C4^2.265C2^3 - GroupNames

G = C₄².₂₆₅C₂³ order 128 = 2⁷

126^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²

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

Aliases: C₄².₂₆₅C₂³, (C₂×C₈)⋊₂₆D₄, C₈⋊₆D₄⋊₃₂C₂, C₈⋊₉D₄⋊₃₀C₂, C₄⋊C₈⋊₅₀C₂², (C₄×C₈)⋊₅₆C₂², C₄.₁₃₂(C₄×D₄), C₈.₁₂₇(C₂×D₄), C₂²≀C₂.₃C₄, C₄⋊D₄.₁₇C₄, C₂⁴.₈₀(C₂×C₄), C₂².₄₅(C₄×D₄), C₈⋊C₄⋊₅₉C₂², C₂²⋊Q₈.₁₇C₄, C₂²⋊C₈⋊₄₄C₂², (C₂×C₈).₄₀₂C₂³, (C₂×C₄).₆₄₉C₂⁴, (C₂²×C₈)⋊₅₁C₂², (C₄×D₄).₅₃C₂², C₄.₁₉₅(C₂²×D₄), C₈○₂M₄(2)⋊₃₁C₂, C₂⁴.₄C₄⋊₃₂C₂, C₂³.₃₃(C₂²×C₄), C₂².D₄.₃C₄, C₂.₁₄(Q₈○M₄(2)), (C₂×M₄(2))⋊₇₇C₂², (C₂²×M₄(2))⋊₂₅C₂, C₂².₁₇₆(C₂³×C₄), (C₂³×C₄).₅₂₆C₂², (C₂²×C₄).₉₁₆C₂³, C₂².₁₉C₂⁴.₁₀C₂, C₄².₆C₂²⋊₃₀C₂, C₄²⋊C₂.₂₉₃C₂², C₂.₄₇(C₂×C₄×D₄), (C₂×C₈○D₄)⋊₂₀C₂, C₄⋊C₄.₁₁₄(C₂×C₄), C₄.₃₀₀(C₂×C₄○D₄), (C₂×D₄).₁₃₆(C₂×C₄), (C₂×C₄).₁₄₁₂(C₂×D₄), C₂²⋊C₄.₁₅(C₂×C₄), (C₂×C₄).₆₆(C₂²×C₄), (C₂×Q₈).₁₅₃(C₂×C₄), (C₂²×C₈)⋊C₂⋊₂₈C₂, (C₂×C₄).₆₈₂(C₄○D₄), (C₂²×C₄).₃₄₀(C₂×C₄), (C₂×C₄○D₄).₂₈₄C₂², SmallGroup(128,1662)

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

Derived series

C₁ — C₂² — C₄².₂₆₅C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×C₈ — C₂²×M₄(2) — C₄².₂₆₅C₂³

Lower central

C₁ — C₂² — C₄².₂₆₅C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₆₅C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₆₅C₂³

Generators and relations for C₄².₂₆₅C₂³
G = < a,b,c,d,e | a⁴=b⁴=e²=1, c²=b², d²=a²b, ab=ba, cac^-1=a^-1b², dad^-1=eae=ab², bc=cb, bd=db, be=eb, dcd^-1=a²c, ce=ec, ede=b²d >

Subgroups: 380 in 246 conjugacy classes, 140 normal (30 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂²×C₈, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂×M₄(2), C₈○D₄, C₂³×C₄, C₂×C₄○D₄, C₈○₂M₄(2), C₂⁴.₄C₄, (C₂²×C₈)⋊C₂, C₄².₆C₂², C₈⋊₉D₄, C₈⋊₆D₄, C₂².₁₉C₂⁴, C₂²×M₄(2), C₂×C₈○D₄, C₄².₂₆₅C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²×C₄, C₂×D₄, C₄○D₄, C₂⁴, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄×D₄, Q₈○M₄(2), C₄².₂₆₅C₂³

Smallest permutation representation of C₄².₂₆₅C₂³
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	···	4N	8A	···	8H	8I	···	8T
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8	8	···	8
size	1	1	1	1	2	2	4	4	4	4	1	1	1	1	2	2	4	···	4	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+	+	+	+	+	+	+					+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	C₄○D₄	Q₈○M₄(2)
kernel	C₄².₂₆₅C₂³	C₈○₂M₄(2)	C₂⁴.₄C₄	(C₂²×C₈)⋊C₂	C₄².₆C₂²	C₈⋊₉D₄	C₈⋊₆D₄	C₂².₁₉C₂⁴	C₂²×M₄(2)	C₂×C₈○D₄	C₂²≀C₂	C₄⋊D₄	C₂²⋊Q₈	C₂².D₄	C₂×C₈	C₂×C₄	C₂
# reps	1	1	1	1	1	4	4	1	1	1	4	4	4	4	4	4	4

Matrix representation of C₄².₂₆₅C₂³ ►in GL₆(𝔽₁₇)

0	1	0	0	0	0
16	0	0	0	0	0
0	0	13	0	16	2
0	0	0	0	2	0
0	0	0	9	0	0
0	0	1	13	15	4

16	0	0	0	0	0
0	16	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	13	0
0	0	0	0	0	13

16	0	0	0	0	0
0	1	0	0	0	0
0	0	13	0	0	0
0	0	0	4	0	0
0	0	0	0	13	0
0	0	1	0	13	4

0	13	0	0	0	0
4	0	0	0	0	0
0	0	0	0	1	0
0	0	2	0	9	16
0	0	13	0	0	0
0	0	15	4	2	0

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	16	0
0	0	4	0	0	16

G:=sub<GL(6,GF(17))| [0,16,0,0,0,0,1,0,0,0,0,0,0,0,13,0,0,1,0,0,0,0,9,13,0,0,16,2,0,15,0,0,2,0,0,4],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,13,0,0,1,0,0,0,4,0,0,0,0,0,0,13,13,0,0,0,0,0,4],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,2,13,15,0,0,0,0,0,4,0,0,1,9,0,2,0,0,0,16,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16] >;

C₄².₂₆₅C₂³ in GAP, Magma, Sage, TeX

C_4^2._{265}C_2^3

% in TeX

G:=Group("C4^2.265C2^3");

// GroupNames label

G:=SmallGroup(128,1662);

// by ID

G=gap.SmallGroup(128,1662);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,184,2019,124]);

// Polycyclic

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

// generators/relations

G = C42.265C23 order 128 = 27

126th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₆₅C₂³ order 128 = 2⁷

126^th non-split extension by C₄² of C₂³ acting via C₂³/C₂=C₂²