C2^2.14C2^5 - GroupNames

G = C₂².₁₄C₂⁵ order 128 = 2⁷

10^th central extension by C₂² of C₂⁵

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

Aliases: C₂².₁₄C₂⁵, C₂³.₁₀₈C₂⁴, C₄².₅₉₂C₂³, C₂⁴.₄₇₃C₂³, (C₄×D₄)⋊₈₇C₂², D₄ ○(C₄²⋊C₂), C₄.₄₀(C₂³×C₄), C₂.₁₀(C₂⁴×C₄), (C₄×Q₈)⋊₈₃C₂², Q₈ ○(C₄²⋊C₂), C₄⋊C₄.₅₁₅C₂³, (C₂×C₄).₁₆₀C₂⁴, (C₂×C₄²)⋊₄₀C₂², D₄.₂₄(C₂²×C₄), C₂².₄(C₂³×C₄), Q₈.₂₅(C₂²×C₄), C₄ ○(C₂².₁₁C₂⁴), (C₂×D₄).₄₉₈C₂³, (C₂×Q₈).₄₈₁C₂³, C₂².₁₁C₂⁴⋊₂₅C₂, C₂²⋊C₄.₁₂₇C₂³, C₂.₁(C₂.C₂⁵), C₂³.₁₁₁(C₂²×C₄), (C₂²×C₄).₆₁₇C₂³, (C₂³×C₄).₅₇₇C₂², C₄²⋊C₂⋊₁₀₄C₂², C₄ ○(C₂³.₃₂C₂³), C₄ ○(C₂³.₃₃C₂³), (C₂²×D₄).₅₇₉C₂², (C₂²×Q₈).₄₈₂C₂², C₂³.₃₃C₂³⋊₃₃C₂, C₂³.₃₂C₂³⋊₂₀C₂, C₄⋊C₄ ○(C₄○D₄), (C₄×C₄○D₄)⋊₁₃C₂, C₄○D₄⋊₁₈(C₂×C₄), (C₂×C₄○D₄)⋊₂₄C₄, (C₂×D₄)⋊₅₃(C₂×C₄), C₂²⋊C₄ ○(C₄○D₄), (C₂×Q₈)⋊₄₄(C₂×C₄), C₄⋊C₄ ○(C₄²⋊C₂), (C₂²×C₄)⋊₄₂(C₂×C₄), (C₂×C₄⋊C₄)⋊₁₂₀C₂², C₄○D₄ ○(C₄²⋊C₂), (C₂×C₄²⋊C₂)⋊₅₀C₂, (C₂×C₄).₂₈₃(C₂²×C₄), (C₂²×C₄○D₄).₂₆C₂, (C₂×C₄○D₄).₃₁₇C₂², C₄²⋊C₂ ○(C₄²⋊C₂), (C₂×C₂²⋊C₄).₄₇₆C₂², (C₂×C₄)○(C₂³.₃₂C₂³), C₄²⋊C₂ ○(C₂×C₄○D₄), SmallGroup(128,2160)

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

Derived series

C₁ — C₂ — C₂².₁₄C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂².₁₄C₂⁵

Lower central

C₁ — C₂ — C₂².₁₄C₂⁵

Upper central

C₁ — C₂×C₄ — C₂².₁₄C₂⁵

Jennings

C₁ — C₂² — C₂².₁₄C₂⁵

Generators and relations for C₂².₁₄C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=d²=e²=f²=1, c²=b, g²=a, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, ce=ec, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 940 in 770 conjugacy classes, 684 normal (10 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄²⋊C₂, C₄×C₄○D₄, C₂².₁₁C₂⁴, C₂³.₃₂C₂³, C₂³.₃₃C₂³, C₂²×C₄○D₄, C₂².₁₄C₂⁵
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₂⁴, C₂³×C₄, C₂⁵, C₂⁴×C₄, C₂.C₂⁵, C₂².₁₄C₂⁵

Smallest permutation representation of C₂².₁₄C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

68 conjugacy classes

class	1	2A	2B	2C	2D	···	2Q	4A	4B	4C	4D	4E	···	4AX
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	1	1	1	4
type	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₂.C₂⁵
kernel	C₂².₁₄C₂⁵	C₂×C₄²⋊C₂	C₄×C₄○D₄	C₂².₁₁C₂⁴	C₂³.₃₂C₂³	C₂³.₃₃C₂³	C₂²×C₄○D₄	C₂×C₄○D₄	C₂
# reps	1	6	8	6	2	8	1	32	4

Matrix representation of C₂².₁₄C₂⁵ ►in GL₅(𝔽₅)

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	1	0	0
0	0	0	1	0
0	0	0	0	1

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

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

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

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

C₂².₁₄C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{14}C_2^5

% in TeX

G:=Group("C2^2.14C2^5");

// GroupNames label

G:=SmallGroup(128,2160);

// by ID

G=gap.SmallGroup(128,2160);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,448,477,387,1123,102]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^2=d^2=e^2=f^2=1,c^2=b,g^2=a,a*b=b*a,d*c*d=f*c*f=a*c=c*a,e*d*e=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*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 = C22.14C25 order 128 = 27

10th central extension by C22 of C25

G = C₂².₁₄C₂⁵ order 128 = 2⁷

10^th central extension by C₂² of C₂⁵