C2^2.48C2^5 - GroupNames

G = C₂².₄₈C₂⁵ order 128 = 2⁷

29^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂².₄₈C₂⁵, C₂⁴.₄₈₈C₂³, C₂³.₁₂₀C₂⁴, C₄².₅₄₉C₂³, C₄.₁₈₄2₊¹⁺⁴, (C₄×D₄)⋊₉₈C₂², C₂³⋊₃(C₄○D₄), (C₂×C₄).₅₀C₂⁴, (C₄×Q₈)⋊₃₃C₂², C₄ ○₂(C₂³⋊₃D₄), C₄⋊D₄⋊₆₆C₂², C₂³⋊₃D₄⋊₁₉C₂, C₄⋊C₄.₂₈₈C₂³, (C₂³×C₄)⋊₃₃C₂², (C₂×C₄²)⋊₄₈C₂², C₄ ○₂(C₂³⋊₂Q₈), C₂³⋊₂Q₈⋊₁₁C₂, C₂²⋊Q₈⋊₈₀C₂², C₂²≀C₂⋊₂₉C₂², (C₂×D₄).₂₉₄C₂³, C₄.₄D₄⋊₆₈C₂², C₂²⋊C₄.₇₈C₂³, (C₂×Q₈).₂₇₈C₂³, C₄².C₂⋊₄₅C₂², C₄ ○₂(C₂².₃₂C₂⁴), C₂².₁₉C₂⁴⋊₁₅C₂, C₄²⋊₂C₂⋊₂₆C₂², C₄²⋊C₂⋊₂₈C₂², C₂².₃₂C₂⁴⋊₂₉C₂, C₂.₇(C₂.C₂⁵), C₂.₁₃(C₂×2₊¹⁺⁴), (C₂²×C₄).₁₁₈₇C₂³, (C₂²×D₄).₅₈₇C₂², C₂².D₄⋊₃₈C₂², C₄ ○₂(C₂².₃₃C₂⁴), C₂³.₃₆C₂³⋊₁₈C₂, C₂².₃₃C₂⁴⋊₂₉C₂, (C₂×C₄×D₄)⋊₇₆C₂, (C₂×C₄⋊C₄)⋊₁₃₁C₂², (C₂×C₄○D₄)⋊₁₉C₂², (C₂×C₄)○(C₂³⋊₂Q₈), C₂².₁₁(C₂×C₄○D₄), C₂.₂₂(C₂²×C₄○D₄), (C₂×C₂²⋊C₄)⋊₈₆C₂², (C₂×C₄)○(C₂².₃₃C₂⁴), SmallGroup(128,2191)

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

Derived series

C₁ — C₂² — C₂².₄₈C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₄×D₄ — C₂².₄₈C₂⁵

Lower central

C₁ — C₂² — C₂².₄₈C₂⁵

Upper central

C₁ — C₂×C₄ — C₂².₄₈C₂⁵

Jennings

C₁ — C₂² — C₂².₄₈C₂⁵

Generators and relations for C₂².₄₈C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=b, ab=ba, dcd=fcf=ac=ca, ede=ad=da, ae=ea, af=fa, ag=ga, ece=bc=cb, bd=db, be=eb, bf=fb, bg=gb, cg=gc, df=fd, dg=gd, ef=fe, eg=ge, fg=gf >

Subgroups: 916 in 580 conjugacy classes, 390 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₄×Q₈, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄².C₂, C₄²⋊₂C₂, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₄×D₄, C₂².₁₉C₂⁴, C₂³.₃₆C₂³, C₂³⋊₃D₄, C₂².₃₂C₂⁴, C₂².₃₃C₂⁴, C₂³⋊₂Q₈, C₂².₄₈C₂⁵
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₂×C₄○D₄, 2₊¹⁺⁴, C₂⁵, C₂²×C₄○D₄, C₂×2₊¹⁺⁴, C₂.C₂⁵, C₂².₄₈C₂⁵

Smallest permutation representation of C₂².₄₈C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

(5 30)(6 31)(7 32)(8 29)(13 20)(14 17)(15 18)(16 19)(21 23)(22 24)(25 27)(26 28)

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2O	4A	4B	4C	4D	4E	···	4J	4K	···	4AB
order	1	2	2	2	2	···	2	2	···	2	4	4	4	4	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	···	4	1	1	1	1	2	···	2	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	2	4	4
type	+	+	+	+	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₄○D₄	2₊¹⁺⁴	C₂.C₂⁵
kernel	C₂².₄₈C₂⁵	C₂×C₄×D₄	C₂².₁₉C₂⁴	C₂³.₃₆C₂³	C₂³⋊₃D₄	C₂².₃₂C₂⁴	C₂².₃₃C₂⁴	C₂³⋊₂Q₈	C₂³	C₄	C₂
# reps	1	3	6	6	3	6	6	1	8	2	2

Matrix representation of C₂².₄₈C₂⁵ ►in GL₆(𝔽₅)

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	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	0
0	0	0	0	0	4

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

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

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

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

3	0	0	0	0	0
0	3	0	0	0	0
0	0	3	0	0	0
0	0	0	3	0	0
0	0	0	0	3	0
0	0	0	0	0	3

G:=sub<GL(6,GF(5))| [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],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,3,0,0,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,3,0,0,0,0,0,1,0,0,0,0,4,4,1,4,0,0,0,0,0,4],[1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,4,4,0,0,0,0,1,0,4,0,0,0,1,0,3,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,4,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,3] >;

C₂².₄₈C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{48}C_2^5

% in TeX

G:=Group("C2^2.48C2^5");

// GroupNames label

G:=SmallGroup(128,2191);

// by ID

G=gap.SmallGroup(128,2191);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123,172]);

// 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,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,e*c*e=b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,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 = C22.48C25 order 128 = 27

29th central stem extension by C22 of C25

G = C₂².₄₈C₂⁵ order 128 = 2⁷

29^th central stem extension by C₂² of C₂⁵