C2^5.85C2^2 - GroupNames

G = C₂⁵.₈₅C₂² order 128 = 2⁷

6^th non-split extension by C₂⁵ of C₂² acting via C₂²/C₂=C₂

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

Aliases: C₂⁵.₈₅C₂², C₂³.₁₆₂C₂⁴, C₂⁴.₅₂₈C₂³, (C₂³×C₄)⋊₁₇C₄, C₄ ○(C₂⁴⋊₃C₄), (C₂⁴×C₄).₁₀C₂, (C₂×C₄²)⋊₂C₂², C₂⁴.₁₂₂(C₂×C₄), (C₂²×C₄).₇₆₈D₄, C₂³.₅₉₇(C₂×D₄), C₄ ○(C₂³.₇Q₈), C₂⁴⋊₃C₄.₁₈C₂, C₄ ○(C₂³.₃₄D₄), C₂².₅₃(C₂³×C₄), C₂².₆₂(C₂²×D₄), C₂²⋊₄(C₄²⋊C₂), C₂³.₂₁₃(C₄○D₄), C₂³.₃₄D₄⋊₆₅C₂, C₂³.₂₀₇(C₂²×C₄), (C₂³×C₄).₆₄₃C₂², (C₂²×C₄).₄₄₀C₂³, C₂³.₇Q₈⋊₁₁₈C₂, C₂.C₄²⋊₅₀C₂², C₂.₁(C₂².₁₉C₂⁴), (C₄×C₂²⋊C₄)⋊₄C₂, (C₂×C₄⋊C₄)⋊₉₇C₂², C₄.₉₆(C₂×C₂²⋊C₄), (C₂×C₄)⋊₁₃(C₂²⋊C₄), (C₂×C₄)○(C₂⁴⋊₃C₄), (C₂×C₄²⋊C₂)⋊₄C₂, (C₂×C₄).₁₅₅₄(C₂×D₄), C₂.₈(C₂×C₄²⋊C₂), C₂².₅₅(C₂×C₄○D₄), C₂.₇(C₂²×C₂²⋊C₄), (C₂²×C₄).₄₈₉(C₂×C₄), (C₂×C₄).₆₂₂(C₂²×C₄), C₂².₇₃(C₂×C₂²⋊C₄), (C₂×C₄)○(C₂³.₇Q₈), (C₂²×C₄)○(C₂⁴⋊₃C₄), (C₂×C₂²⋊C₄).₄₁₁C₂², (C₂²×C₄)○(C₂³.₇Q₈), SmallGroup(128,1012)

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

Derived series

C₁ — C₂² — C₂⁵.₈₅C₂²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂⁴×C₄ — C₂⁵.₈₅C₂²

Lower central

C₁ — C₂² — C₂⁵.₈₅C₂²

Upper central

C₁ — C₂²×C₄ — C₂⁵.₈₅C₂²

Jennings

C₁ — C₂³ — C₂⁵.₈₅C₂²

Generators and relations for C₂⁵.₈₅C₂²
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=1, f²=e, g²=c, ab=ba, faf^-1=ac=ca, ad=da, ae=ea, ag=ga, bc=cb, fbf^-1=bd=db, be=eb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 908 in 544 conjugacy classes, 196 normal (12 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂⁴, C₂⁴, C₂.C₄², C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂³×C₄, C₂³×C₄, C₂³×C₄, C₂⁵, C₄×C₂²⋊C₄, C₂⁴⋊₃C₄, C₂³.₇Q₈, C₂³.₃₄D₄, C₂×C₄²⋊C₂, C₂⁴×C₄, C₂⁵.₈₅C₂²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₄○D₄, C₂⁴, C₂×C₂²⋊C₄, C₄²⋊C₂, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂²×C₂²⋊C₄, C₂×C₄²⋊C₂, C₂².₁₉C₂⁴, C₂⁵.₈₅C₂²

Smallest permutation representation of C₂⁵.₈₅C₂²
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

56 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4H	4I	···	4T	4U	···	4AJ
order	1	2	···	2	2	···	2	4	···	4	4	···	4	4	···	4
size	1	1	···	1	2	···	2	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2
type	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₄	D₄	C₄○D₄
kernel	C₂⁵.₈₅C₂²	C₄×C₂²⋊C₄	C₂⁴⋊₃C₄	C₂³.₇Q₈	C₂³.₃₄D₄	C₂×C₄²⋊C₂	C₂⁴×C₄	C₂³×C₄	C₂²×C₄	C₂³
# reps	1	4	2	4	2	2	1	16	8	16

Matrix representation of C₂⁵.₈₅C₂² ►in GL₅(𝔽₅)

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

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

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

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

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

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

C₂⁵.₈₅C₂² in GAP, Magma, Sage, TeX

C_2^5._{85}C_2^2

% in TeX

G:=Group("C2^5.85C2^2");

// GroupNames label

G:=SmallGroup(128,1012);

// by ID

G=gap.SmallGroup(128,1012);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,80]);

// Polycyclic

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

// generators/relations

G = C25.85C22 order 128 = 27

6th non-split extension by C25 of C22 acting via C22/C2=C2

G = C₂⁵.₈₅C₂² order 128 = 2⁷

6^th non-split extension by C₂⁵ of C₂² acting via C₂²/C₂=C₂