C2xC2^3.7D4 - GroupNames

G = C₂×C₂³.₇D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₇D₄

direct product, p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂×C₂³.₇D₄, C₂⁴.₄₅D₄, C₂³.₆C₂⁴, C₂⁴.₄₇₀C₂³, 2₊¹⁺⁴.₁₂C₂², (C₂×D₄).₁₄₈D₄, C₂²⋊C₄⋊₂C₂³, C₂³⋊C₄⋊₆C₂², (C₂²×C₄)⋊₈C₂³, C₂³.₂₅(C₂×D₄), (C₂³×C₄)⋊₁₄C₂², (C₂×D₄).₄₀C₂³, (C₂²×C₄).₁₁₄D₄, C₂².₂₆C₂²≀C₂, C₂².₄₀(C₂²×D₄), (C₂×2₊¹⁺⁴).₉C₂, (C₂²×D₄).₃₃₃C₂², C₂².D₄⋊₂₈C₂², (C₂×C₄).₂₆(C₂×D₄), (C₂×C₂³⋊C₄)⋊₁₇C₂, C₂.₆₁(C₂×C₂²≀C₂), (C₂×C₂²⋊C₄)⋊₃₇C₂², (C₂×C₂².D₄)⋊₄₇C₂, SmallGroup(128,1756)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂³ — C₂×C₂³.₇D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂×2₊¹⁺⁴ — C₂×C₂³.₇D₄

Lower central

C₁ — C₂ — C₂³ — C₂×C₂³.₇D₄

Upper central

C₁ — C₂² — C₂⁴ — C₂×C₂³.₇D₄

Jennings

C₁ — C₂ — C₂³ — C₂×C₂³.₇D₄

Generators and relations for C₂×C₂³.₇D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=fbf^-1=bcd, ece^-1=cd=dc, cf=fc, de=ed, df=fd, fef^-1=de^-1 >

Subgroups: 836 in 397 conjugacy classes, 108 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂².D₄, C₂².D₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, 2₊¹⁺⁴, 2₊¹⁺⁴, C₂×C₂³⋊C₄, C₂³.₇D₄, C₂×C₂².D₄, C₂×2₊¹⁺⁴, C₂×C₂³.₇D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂³.₇D₄, C₂×C₂²≀C₂, C₂×C₂³.₇D₄

Smallest permutation representation of C₂×C₂³.₇D₄
►On 32 points

Generators in S₃₂

(1 30)(2 31)(3 32)(4 29)(5 9)(6 10)(7 11)(8 12)(13 23)(14 24)(15 21)(16 22)(17 27)(18 28)(19 25)(20 26)

(2 31)(3 5)(4 10)(6 29)(8 12)(9 32)(14 27)(15 18)(16 22)(17 24)(19 25)(21 28)

(1 11)(2 31)(3 9)(4 29)(5 32)(6 10)(7 30)(8 12)(13 23)(14 27)(15 21)(16 25)(17 24)(18 28)(19 22)(20 26)

(1 7)(2 8)(3 5)(4 6)(9 32)(10 29)(11 30)(12 31)(13 20)(14 17)(15 18)(16 19)(21 28)(22 25)(23 26)(24 27)

(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)

(1 25 7 22)(2 21 8 28)(3 27 5 24)(4 23 6 26)(9 14 32 17)(10 20 29 13)(11 16 30 19)(12 18 31 15)

G:=sub<Sym(32)| (1,30)(2,31)(3,32)(4,29)(5,9)(6,10)(7,11)(8,12)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (2,31)(3,5)(4,10)(6,29)(8,12)(9,32)(14,27)(15,18)(16,22)(17,24)(19,25)(21,28), (1,11)(2,31)(3,9)(4,29)(5,32)(6,10)(7,30)(8,12)(13,23)(14,27)(15,21)(16,25)(17,24)(18,28)(19,22)(20,26), (1,7)(2,8)(3,5)(4,6)(9,32)(10,29)(11,30)(12,31)(13,20)(14,17)(15,18)(16,19)(21,28)(22,25)(23,26)(24,27), (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), (1,25,7,22)(2,21,8,28)(3,27,5,24)(4,23,6,26)(9,14,32,17)(10,20,29,13)(11,16,30,19)(12,18,31,15)>;

