C2xC2^3.C2^3 - GroupNames

G = C₂×C₂³.C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.C₂³

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

Aliases: C₂×C₂³.C₂³, C₂³.₂C₂⁴, C₂⁴.₄₆₉C₂³, (C₂³×C₄)⋊₁₁C₄, (C₂²×D₄)⋊₂₅C₄, C₂⁴.₃₃(C₂×C₄), (C₂²×Q₈)⋊₁₉C₄, C₂³⋊C₄⋊₁₁C₂², C₂³.₆₃₆(C₂×D₄), (C₂²×C₄).₇₇₄D₄, (C₂×D₄).₃₅₀C₂³, C₂²⋊C₄.₆₆C₂³, C₂².₁₁(C₂³×C₄), C₂³.₅₇(C₂²×C₄), C₂².₂₄(C₂²×D₄), C₄²⋊C₂⋊₇₂C₂², C₄ ○(C₂³.C₂³), (C₂³×C₄).₅₀₈C₂², (C₂²×C₄).₈₉₄C₂³, (C₂²×D₄).₅₄₇C₂², C₄ ○(C₂×C₂³⋊C₄), (C₂×C₄○D₄)⋊₁₇C₄, (C₂×D₄)⋊₄₄(C₂×C₄), (C₂×C₄)○(C₂³⋊C₄), (C₂²×C₄)⋊₈(C₂×C₄), (C₂×Q₈)⋊₃₅(C₂×C₄), (C₂×C₂³⋊C₄)⋊₁₉C₂, C₄.₆₈(C₂×C₂²⋊C₄), (C₂×C₄).₁₃₉₈(C₂×D₄), (C₂×C₄²⋊C₂)⋊₃₉C₂, (C₂×C₄).₁₀₅(C₂²×C₄), (C₂²×C₄○D₄).₁₆C₂, C₂.₂₅(C₂²×C₂²⋊C₄), (C₂×C₄).₂₇₈(C₂²⋊C₄), (C₂×C₄○D₄).₂₇₀C₂², C₂².₁₃₅(C₂×C₂²⋊C₄), (C₂×C₂²⋊C₄).₄₇₅C₂², SmallGroup(128,1614)

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

Derived series

C₁ — C₂² — C₂×C₂³.C₂³

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₂²×C₄○D₄ — C₂×C₂³.C₂³

Lower central

C₁ — C₂ — C₂² — C₂×C₂³.C₂³

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂×C₂³.C₂³

Jennings

C₁ — C₂ — C₂³ — C₂×C₂³.C₂³

Generators and relations for C₂×C₂³.C₂³
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=f²=1, e²=c, g²=d, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ebe^-1=bd=db, bf=fb, bg=gb, fcf=cd=dc, ce=ec, cg=gc, de=ed, df=fd, dg=gd, fef=bde, eg=ge, fg=gf >

Subgroups: 700 in 394 conjugacy classes, 172 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂³⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₂³⋊C₄, C₂³.C₂³, C₂×C₄²⋊C₂, C₂²×C₄○D₄, C₂×C₂³.C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂³.C₂³, C₂²×C₂²⋊C₄, C₂×C₂³.C₂³

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

Generators in S₃₂

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

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4J	4K	···	4AD
order	1	2	2	2	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	4	4	1	1	1	1	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₂³.C₂³
kernel	C₂×C₂³.C₂³	C₂×C₂³⋊C₄	C₂³.C₂³	C₂×C₄²⋊C₂	C₂²×C₄○D₄	C₂³×C₄	C₂²×D₄	C₂²×Q₈	C₂×C₄○D₄	C₂²×C₄	C₂
# reps	1	4	8	2	1	4	2	2	8	8	4

Matrix representation of C₂×C₂³.C₂³ ►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

4	0	0	0	0	0
0	4	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

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

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

1	0	0	0	0	0
0	1	0	0	0	0
0	0	2	0	0	0
0	0	0	2	0	0
0	0	0	0	2	0
0	0	0	0	0	2

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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[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],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,3,4,0,0,1,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,2,2,0,0,0,0,1,3,0,0,0,0,0,0,2,2,0,0,0,0,1,3],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,2] >;

C₂×C₂³.C₂³ in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_2^3

% in TeX

G:=Group("C2xC2^3.C2^3");

// GroupNames label

G:=SmallGroup(128,1614);

// by ID

G=gap.SmallGroup(128,1614);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,352,2804,2028]);

// Polycyclic

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

// generators/relations

G = C2×C23.C23 order 128 = 27

Direct product of C2 and C23.C23

G = C₂×C₂³.C₂³ order 128 = 2⁷

Direct product of C₂ and C₂³.C₂³