C2xC2.C2^5 - GroupNames

G = C₂×C₂.C₂⁵ order 128 = 2⁷

Direct product of C₂ and C₂.C₂⁵

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂×C₂.C₂⁵, C₂.₆C₂⁶, C₄.₂₄C₂⁵, D₄.₁₂C₂⁴, C₂².₂C₂⁵, Q₈.₁₂C₂⁴, C₂³.₉₈C₂⁴, C₂⁴.₅₁₉C₂³, 2₊¹⁺⁴⋊₁₃C₂², 2_-¹⁺⁴⋊₁₂C₂², C₄○D₄⋊₁₀C₂³, (C₂×D₄)⋊₂₃C₂³, (C₂×Q₈)⋊₂₄C₂³, (C₂×C₄)○2_-¹⁺⁴, C₄ ○(C₂×2_-¹⁺⁴), C₄ ○(C₂×2₊¹⁺⁴), (C₂×C₄)○2₊¹⁺⁴, C₄ ○(C₂.C₂⁵), (C₂×C₄).₆₁₉C₂⁴, (C₂³×C₄)⋊₅₄C₂², (C₂²×C₄)⋊₂₀C₂³, (C₂²×D₄)⋊₆₇C₂², (C₂²×Q₈)⋊₇₂C₂², (C₂×2₊¹⁺⁴)⋊₁₇C₂, (C₂×2_-¹⁺⁴)⋊₁₅C₂, D₄ ○(C₂×C₄○D₄), Q₈ ○(C₂×C₄○D₄), (C₂×D₄)○(C₄○D₄), (C₂×Q₈)○(C₄○D₄), C₄○D₄ ○₂(C₄○D₄), (C₂×C₄○D₄)⋊₈₁C₂², (C₂²×C₄○D₄)⋊₃₀C₂, (C₂×C₄)○(C₂×2_-¹⁺⁴), C₄○D₄ ○(C₂×C₄○D₄), (C₂×Q₈)○(C₂×C₄○D₄), (C₂×C₄○D₄)○(C₂×C₄○D₄), SmallGroup(128,2325)

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

Derived series

C₁ — C₂ — C₂×C₂.C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂×C₂.C₂⁵

Lower central

C₁ — C₂ — C₂×C₂.C₂⁵

Upper central

C₁ — C₂×C₄ — C₂×C₂.C₂⁵

Jennings

C₁ — C₂ — C₂×C₂.C₂⁵

Generators and relations for C₂×C₂.C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, dcd=fcf=bc=cb, ede=bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 3308 in 2978 conjugacy classes, 2828 normal (5 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, 2₊¹⁺⁴, 2_-¹⁺⁴, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂×2_-¹⁺⁴, C₂.C₂⁵, C₂×C₂.C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₂⁴, C₂⁵, C₂.C₂⁵, C₂⁶, C₂×C₂.C₂⁵

Smallest permutation representation of C₂×C₂.C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

(5 7)(6 8)(21 23)(22 24)(25 27)(26 28)(29 31)(30 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)

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

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

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

68 conjugacy classes

class	1	2A	2B	2C	2D	···	2AG	4A	4B	4C	4D	4E	···	4AH
order	1	2	2	2	2	···	2	4	4	4	4	4	···	4
size	1	1	1	1	2	···	2	1	1	1	1	2	···	2

68 irreducible representations

dim	1	1	1	1	1	4
type	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂.C₂⁵
kernel	C₂×C₂.C₂⁵	C₂²×C₄○D₄	C₂×2₊¹⁺⁴	C₂×2_-¹⁺⁴	C₂.C₂⁵	C₂
# reps	1	15	10	6	32	4

Matrix representation of C₂×C₂.C₂⁵ ►in GL₅(𝔽₅)

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

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

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

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

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

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

4	0	0	0	0
0	3	0	0	0
0	0	3	0	0
0	0	0	3	0
0	0	0	0	3

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

C₂×C₂.C₂⁵ in GAP, Magma, Sage, TeX

C_2\times C_2.C_2^5

% in TeX

G:=Group("C2xC2.C2^5");

// GroupNames label

G:=SmallGroup(128,2325);

// by ID

G=gap.SmallGroup(128,2325);

# by ID

G:=PCGroup([7,-2,2,2,2,2,2,-2,925,723,2019,172]);

// Polycyclic

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

// generators/relations

G = C2×C2.C25 order 128 = 27

Direct product of C2 and C2.C25

G = C₂×C₂.C₂⁵ order 128 = 2⁷

Direct product of C₂ and C₂.C₂⁵