D7xC2^2wrC2 - GroupNames

G = D₇×C₂²≀C₂ order 448 = 2⁶·7

Direct product of D₇ and C₂²≀C₂

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

Aliases: D₇×C₂²≀C₂, C₂⁴⋊₉D₁₄, C₂²⋊₅(D₄×D₇), D₁₄⋊₁₁(C₂×D₄), (C₂×D₄)⋊₁₇D₁₄, (C₂×C₂₈)⋊₁C₂³, (D₇×C₂⁴)⋊₂C₂, C₂²⋊D₂₈⋊₈C₂, C₂³⋊D₁₄⋊₂C₂, D₁₄⋊C₄⋊₉C₂², C₂²⋊C₄⋊₂₃D₁₄, (D₄×C₁₄)⋊₅C₂², (C₂²×D₇)⋊₁₄D₄, C₂⁴⋊D₇⋊₅C₂, C₂³⋊₁(C₂²×D₇), (C₂×D₂₈)⋊₁₇C₂², (C₂²×C₁₄)⋊₁C₂³, (C₂³×C₁₄)⋊₈C₂², (C₂×Dic₇)⋊₂C₂³, C₁₄.₅₄(C₂²×D₄), (C₂²×D₇)⋊₂C₂³, (C₂×C₁₄).₁₃₂C₂⁴, (C₂³×D₇)⋊₂₀C₂², C₂³.D₇⋊₁₃C₂², C₂².₁₅₃(C₂³×D₇), (C₂×D₄×D₇)⋊₅C₂, C₂.₂₇(C₂×D₄×D₇), C₇⋊₂(C₂×C₂²≀C₂), (C₂×C₁₄)⋊₅(C₂×D₄), (C₂×C₄×D₇)⋊₅C₂², (D₇×C₂²⋊C₄)⋊₁C₂, (C₂×C₄)⋊₁(C₂²×D₇), (C₇×C₂²≀C₂)⋊₃C₂, (C₂×C₇⋊D₄)⋊₇C₂², (C₇×C₂²⋊C₄)⋊₃C₂², SmallGroup(448,1041)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₄ — D₇×C₂²≀C₂

Chief series

C₁ — C₇ — C₁₄ — C₂×C₁₄ — C₂²×D₇ — C₂³×D₇ — D₇×C₂⁴ — D₇×C₂²≀C₂

Lower central

C₇ — C₂×C₁₄ — D₇×C₂²≀C₂

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for D₇×C₂²≀C₂
G = < a,b,c,d,e,f,g | a⁷=b²=c²=d²=e²=f²=g²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 3916 in 662 conjugacy classes, 131 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₇, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₇, D₇, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₇, C₂₈, D₁₄, D₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₁₄, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²≀C₂, C₂²×D₄, C₂⁵, C₄×D₇, D₂₈, C₂×Dic₇, C₇⋊D₄, C₂×C₂₈, C₇×D₄, C₂²×D₇, C₂²×D₇, C₂²×D₇, C₂²×C₁₄, C₂²×C₁₄, C₂²×C₁₄, C₂×C₂²≀C₂, D₁₄⋊C₄, C₂³.D₇, C₇×C₂²⋊C₄, C₂×C₄×D₇, C₂×D₂₈, D₄×D₇, C₂×C₇⋊D₄, D₄×C₁₄, C₂³×D₇, C₂³×D₇, C₂³×D₇, C₂³×C₁₄, D₇×C₂²⋊C₄, C₂²⋊D₂₈, C₂³⋊D₁₄, C₂⁴⋊D₇, C₇×C₂²≀C₂, C₂×D₄×D₇, D₇×C₂⁴, D₇×C₂²≀C₂
Quotients: C₁, C₂, C₂², D₄, C₂³, D₇, C₂×D₄, C₂⁴, D₁₄, C₂²≀C₂, C₂²×D₄, C₂²×D₇, C₂×C₂²≀C₂, D₄×D₇, C₂³×D₇, C₂×D₄×D₇, D₇×C₂²≀C₂

Smallest permutation representation of D₇×C₂²≀C₂
►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 26)(2 25)(3 24)(4 23)(5 22)(6 28)(7 27)(8 17)(9 16)(10 15)(11 21)(12 20)(13 19)(14 18)(29 52)(30 51)(31 50)(32 56)(33 55)(34 54)(35 53)(36 45)(37 44)(38 43)(39 49)(40 48)(41 47)(42 46)

(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)

(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)

(1 13)(2 14)(3 8)(4 9)(5 10)(6 11)(7 12)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(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 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)

(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 36)(9 37)(10 38)(11 39)(12 40)(13 41)(14 42)(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)

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

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

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

70 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	2N	2O	···	2T	2U	4A	4B	4C	4D	4E	4F	7A	7B	7C	14A	···	14I	14J	···	14AA	14AB	14AC	14AD	28A	···	28I
order	1	2	2	2	2	···	2	2	2	2	2	2	2	···	2	2	4	4	4	4	4	4	7	7	7	14	···	14	14	···	14	14	14	14	28	···	28
size	1	1	1	1	2	···	2	4	7	7	7	7	14	···	14	28	4	4	4	28	28	28	2	2	2	2	···	2	4	···	4	8	8	8	8	···	8

70 irreducible representations

dim

type

image

C₁

C₂

D₄

D₇

D₁₄

D₄×D₇

kernel

D₇×C₂²≀C₂

D₇×C₂²⋊C₄

C₂²⋊D₂₈

C₂³⋊D₁₄

C₂⁴⋊D₇

C₇×C₂²≀C₂

C₂×D₄×D₇

D₇×C₂⁴

C₂²×D₇

C₂²≀C₂

C₂²⋊C₄

C₂×D₄

C₂⁴

C₂²

# reps

Matrix representation of D₇×C₂²≀C₂ ►in GL₆(𝔽₂₉)

26	1	0	0	0	0
23	21	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

21	28	0	0	0	0
5	8	0	0	0	0
0	0	28	0	0	0
0	0	0	28	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	1	0	0	0
0	0	0	28	0	0
0	0	0	0	1	0
0	0	0	0	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	1	0
0	0	0	0	0	28

1	0	0	0	0	0
0	1	0	0	0	0
0	0	28	0	0	0
0	0	0	28	0	0
0	0	0	0	28	0
0	0	0	0	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	0	0	0	28

28	0	0	0	0	0
0	28	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(29))| [26,23,0,0,0,0,1,21,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[21,5,0,0,0,0,28,8,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₇×C₂²≀C₂ in GAP, Magma, Sage, TeX

D_7\times C_2^2\wr C_2

% in TeX

G:=Group("D7xC2^2wrC2");

// GroupNames label

G:=SmallGroup(448,1041);

// by ID

G=gap.SmallGroup(448,1041);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-7,570,185,18822]);

// Polycyclic

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

// generators/relations

G = D7×C22≀C2 order 448 = 26·7

Direct product of D7 and C22≀C2

G = D₇×C₂²≀C₂ order 448 = 2⁶·7

Direct product of D₇ and C₂²≀C₂