C2^4.15D4 - GroupNames

G = C₂⁴.₁₅D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₅D₄, C₂³⋊C₈⋊₇C₂, C₄⋊C₄.₁₅D₄, (C₂×D₄).₁₈D₄, (C₂×Q₈).₁₈D₄, (C₂²×C₄).₁₇D₄, C₂³.₅₂₉(C₂×D₄), C₂².D₈⋊₃C₂, C₂.₁₁(D₄⋊D₄), C₂².₂₅(C₄○D₈), C₂³.₁₁D₄⋊₂C₂, C₄⋊D₄.₁₂C₂², C₂³.₃₁D₄⋊₈C₂, C₂².SD₁₆⋊₁₄C₂, (C₂²×C₄).₁₈C₂³, C₂²⋊Q₈.₁₂C₂², C₂.₁₁(D₄.₉D₄), C₂².₁₃₉C₂²≀C₂, C₂³.₄₇D₄⋊₂₇C₂, C₂²⋊C₈.₁₁₇C₂², C₂².₂₁(C₈⋊C₂²), C₂².₃₂C₂⁴.₂C₂, C₂.₉(C₂³.₇D₄), C₂.C₄².₂₅C₂², (C₂×C₄).₂₀₇(C₂×D₄), (C₂×C₄⋊C₄).₂₃C₂², (C₂×C₂²⋊C₄).₉₈C₂², SmallGroup(128,344)

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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₁₅D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₂²⋊C₄ — C₂².₃₂C₂⁴ — C₂⁴.₁₅D₄

Lower central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₅D₄

Upper central

C₁ — C₂² — C₂²×C₄ — C₂⁴.₁₅D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₁₅D₄

Generators and relations for C₂⁴.₁₅D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=c, f²=d, 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=cde³ >

Subgroups: 308 in 116 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, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₂²⋊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, C₂³.₁₁D₄, C₂².D₈, C₂³.₄₇D₄, C₂².₃₂C₂⁴, C₂⁴.₁₅D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂²≀C₂, C₄○D₈, C₈⋊C₂², D₄⋊D₄, D₄.₉D₄, C₂³.₇D₄, C₂⁴.₁₅D₄

Character table of C₂⁴.₁₅D₄

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	4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from C₈⋊C₂²
ρ₂₀	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
ρ₂₁	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
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 C₂⁴.₁₅D₄
►On 32 points

Generators in S₃₂

(3 31)(4 32)(7 27)(8 28)(9 13)(10 14)(11 24)(12 17)(15 20)(16 21)(18 22)(19 23)

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

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

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

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

Matrix representation of C₂⁴.₁₅D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
1	16	0	0	0	0
0	0	1	0	0	1
0	0	0	16	0	0
0	0	0	0	1	16
0	0	0	0	0	16

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	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

9	0	0	0	0	0
12	2	0	0	0	0
0	0	0	13	0	0
0	0	0	13	13	0
0	0	13	4	0	0
0	0	0	8	0	4

2	13	0	0	0	0
5	15	0	0	0	0
0	0	13	0	13	0
0	0	0	13	13	0
0	0	0	0	4	0
0	0	0	0	8	13

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

C₂⁴.₁₅D₄ in GAP, Magma, Sage, TeX

C_2^4._{15}D_4

% in TeX

G:=Group("C2^4.15D4");

// GroupNames label

G:=SmallGroup(128,344);

// by ID

G=gap.SmallGroup(128,344);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,448,141,422,520,1123,570,521,136,1411]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂⁴.₁₅D₄ in TeX

G = C24.15D4 order 128 = 27

15th non-split extension by C24 of D4 acting faithfully

G = C₂⁴.₁₅D₄ order 128 = 2⁷

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