C4^2.259C2^3 - GroupNames

G = C₄².₂₅₉C₂³ order 128 = 2⁷

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

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

Aliases: C₄².₂₅₉C₂³, C₄⋊C₈⋊₄₉C₂², (C₄×C₈)⋊₅₃C₂², C₂⁴.₇₇(C₂×C₄), C₈⋊C₄⋊₅₅C₂², (C₄×M₄(2))⋊₃₀C₂, (C₂×C₄).₆₄₀C₂⁴, C₄².₂₀₃(C₂×C₄), (C₂×C₈).₃₉₈C₂³, C₄⋊M₄(2)⋊₃₂C₂, C₄²⋊C₂.₂₉C₄, C₂³.₉₉(C₂²×C₄), C₄.₆₆(C₄²⋊C₂), C₂²⋊C₈.₁₃₇C₂², C₂.₁₀(Q₈○M₄(2)), C₂⁴.₄C₄.₂₃C₂, (C₂²×C₄).₉₁₁C₂³, (C₂³×C₄).₅₂₂C₂², C₂².₁₆₈(C₂³×C₄), (C₂×C₄²).₇₅₄C₂², C₄².₇C₂²⋊₁₉C₂, C₄²⋊C₂.₂₈₉C₂², C₂².₃₈(C₄²⋊C₂), (C₂×M₄(2)).₃₄₂C₂², (C₂×C₄⋊C₄).₆₈C₄, C₄⋊C₄.₂₁₆(C₂×C₄), C₄.₂₉₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₄₅C₄, C₂²⋊C₄.₆₇(C₂×C₄), (C₂×C₄).₆₇₈(C₄○D₄), (C₂×C₄).₂₅₆(C₂²×C₄), (C₂²×C₄).₃₃₄(C₂×C₄), C₂.₄₀(C₂×C₄²⋊C₂), (C₂×C₄²⋊C₂).₅₇C₂, SmallGroup(128,1653)

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

Derived series

C₁ — C₂² — C₄².₂₅₉C₂³

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄² — C₂×C₄²⋊C₂ — C₄².₂₅₉C₂³

Lower central

C₁ — C₂² — C₄².₂₅₉C₂³

Upper central

C₁ — C₂×C₄ — C₄².₂₅₉C₂³

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₄².₂₅₉C₂³

Generators and relations for C₄².₂₅₉C₂³
G = < a,b,c,d,e | a⁴=b⁴=d²=e²=1, c²=b, ab=ba, cac^-1=a^-1b², dad=ab², ae=ea, bc=cb, bd=db, be=eb, dcd=a²b²c, ece=b²c, de=ed >

Subgroups: 284 in 196 conjugacy classes, 132 normal (14 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₄×C₈, C₈⋊C₄, C₂²⋊C₈, C₄⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂×M₄(2), C₂³×C₄, C₄×M₄(2), C₂⁴.₄C₄, C₄⋊M₄(2), C₄².₇C₂², C₂×C₄²⋊C₂, C₄².₂₅₉C₂³
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, C₂²×C₄, C₄○D₄, C₂⁴, C₄²⋊C₂, C₂³×C₄, C₂×C₄○D₄, C₂×C₄²⋊C₂, Q₈○M₄(2), C₄².₂₅₉C₂³

Smallest permutation representation of C₄².₂₅₉C₂³
►On 32 points

Generators in S₃₂

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

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

(2 32)(4 26)(6 28)(8 30)(9 24)(10 14)(11 18)(12 16)(13 20)(15 22)(17 21)(19 23)

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

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

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

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

44 conjugacy classes

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

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	4
type	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄○D₄	Q₈○M₄(2)
kernel	C₄².₂₅₉C₂³	C₄×M₄(2)	C₂⁴.₄C₄	C₄⋊M₄(2)	C₄².₇C₂²	C₂×C₄²⋊C₂	C₂×C₂²⋊C₄	C₂×C₄⋊C₄	C₄²⋊C₂	C₂×C₄	C₂
# reps	1	2	2	2	8	1	4	4	8	8	4

Matrix representation of C₄².₂₅₉C₂³ ►in GL₆(𝔽₁₇)

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

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

4	9	0	0	0	0
4	13	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	4	0	0	0
0	0	0	4	0	0

1	0	0	0	0	0
1	16	0	0	0	0
0	0	1	0	0	0
0	0	0	16	0	0
0	0	0	0	1	0
0	0	0	0	0	16

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	0	0	0	16

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

C₄².₂₅₉C₂³ in GAP, Magma, Sage, TeX

C_4^2._{259}C_2^3

% in TeX

G:=Group("C4^2.259C2^3");

// GroupNames label

G:=SmallGroup(128,1653);

// by ID

G=gap.SmallGroup(128,1653);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,723,100,2019,124]);

// Polycyclic

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

// generators/relations

G = C42.259C23 order 128 = 27

120th non-split extension by C42 of C23 acting via C23/C2=C22

G = C₄².₂₅₉C₂³ order 128 = 2⁷

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