C2^4.17D4 - GroupNames

G = C₂⁴.₁₇D₄ order 128 = 2⁷

17^th non-split extension by C₂⁴ of D₄ acting faithfully

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

Aliases: C₂⁴.₁₇D₄, C₂³.₃Q₁₆, C₄⋊C₄.₁₇D₄, C₂³⋊C₈.₇C₂, (C₂×Q₈).₁₉D₄, (C₂²×C₄).₅₃D₄, C₂³.₅₃₁(C₂×D₄), C₂²⋊C₈.₇C₂², C₂.₁₀(D₄⋊₄D₄), C₂³⋊₂Q₈.₃C₂, C₂².₁₃(C₂×Q₁₆), C₂³.₃₁D₄⋊₉C₂, C₂³.₄₈D₄⋊₁C₂, (C₂²×C₄).₂₀C₂³, C₂.₈(C₂²⋊Q₁₆), C₂²⋊Q₈.₁₃C₂², C₂².₁₄₁C₂²≀C₂, C₂³.₄Q₈.₂C₂, C₂.₁₁(C₂³.₇D₄), C₂².₃₃(C₈.C₂²), C₂.C₄².₂₆C₂², (C₂×C₄).₂₀₉(C₂×D₄), (C₂×C₄⋊C₄).₂₄C₂², (C₂×C₂²⋊C₄).₁₀₀C₂², SmallGroup(128,346)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₇D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³⋊₂Q₈ — C₂⁴.₁₇D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₇D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₇D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₇D₄

Generators and relations for C₂⁴.₁₇D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=c, f²=dc=cd, eae^-1=faf^-1=ab=ba, ac=ca, ad=da, bc=cb, ebe^-1=bd=db, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=cde³ >

Subgroups: 292 in 118 conjugacy classes, 34 normal (18 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂³⋊C₈, C₂³.₃₁D₄, C₂³.₄Q₈, C₂³.₄₈D₄, C₂³⋊₂Q₈, C₂⁴.₁₇D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂²≀C₂, C₂×Q₁₆, C₈.C₂², C₂²⋊Q₁₆, D₄⋊₄D₄, C₂³.₇D₄, C₂⁴.₁₇D₄

Character table of C₂⁴.₁₇D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	8A	8B	8C	8D
size	1	1	1	1	2	2	4	4	4	4	8	8	8	8	8	8	8	8	8	8	8	8	8
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	1	1	1	1	1	-1	-1	1	1	-1	1	-1	1	-1	1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	-1	-1	linear of order 2
ρ₄	1	1	1	1	1	1	-1	-1	1	1	-1	-1	1	-1	1	1	-1	1	-1	1	-1	1	-1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	1	1	-1	1	1	1	1	-1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₆	1	1	1	1	1	1	-1	-1	1	1	1	1	-1	1	-1	-1	-1	-1	1	1	-1	1	-1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	1	1	1	1	linear of order 2
ρ₈	1	1	1	1	1	1	-1	-1	1	1	1	-1	1	-1	1	-1	-1	-1	1	-1	1	-1	1	linear of order 2
ρ₉	2	2	2	2	-2	-2	0	0	-2	2	0	0	-2	0	2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	2	2	2	2	-2	-2	0	0	0	0	0	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	0	2	-2	0	2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	2	-2	0	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	0	0	-2	2	0	0	2	0	-2	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√2	-√2	-√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₆	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	√2	√2	-√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₇	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√2	√2	√2	-√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₈	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√2	-√2	√2	√2	symplectic lifted from Q₁₆, Schur index 2
ρ₁₉	4	-4	-4	4	0	0	0	0	0	0	2	0	0	0	0	0	0	0	-2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₀	4	-4	-4	4	0	0	0	0	0	0	-2	0	0	0	0	0	0	0	2	0	0	0	0	orthogonal lifted from D₄⋊₄D₄
ρ₂₁	4	4	-4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from C₈.C₂², Schur index 2
ρ₂₂	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	0	0	complex lifted from C₂³.₇D₄
ρ₂₃	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	0	0	complex lifted from C₂³.₇D₄

Smallest permutation representation of C₂⁴.₁₇D₄
►On 32 points

Generators in S₃₂

(2 6)(3 27)(4 32)(7 31)(8 28)(9 20)(10 17)(11 15)(13 24)(14 21)(18 22)(26 30)

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

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

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

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

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

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

Matrix representation of C₂⁴.₁₇D₄ ►in GL₆(𝔽₁₇)

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	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

16	0	0	0	0	0
0	16	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	16	0	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	0	16

6	11	0	0	0	0
3	0	0	0	0	0
0	0	0	0	0	13
0	0	0	0	13	0
0	0	13	0	0	0
0	0	0	13	0	0

10	7	0	0	0	0
5	7	0	0	0	0
0	0	13	0	0	0
0	0	0	13	0	0
0	0	0	0	0	13
0	0	0	0	13	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[6,3,0,0,0,0,11,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,13,0,0,0,0,13,0,0,0],[10,5,0,0,0,0,7,7,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,13,0,0,0,0,13,0] >;

C₂⁴.₁₇D₄ in GAP, Magma, Sage, TeX

C_2^4._{17}D_4

% in TeX

G:=Group("C2^4.17D4");

// GroupNames label

G:=SmallGroup(128,346);

// by ID

G=gap.SmallGroup(128,346);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,232,422,1123,570,521,136,1411]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=c,f^2=d*c=c*d,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*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;

// generators/relations

Export

Character table of C₂⁴.₁₇D₄ in TeX

G = C24.17D4 order 128 = 27

17th non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₇D₄ order 128 = 2⁷

17^th non-split extension by C₂⁴ of D₄ acting faithfully