C2^2.38C2^5 - GroupNames

G = C₂².₃₈C₂⁵ order 128 = 2⁷

19^th central stem extension by C₂² of C₂⁵

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

Aliases: C₂³.₁₈C₂⁴, C₂².₃₈C₂⁵, C₄².₅₄₁C₂³, C₂⁴.₄₈₀C₂³, (C₂²×C₄)⋊₄₄D₄, C₄⋊Q₈⋊₇₄C₂², (C₄×D₄)⋊₃₀C₂², C₄ ○(C₂³⋊₃D₄), (C₂×C₄).₄₁C₂⁴, C₂.₁₇(D₄×C₂³), C₂³⋊₃D₄⋊₁₈C₂, C₄⋊D₄⋊₆₃C₂², C₄⋊₁D₄⋊₄₃C₂², C₄⋊C₄.₂₈₂C₂³, (C₂×C₄²)⋊₄₅C₂², (C₂³×C₄)⋊₃₁C₂², C₂³.₃₅₅(C₂×D₄), C₄.₁₇₃(C₂²×D₄), C₂²⋊C₄.₆C₂³, C₂²⋊Q₈⋊₇₄C₂², C₂²≀C₂⋊₂₇C₂², C₂².₃(C₂²×D₄), C₄ ○(C₂².₂₉C₂⁴), (C₂×D₄).₂₉₁C₂³, C₄.₄D₄⋊₆₄C₂², (C₂²×D₄)⋊₆₀C₂², (C₂×Q₈).₄₂₃C₂³, (C₂²×Q₈)⋊₆₀C₂², C₂².₁₉C₂⁴⋊₁₃C₂, C₂².₂₉C₂⁴⋊₃₇C₂, C₄²⋊C₂⋊₈₈C₂², C₂.₄(C₂.C₂⁵), C₄ ○(C₂³.₃₈C₂³), C₄ ○(C₂².₃₁C₂⁴), C₂².₂₆C₂⁴⋊₂₆C₂, (C₂²×C₄).₁₁₇₉C₂³, C₂².D₄⋊₃₂C₂², C₂³.₃₈C₂³⋊₃₉C₂, C₂².₃₁C₂⁴⋊₃₀C₂, (C₂×C₄).₆₆₂(C₂×D₄), (C₂×C₄⋊C₄)⋊₁₂₉C₂², (C₂²×C₄○D₄)⋊₁₅C₂, (C₂×C₄○D₄)⋊₆₉C₂², (C₂×C₄)○(C₂³⋊₃D₄), (C₂×C₄²⋊C₂)⋊₅₅C₂, (C₂×C₄)○(C₂².₂₉C₂⁴), (C₂×C₂²⋊C₄).₅₂₉C₂², (C₂×C₄)○(C₂².₃₁C₂⁴), SmallGroup(128,2181)

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

Derived series

C₁ — C₂² — C₂².₃₈C₂⁵

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂².₃₈C₂⁵

Lower central

C₁ — C₂² — C₂².₃₈C₂⁵

Upper central

C₁ — C₂×C₄ — C₂².₃₈C₂⁵

Jennings

C₁ — C₂² — C₂².₃₈C₂⁵

Generators and relations for C₂².₃₈C₂⁵
G = < a,b,c,d,e,f,g | a²=b²=c²=d²=e²=f²=1, g²=a, 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: 1244 in 778 conjugacy classes, 428 normal (12 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, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄⋊₁D₄, C₄⋊Q₈, C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄²⋊C₂, C₂².₁₉C₂⁴, C₂².₂₆C₂⁴, C₂³⋊₃D₄, C₂².₂₉C₂⁴, C₂³.₃₈C₂³, C₂².₃₁C₂⁴, C₂²×C₄○D₄, C₂².₃₈C₂⁵
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²×D₄, C₂⁵, D₄×C₂³, C₂.C₂⁵, C₂².₃₈C₂⁵

Smallest permutation representation of C₂².₃₈C₂⁵
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	···	2Q	4A	4B	4C	4D	4E	···	4J	4K	···	4Z
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	1	2	4
type	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₂	C₂	C₂	D₄	C₂.C₂⁵
kernel	C₂².₃₈C₂⁵	C₂×C₄²⋊C₂	C₂².₁₉C₂⁴	C₂².₂₆C₂⁴	C₂³⋊₃D₄	C₂².₂₉C₂⁴	C₂³.₃₈C₂³	C₂².₃₁C₂⁴	C₂²×C₄○D₄	C₂²×C₄	C₂
# reps	1	1	8	4	4	4	4	4	2	8	4

Matrix representation of C₂².₃₈C₂⁵ ►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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

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

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

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

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

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

C₂².₃₈C₂⁵ in GAP, Magma, Sage, TeX

C_2^2._{38}C_2^5

% in TeX

G:=Group("C2^2.38C2^5");

// GroupNames label

G:=SmallGroup(128,2181);

// by ID

G=gap.SmallGroup(128,2181);

# by ID

G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123,102]);

// 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=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,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 = C22.38C25 order 128 = 27

19th central stem extension by C22 of C25

G = C₂².₃₈C₂⁵ order 128 = 2⁷

19^th central stem extension by C₂² of C₂⁵