C2^4.18D4 - GroupNames

G = C₂⁴.₁₈D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₈D₄, C₂³⋊C₈⋊₈C₂, C₄⋊C₄.₂₁D₄, (C₂×D₄).₂₂D₄, (C₂×Q₈).₂₂D₄, (C₂²×C₄).₁₉D₄, C₂³.₅₃₅(C₂×D₄), C₂².SD₁₆⋊₉C₂, C₂.₁₁(D₄⋊₄D₄), C₂³.₄Q₈⋊₂C₂, C₂².₂₇(C₄○D₈), C₂³.₄₈D₄⋊₃C₂, C₄⋊D₄.₁₆C₂², C₂²⋊C₈.₁₀C₂², (C₂²×C₄).₂₄C₂³, C₂²⋊Q₈.₁₆C₂², C₂².₁₄₅C₂²≀C₂, C₂.₁₁(D₄.₇D₄), C₂³.₄₆D₄⋊₂₇C₂, C₂³.₃₁D₄⋊₁₆C₂, C₂².₃₂C₂⁴.₃C₂, C₂.₁₅(C₂³.₇D₄), C₂².₁₇(C₈.C₂²), C₂.C₄².₃₀C₂², (C₂×C₄).₂₁₃(C₂×D₄), (C₂×C₄⋊C₄).₂₈C₂², (C₂×C₂²⋊C₄).₁₀₁C₂², SmallGroup(128,350)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₈D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂².₃₂C₂⁴ — C₂⁴.₁₈D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₈D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₈D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₈D₄

Generators and relations for C₂⁴.₁₈D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=c, f²=dc=cd, eae^-1=ab=ba, ac=ca, ad=da, faf^-1=abc, bc=cb, ebe^-1=bd=db, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef^-1=de³ >

Subgroups: 316 in 120 conjugacy classes, 32 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂⁴, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂³⋊C₈, C₂².SD₁₆, C₂³.₃₁D₄, C₂³.₄Q₈, C₂³.₄₆D₄, C₂³.₄₈D₄, C₂².₃₂C₂⁴, C₂⁴.₁₈D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈.C₂², D₄.₇D₄, D₄⋊₄D₄, C₂³.₇D₄, C₂⁴.₁₈D₄

Character table of C₂⁴.₁₈D₄

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	8	8	4	4	4	4	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	-2	0	-2	0	0	2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	0	0	2	0	0	-2	0	0	-2	0	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	2	2	0	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	2	0	0	-2	0	0	2	0	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	2	2	0	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	2	0	-2	0	0	2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	-2	-2	-2	2	0	0	0	2i	-2i	0	0	0	0	0	0	0	0	√2	-√-2	-√2	√-2	complex lifted from C₄○D₈
ρ₁₆	2	2	-2	-2	-2	2	0	0	0	-2i	2i	0	0	0	0	0	0	0	0	√2	√-2	-√2	-√-2	complex lifted from C₄○D₈
ρ₁₇	2	2	-2	-2	-2	2	0	0	0	-2i	2i	0	0	0	0	0	0	0	0	-√2	-√-2	√2	√-2	complex lifted from C₄○D₈
ρ₁₈	2	2	-2	-2	-2	2	0	0	0	2i	-2i	0	0	0	0	0	0	0	0	-√2	√-2	√2	-√-2	complex lifted from C₄○D₈
ρ₁₉	4	-4	-4	4	0	0	0	0	0	0	0	0	-2	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	0	0	2	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 C₂⁴.₁₈D₄
►On 32 points

Generators in S₃₂

(3 25)(4 26)(7 29)(8 30)(9 13)(10 14)(11 24)(12 17)(15 20)(16 21)(18 22)(19 23)

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

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

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

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

Matrix representation of C₂⁴.₁₈D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	16	0	0	0	0
0	0	16	0	0	0
0	0	13	1	0	0
0	0	0	0	16	0
0	0	13	0	0	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	1	0	16	0
0	0	4	0	0	16

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

2	0	0	0	0	0
0	8	0	0	0	0
0	0	13	0	0	2
0	0	1	0	16	4
0	0	6	0	0	1
0	0	11	16	0	4

0	8	0	0	0	0
2	0	0	0	0	0
0	0	13	2	0	0
0	0	1	4	0	0
0	0	6	1	1	0
0	0	11	4	0	16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,13,0,13,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,4,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[2,0,0,0,0,0,0,8,0,0,0,0,0,0,13,1,6,11,0,0,0,0,0,16,0,0,0,16,0,0,0,0,2,4,1,4],[0,2,0,0,0,0,8,0,0,0,0,0,0,0,13,1,6,11,0,0,2,4,1,4,0,0,0,0,1,0,0,0,0,0,0,16] >;

C₂⁴.₁₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{18}D_4

% in TeX

G:=Group("C2^4.18D4");

// GroupNames label

G:=SmallGroup(128,350);

// by ID

G=gap.SmallGroup(128,350);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,422,520,1123,570,521,136,1411]);

// 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=a*b=b*a,a*c=c*a,a*d=d*a,f*a*f^-1=a*b*c,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=d*e^3>;

// generators/relations

Export

Character table of C₂⁴.₁₈D₄ in TeX

G = C24.18D4 order 128 = 27

18th non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₈D₄ order 128 = 2⁷

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