G:=Group( (1,30)(2,31)(3,32)(4,29)(5,9)(6,10)(7,11)(8,12)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26), (2,31)(3,5)(4,10)(6,29)(8,12)(9,32)(14,27)(15,18)(16,22)(17,24)(19,25)(21,28), (1,11)(2,31)(3,9)(4,29)(5,32)(6,10)(7,30)(8,12)(13,23)(14,27)(15,21)(16,25)(17,24)(18,28)(19,22)(20,26), (1,7)(2,8)(3,5)(4,6)(9,32)(10,29)(11,30)(12,31)(13,20)(14,17)(15,18)(16,19)(21,28)(22,25)(23,26)(24,27), (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), (1,25,7,22)(2,21,8,28)(3,27,5,24)(4,23,6,26)(9,14,32,17)(10,20,29,13)(11,16,30,19)(12,18,31,15) );

G=PermutationGroup([[(1,30),(2,31),(3,32),(4,29),(5,9),(6,10),(7,11),(8,12),(13,23),(14,24),(15,21),(16,22),(17,27),(18,28),(19,25),(20,26)], [(2,31),(3,5),(4,10),(6,29),(8,12),(9,32),(14,27),(15,18),(16,22),(17,24),(19,25),(21,28)], [(1,11),(2,31),(3,9),(4,29),(5,32),(6,10),(7,30),(8,12),(13,23),(14,27),(15,21),(16,25),(17,24),(18,28),(19,22),(20,26)], [(1,7),(2,8),(3,5),(4,6),(9,32),(10,29),(11,30),(12,31),(13,20),(14,17),(15,18),(16,19),(21,28),(22,25),(23,26),(24,27)], [(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)], [(1,25,7,22),(2,21,8,28),(3,27,5,24),(4,23,6,26),(9,14,32,17),(10,20,29,13),(11,16,30,19),(12,18,31,15)]])

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2O	4A	···	4J	4K	···	4P
order	1	2	2	2	2	···	2	2	···	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	2	2	2	4
type	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	D₄	D₄	D₄	C₂³.₇D₄
kernel	C₂×C₂³.₇D₄	C₂×C₂³⋊C₄	C₂³.₇D₄	C₂×C₂².D₄	C₂×2₊¹⁺⁴	C₂²×C₄	C₂×D₄	C₂⁴	C₂
# reps	1	3	8	3	1	3	6	3	4

Matrix representation of C₂×C₂³.₇D₄ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	4	0	0
0	0	0	1	0	0
0	0	0	0	4	4
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	1	0	0
0	0	0	0	4	0
0	0	0	0	0	4

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

0	4	0	0	0	0
1	0	0	0	0	0
0	0	0	0	2	2
0	0	0	0	0	3
0	0	3	3	0	0
0	0	4	2	0	0

4	0	0	0	0	0
0	1	0	0	0	0
0	0	3	0	0	0
0	0	4	2	0	0
0	0	0	0	3	0
0	0	0	0	0	3

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1,0,0,0,0,0,0,4,0,0,0,0,0,4,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,3,4,0,0,0,0,3,2,0,0,2,0,0,0,0,0,2,3,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂×C₂³.₇D₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3._7D_4

% in TeX

G:=Group("C2xC2^3.7D4");

// GroupNames label

G:=SmallGroup(128,1756);

// by ID

G=gap.SmallGroup(128,1756);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,718,2028]);

// Polycyclic

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

// generators/relations

G = C2×C23.7D4 order 128 = 27

Direct product of C2 and C23.7D4

G = C₂×C₂³.₇D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₇D₄