C7xC2wrC4 - GroupNames

G = C₇×C₂≀C₄ order 448 = 2⁶·7

Direct product of C₇ and C₂≀C₄

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C₇×C₂≀C₄, C₂⁴⋊₂C₂₈, C₂²⋊C₄⋊₁C₂₈, C₂³⋊C₄⋊₁C₁₄, (C₂³×C₁₄)⋊₁C₄, (C₂×C₂₈).₁₇D₄, C₂³.₁(C₇×D₄), C₄.D₄⋊₅C₁₄, C₂³.₁(C₂×C₂₈), (C₂²×C₁₄).₁D₄, C₂²≀C₂.₁C₁₄, C₁₄.₃₂(C₂³⋊C₄), (D₄×C₁₄).₁₇₄C₂², (C₂×C₄).₁(C₇×D₄), (C₇×C₂³⋊C₄)⋊₇C₂, (C₇×C₂²⋊C₄)⋊₃C₄, C₂.₆(C₇×C₂³⋊C₄), (C₂×D₄).₁(C₂×C₁₄), (C₇×C₂²≀C₂).₃C₂, (C₇×C₄.D₄)⋊₁₂C₂, (C₂²×C₁₄).₈(C₂×C₄), C₂².₁₀(C₇×C₂²⋊C₄), (C₂×C₁₄).₇₃(C₂²⋊C₄), SmallGroup(448,155)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂³ — C₇×C₂≀C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂×D₄ — D₄×C₁₄ — C₇×C₂³⋊C₄ — C₇×C₂≀C₄

Lower central

C₁ — C₂ — C₂² — C₂³ — C₇×C₂≀C₄

Upper central

C₁ — C₁₄ — C₂×C₁₄ — D₄×C₁₄ — C₇×C₂≀C₄

Generators and relations for C₇×C₂≀C₄
G = < a,b,c,d,e,f | a⁷=b²=c²=d²=e²=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf^-1=bcde, cd=dc, ce=ec, fcf^-1=cde, fdf^-1=de=ed, ef=fe >

Subgroups: 258 in 94 conjugacy classes, 26 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₇, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₁₄, C₁₄, C₂²⋊C₄, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, C₂₈, C₂×C₁₄, C₂×C₁₄, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₅₆, C₂×C₂₈, C₂×C₂₈, C₇×D₄, C₂²×C₁₄, C₂²×C₁₄, C₂≀C₄, C₇×C₂²⋊C₄, C₇×C₂²⋊C₄, C₇×M₄(2), D₄×C₁₄, D₄×C₁₄, C₂³×C₁₄, C₇×C₂³⋊C₄, C₇×C₄.D₄, C₇×C₂²≀C₂, C₇×C₂≀C₄
Quotients: C₁, C₂, C₄, C₂², C₇, C₂×C₄, D₄, C₁₄, C₂²⋊C₄, C₂₈, C₂×C₁₄, C₂³⋊C₄, C₂×C₂₈, C₇×D₄, C₂≀C₄, C₇×C₂²⋊C₄, C₇×C₂³⋊C₄, C₇×C₂≀C₄

Smallest permutation representation of C₇×C₂≀C₄
►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)

(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 53)(16 54)(17 55)(18 56)(19 50)(20 51)(21 52)

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

(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 55)(23 56)(24 50)(25 51)(26 52)(27 53)(28 54)

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

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

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

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

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

91 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	7A	···	7F	8A	8B	14A	···	14F	14G	···	14L	14M	···	14AJ	28A	···	28F	28G	···	28X	56A	···	56L
order	1	2	2	2	2	2	2	4	4	4	4	7	···	7	8	8	14	···	14	14	···	14	14	···	14	28	···	28	28	···	28	56	···	56
size	1	1	2	4	4	4	4	4	8	8	8	1	···	1	8	8	1	···	1	2	···	2	4	···	4	4	···	4	8	···	8	8	···	8

91 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	2	2	2	2	4	4	4	4
type	+	+	+	+									+	+			+	+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₇	C₁₄	C₁₄	C₁₄	C₂₈	C₂₈	D₄	D₄	C₇×D₄	C₇×D₄	C₂³⋊C₄	C₂≀C₄	C₇×C₂³⋊C₄	C₇×C₂≀C₄
kernel	C₇×C₂≀C₄	C₇×C₂³⋊C₄	C₇×C₄.D₄	C₇×C₂²≀C₂	C₇×C₂²⋊C₄	C₂³×C₁₄	C₂≀C₄	C₂³⋊C₄	C₄.D₄	C₂²≀C₂	C₂²⋊C₄	C₂⁴	C₂×C₂₈	C₂²×C₁₄	C₂×C₄	C₂³	C₁₄	C₇	C₂	C₁
# reps	1	1	1	1	2	2	6	6	6	6	12	12	1	1	6	6	1	2	6	12

Matrix representation of C₇×C₂≀C₄ ►in GL₄(𝔽₁₁₃) generated by

106	0	0	0
0	106	0	0
0	0	106	0
0	0	0	106

1	0	1	0
0	1	1	0
0	0	112	0
0	0	111	1

0	1	1	0
1	0	1	0
0	0	112	0
0	0	111	1

1	0	0	1
0	1	0	1
0	0	112	0
0	0	0	112

112	0	0	0
0	112	0	0
0	0	112	0
0	0	0	112

0	112	1	112
0	112	0	112
1	1	0	1
0	2	0	1

G:=sub<GL(4,GF(113))| [106,0,0,0,0,106,0,0,0,0,106,0,0,0,0,106],[1,0,0,0,0,1,0,0,1,1,112,111,0,0,0,1],[0,1,0,0,1,0,0,0,1,1,112,111,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,112,0,1,1,0,112],[112,0,0,0,0,112,0,0,0,0,112,0,0,0,0,112],[0,0,1,0,112,112,1,2,1,0,0,0,112,112,1,1] >;

C₇×C₂≀C₄ in GAP, Magma, Sage, TeX

C_7\times C_2\wr C_4

% in TeX

G:=Group("C7xC2wrC4");

// GroupNames label

G:=SmallGroup(448,155);

// by ID

G=gap.SmallGroup(448,155);

# by ID

G:=PCGroup([7,-2,-2,-7,-2,-2,-2,-2,392,421,3923,2951,375,14117]);

// Polycyclic

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

// generators/relations

G = C7×C2≀C4 order 448 = 26·7

Direct product of C7 and C2≀C4

G = C₇×C₂≀C₄ order 448 = 2⁶·7

Direct product of C₇ and C₂≀C₄