,2,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,28,0,0,0,1,0,28,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,28,28,1,0,0,1,0,28,0,0,0,0,0,28,1,0,0,0,0,27,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,28,0,0,0,1,0,28,0,0,0,0,0,28,0,0,0,0,0,27,1] >;

D₇×2₊¹⁺⁴ in GAP, Magma, Sage, TeX

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

% in TeX

G:=Group("D7xES+(2,2)");

// GroupNames label

G:=SmallGroup(448,1379);

// by ID

G=gap.SmallGroup(448,1379);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,297,851,18822]);

// Polycyclic

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

// generators/relations

G = D7×2+ 1+4 order 448 = 26·7

Direct product of D7 and 2+ 1+4

G = D₇×2₊¹⁺⁴ order 448 = 2⁶·7

Direct product of D₇ and 2₊¹⁺⁴