C2^4.14D4 - GroupNames

G = C₂⁴.₁₄D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₄D₄, C₂³.₇SD₁₆, C₄⋊C₄.₁₁D₄, C₂³⋊C₈.₈C₂, (C₂×Q₈).₁₅D₄, (C₂²×C₄).₄₉D₄, C₂.₈(Q₈⋊D₄), C₂³.₅₂₅(C₂×D₄), C₂³⋊₂Q₈.₂C₂, C₂²⋊Q₈.₉C₂², (C₂²×C₄).₁₄C₂³, C₂².₁₄(C₂×SD₁₆), C₂².₁₃₅C₂²≀C₂, C₂.₁₀(D₄.₉D₄), C₂³.₄₇D₄⋊₂₅C₂, C₂³.₃₁D₄⋊₁₄C₂, C₂²⋊C₈.₁₁₄C₂², C₂³.₁₁D₄.₂C₂, C₂.₅(C₂³.₇D₄), C₂².₃₁(C₈.C₂²), C₂.C₄².₂₁C₂², (C₂×C₄).₂₀₃(C₂×D₄), (C₂×C₄⋊C₄).₁₉C₂², (C₂×C₂²⋊C₄).₉₇C₂², SmallGroup(128,340)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₄D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂³⋊₂Q₈ — C₂⁴.₁₄D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₄D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₄D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₄D₄

Generators and relations for C₂⁴.₁₄D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=c, f²=d, 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=de³ >

Subgroups: 284 in 114 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, C₂.C₄², C₂²⋊C₈, Q₈⋊C₄, C₄.Q₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂³⋊C₈, C₂³.₃₁D₄, C₂³.₁₁D₄, C₂³.₄₇D₄, C₂³⋊₂Q₈, C₂⁴.₁₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₂²≀C₂, C₂×SD₁₆, C₈.C₂², Q₈⋊D₄, D₄.₉D₄, C₂³.₇D₄, C₂⁴.₁₄D₄

Character table of C₂⁴.₁₄D₄

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	complex lifted from SD₁₆
ρ₁₆	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	-√-2	√-2	√-2	-√-2	complex lifted from SD₁₆
ρ₁₇	2	2	-2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	-√-2	-√-2	√-2	√-2	complex lifted from SD₁₆
ρ₁₈	2	2	-2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	√-2	-√-2	-√-2	√-2	complex lifted from SD₁₆
ρ₁₉	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
ρ₂₀	4	-4	-4	4	0	0	0	0	0	0	2i	0	0	0	0	0	0	0	-2i	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₁	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
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	-2i	0	0	0	0	0	0	0	2i	0	0	0	0	complex lifted from D₄.₉D₄

Smallest permutation representation of C₂⁴.₁₄D₄
►On 32 points

Generators in S₃₂

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

(1 32)(3 26)(5 28)(7 30)(10 19)(12 21)(14 23)(16 17)

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

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

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

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

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

Matrix representation of C₂⁴.₁₄D₄ ►in GL₆(𝔽₁₇)

16	0	0	0	0	0
0	16	0	0	0	0
0	0	16	15	0	0
0	0	0	1	0	0
0	0	0	0	16	15
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	1	0
0	0	0	0	0	1

16	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	0
0	0	0	0	0	16

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

0	7	0	0	0	0
5	7	0	0	0	0
0	0	0	0	4	8
0	0	0	0	0	13
0	0	16	15	0	0
0	0	1	1	0	0

16	0	0	0	0	0
16	1	0	0	0	0
0	0	1	2	0	0
0	0	16	16	0	0
0	0	0	0	4	8
0	0	0	0	0	13

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,15,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[0,5,0,0,0,0,7,7,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,4,0,0,0,0,0,8,13,0,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,1,16,0,0,0,0,2,16,0,0,0,0,0,0,4,0,0,0,0,0,8,13] >;

C₂⁴.₁₄D₄ in GAP, Magma, Sage, TeX

C_2^4._{14}D_4

% in TeX

G:=Group("C2^4.14D4");

// GroupNames label

G:=SmallGroup(128,340);

// by ID

G=gap.SmallGroup(128,340);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,232,422,1123,570,521,136,1411]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₁₄D₄ in TeX

G = C24.14D4 order 128 = 27

14th non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₄D₄ order 128 = 2⁷

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