C2^4.22D4 - GroupNames

G = C₂⁴.₂₂D₄ order 128 = 2⁷

22^nd non-split extension by C₂⁴ of D₄ acting faithfully

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C₂⁴.₂₂D₄, C₂⁴.₁₇₀C₂³, C₂³⋊(C₄⋊C₄), C₂³⋊C₄⋊₄C₄, (C₂×D₄).₈Q₈, C₂.₂C₂≀C₂², C₂³.₄(C₂×Q₈), C₂².₃₃(C₄×D₄), (C₂²×C₄).₂₂D₄, C₂³.₆(C₄○D₄), C₂³.₉D₄⋊₅C₂, C₂³.₅₅₆(C₂×D₄), C₂³.₅(C₂²×C₄), C₂³.₈Q₈⋊₁C₂, C₂².₈₈C₂²≀C₂, (C₂³×C₄).₁₉C₂², C₂².₁₁C₂⁴.₄C₂, C₂³.₂₃D₄.₂C₂, C₂.₂(C₂³.₇D₄), (C₂²×D₄).₁₄C₂², C₂².₅₁(C₂²⋊Q₈), C₂.₂₃(C₂³.₈Q₈), C₂².₄₇(C₂².D₄), (C₂×C₄)⋊(C₄⋊C₄), C₂²⋊C₄⋊₄(C₂×C₄), (C₂×D₄).₆₈(C₂×C₄), (C₂×C₂³⋊C₄).₅C₂, C₂².₂₅(C₂×C₄⋊C₄), (C₂×C₂²⋊C₄).₉C₂², SmallGroup(128,599)

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

Derived series

C₁ — C₂³ — C₂⁴.₂₂D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂²×D₄ — C₂².₁₁C₂⁴ — C₂⁴.₂₂D₄

Lower central

C₁ — C₂ — C₂³ — C₂⁴.₂₂D₄

Upper central

C₁ — C₂² — C₂⁴ — C₂⁴.₂₂D₄

Jennings

C₁ — C₂ — C₂⁴ — C₂⁴.₂₂D₄

Generators and relations for C₂⁴.₂₂D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=e⁴=1, f²=c, eae^-1=faf^-1=ab=ba, ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e^-1 >

Subgroups: 436 in 187 conjugacy classes, 56 normal (20 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂.C₄², C₂³⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂³×C₄, C₂²×D₄, C₂³.₉D₄, C₂³.₈Q₈, C₂³.₂₃D₄, C₂×C₂³⋊C₄, C₂².₁₁C₂⁴, C₂⁴.₂₂D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₂³, C₄⋊C₄, C₂²×C₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₂²⋊Q₈, C₂².D₄, C₂³.₈Q₈, C₂≀C₂², C₂³.₇D₄, C₂⁴.₂₂D₄

Smallest permutation representation of C₂⁴.₂₂D₄
►On 32 points

Generators in S₃₂

(2 12)(3 8)(4 31)(5 10)(7 29)(9 30)(13 22)(14 26)(15 17)(19 25)(20 23)(24 27)

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	4A	···	4N	4O	···	4T
order	1	2	2	2	2	···	2	2	2	4	···	4	4	···	4
size	1	1	1	1	2	···	2	4	4	4	···	4	8	···	8

32 irreducible representations

dim	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+	+		+	-	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	D₄	Q₈	D₄	C₄○D₄	C₂≀C₂²	C₂³.₇D₄
kernel	C₂⁴.₂₂D₄	C₂³.₉D₄	C₂³.₈Q₈	C₂³.₂₃D₄	C₂×C₂³⋊C₄	C₂².₁₁C₂⁴	C₂³⋊C₄	C₂²×C₄	C₂×D₄	C₂⁴	C₂³	C₂	C₂
# reps	1	2	2	1	1	1	8	5	2	1	4	2	2

Matrix representation of C₂⁴.₂₂D₄ ►in GL₆(𝔽₅)

4	0	0	0	0	0
0	4	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	1	1

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	4	0
0	0	1	1	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	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

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

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

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,4,0,0,0,0,4,1,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,4,0,0,0,0,0,0,4],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,4,0,0,0,4,1,0,0,0,0,0,2,0,1],[0,3,0,0,0,0,3,0,0,0,0,0,0,0,4,0,4,0,0,0,0,4,4,0,0,0,0,0,1,0,0,0,0,0,2,4] >;

C₂⁴.₂₂D₄ in GAP, Magma, Sage, TeX

C_2^4._{22}D_4

% in TeX

G:=Group("C2^4.22D4");

// GroupNames label

G:=SmallGroup(128,599);

// by ID

G=gap.SmallGroup(128,599);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,232,422,521,1411]);

// Polycyclic

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

// generators/relations

G = C24.22D4 order 128 = 27

22nd non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₂₂D₄ order 128 = 2⁷

22^nd non-split extension by C₂⁴ of D₄ acting faithfully