C2^4.12D4 - GroupNames

G = C₂⁴.₁₂D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₂D₄, C₄⋊C₄.₉D₄, C₂³⋊C₈⋊₆C₂, (C₂×D₄).₁₄D₄, (C₂×Q₈).₁₄D₄, Q₈⋊D₄⋊₂₇C₂, C₂.₁₅C₂≀C₂², C₂²⋊Q₁₆⋊₃C₂, C₂³⋊Q₈⋊₂C₂, (C₂²×C₄).₁₅D₄, C₂².SD₁₆⋊₆C₂, C₄⋊D₄.₈C₂², C₂³.₅₂₃(C₂×D₄), C₂²⋊C₈.₅C₂², C₂²⋊Q₈.₈C₂², C₂.₈(D₄.₉D₄), C₂.₉(D₄.₇D₄), C₂².₂₃(C₄○D₈), (C₂²×C₄).₁₂C₂³, C₂².₁₃₃C₂²≀C₂, (C₂²×Q₈).₆C₂², C₂³.₃₁D₄⋊₁₃C₂, C₂².₃₂C₂⁴.₁C₂, C₂².₁₅(C₈.C₂²), C₂.C₄².₂₀C₂², (C₂×C₄).₂₀₁(C₂×D₄), (C₂×C₂²⋊C₄).₉₅C₂², SmallGroup(128,338)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₂D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂².₃₂C₂⁴ — C₂⁴.₁₂D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₂D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₂D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₂D₄

Generators and relations for C₂⁴.₁₂D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=f²=c, eae^-1=ab=ba, ac=ca, ad=da, faf^-1=abc, bc=cb, ebe^-1=bd=db, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=e³ >

Subgroups: 348 in 130 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, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, Q₈⋊C₄, C₂×C₂²⋊C₄, C₂×C₂²⋊C₄, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₄.₄D₄, C₄²⋊₂C₂, C₂×SD₁₆, C₂×Q₁₆, C₂²×Q₈, C₂³⋊C₈, C₂².SD₁₆, C₂³.₃₁D₄, C₂³⋊Q₈, Q₈⋊D₄, C₂²⋊Q₁₆, C₂².₃₂C₂⁴, C₂⁴.₁₂D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈.C₂², D₄.₇D₄, D₄.₉D₄, C₂≀C₂², C₂⁴.₁₂D₄

Character table of C₂⁴.₁₂D₄

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	0	0	0	2	0	-2	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
ρ₂₀	4	-4	4	-4	0	0	0	0	0	0	0	0	0	0	0	-2	0	2	0	0	0	0	0	orthogonal lifted from C₂≀C₂²
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	-2i	0	0	0	0	0	2i	0	0	0	0	complex lifted from D₄.₉D₄
ρ₂₃	4	-4	-4	4	0	0	0	0	0	0	0	0	2i	0	0	0	0	0	-2i	0	0	0	0	complex lifted from D₄.₉D₄

Smallest permutation representation of C₂⁴.₁₂D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

Matrix representation of C₂⁴.₁₂D₄ ►in GL₆(𝔽₁₇)

0	13	0	0	0	0
4	0	0	0	0	0
0	0	16	0	0	0
0	0	4	1	0	0
0	0	0	0	16	0
0	0	8	0	9	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	15	0	1	0
0	0	0	0	0	1

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

5	12	0	0	0	0
5	5	0	0	0	0
0	0	2	0	15	0
0	0	0	0	0	1
0	0	0	16	15	0
0	0	1	13	0	0

4	0	0	0	0	0
0	13	0	0	0	0
0	0	4	2	0	0
0	0	1	13	0	0
0	0	3	2	1	0
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [0,4,0,0,0,0,13,0,0,0,0,0,0,0,16,4,0,8,0,0,0,1,0,0,0,0,0,0,16,9,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,15,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[5,5,0,0,0,0,12,5,0,0,0,0,0,0,2,0,0,1,0,0,0,0,16,13,0,0,15,0,15,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,1,3,0,0,0,2,13,2,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂⁴.₁₂D₄ in GAP, Magma, Sage, TeX

C_2^4._{12}D_4

% in TeX

G:=Group("C2^4.12D4");

// GroupNames label

G:=SmallGroup(128,338);

// by ID

G=gap.SmallGroup(128,338);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,224,141,422,520,1123,570,521,136,1411]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₁₂D₄ in TeX

G = C24.12D4 order 128 = 27

12nd non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₂D₄ order 128 = 2⁷

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