M4(2).10D4 - GroupNames

G = M₄(2).₁₀D₄ order 128 = 2⁷

10^th non-split extension by M₄(2) of D₄ acting via D₄/C₂=C₂²

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

Aliases: M₄(2).₁₀D₄, (C₂×C₈).₁₅₈D₄, C₄.₇₃C₂²≀C₂, (C₂×D₄).₁₁₂D₄, (C₂×Q₈).₁₀₃D₄, C₄.₁₄₉(C₄⋊D₄), C₂².C₄²⋊₉C₂, C₂.₂₆(D₄.₃D₄), C₂.₂₀(D₄.₄D₄), C₂³.₂₇₈(C₄○D₄), C₂³.₃₆D₄⋊₃₅C₂, (C₂²×C₄).₇₃₀C₂³, (C₂²×C₈).₁₆₈C₂², (C₂²×D₄).₈₅C₂², C₂².₂₃₆(C₄⋊D₄), C₄.₇₄(C₂².D₄), C₂².₁₂(C₄.₄D₄), C₂.₂₄(C₂³.₁₀D₄), (C₂×M₄(2)).₂₃₃C₂², C₂².₁₃(C₂².D₄), (C₂×C₈⋊C₂²).₈C₂, (C₂×D₄⋊C₄)⋊₂₇C₂, (C₂×C₈.C₄)⋊₁₄C₂, (C₂²×C₈)⋊C₂⋊₂C₂, (C₂×C₄.D₄)⋊₂₅C₂, (C₂×C₄).₁₃₇₁(C₂×D₄), (C₂×C₄).₃₄₆(C₄○D₄), (C₂×C₄⋊C₄).₁₂₉C₂², (C₂×C₄○D₄).₆₄C₂², SmallGroup(128,783)

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

Derived series

C₁ — C₂²×C₄ — M₄(2).₁₀D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×M₄(2) — C₂×C₄.D₄ — M₄(2).₁₀D₄

Lower central

C₁ — C₂ — C₂²×C₄ — M₄(2).₁₀D₄

Upper central

C₁ — C₂² — C₂²×C₄ — M₄(2).₁₀D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — M₄(2).₁₀D₄

Generators and relations for M₄(2).₁₀D₄
G = < a,b,c,d | a⁸=b²=c⁴=d²=1, bab=a⁵, cac^-1=a⁵b, dad=a^-1, bc=cb, dbd=a⁴b, dcd=a²c^-1 >

Subgroups: 352 in 141 conjugacy classes, 42 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), M₄(2), D₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊C₈, C₄.D₄, D₄⋊C₄, Q₈⋊C₄, C₈.C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂×C₄○D₄, C₂².C₄², (C₂²×C₈)⋊C₂, C₂×C₄.D₄, C₂×D₄⋊C₄, C₂³.₃₆D₄, C₂×C₈.C₄, C₂×C₈⋊C₂², M₄(2).₁₀D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂³.₁₀D₄, D₄.₃D₄, D₄.₄D₄, M₄(2).₁₀D₄

Character table of M₄(2).₁₀D₄

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	4B	4C	4D	4E	4F	4G	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	8	8	8	2	2	2	2	8	8	8	4	4	4	4	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	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	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	-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	-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	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	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	-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	-1	1	1	linear of order 2
ρ₉	2	2	2	2	-2	-2	-2	0	0	2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	2	-2	-2	2	0	2	-2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	-2	2	-2	2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	-2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	0	0	0	-2	2	-2	2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	-2	2	-2	2	-2	0	0	0	2	-2	-2	2	0	0	0	0	0	0	0	2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₄	2	-2	2	-2	-2	2	0	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	2	-2	-2	0	0	0	-2	2	-2	2	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	2	2	-2	-2	2	0	0	2	-2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	-2	2	-2	-2	2	0	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	complex lifted from C₄○D₄
ρ₁₈	2	-2	2	-2	2	-2	0	0	0	-2	2	2	-2	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	2	-2	-2	2	0	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	-2i	0	0	0	2i	complex lifted from C₄○D₄
ρ₂₀	2	2	2	2	2	2	0	0	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	2	-2	0	0	0	-2	2	2	-2	0	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	2	2	2	2	0	0	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	complex lifted from C₄○D₄
ρ₂₃	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	-2√2	2√2	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₄	4	4	-4	-4	0	0	0	0	0	0	0	0	0	0	0	0	0	2√2	-2√2	0	0	0	0	0	0	0	orthogonal lifted from D₄.₄D₄
ρ₂₅	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	-2√-2	0	0	2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄
ρ₂₆	4	-4	-4	4	0	0	0	0	0	0	0	0	0	0	0	0	2√-2	0	0	-2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄

Smallest permutation representation of M₄(2).₁₀D₄
►On 32 points

Generators in S₃₂

(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)

(2 6)(4 8)(10 14)(12 16)(18 22)(20 24)(26 30)(28 32)

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

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

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

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

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

Matrix representation of M₄(2).₁₀D₄ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
4	4	0	0	0	0
0	0	0	0	11	11
0	0	0	0	3	0
0	0	11	11	0	0
0	0	3	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	0	0
0	0	0	0	16	0
0	0	0	0	0	16

1	2	0	0	0	0
16	16	0	0	0	0
0	0	11	11	0	0
0	0	3	6	0	0
0	0	0	0	0	6
0	0	0	0	3	0

4	8	0	0	0	0
13	13	0	0	0	0
0	0	0	0	11	11
0	0	0	0	3	6
0	0	11	11	0	0
0	0	3	6	0	0

G:=sub<GL(6,GF(17))| [13,4,0,0,0,0,0,4,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,0,0,0,11,3,0,0,0,0,11,0,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,16,0,0,0,0,2,16,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,0,0,0,0,0,3,0,0,0,0,6,0],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,0,0,11,3,0,0,0,0,11,6,0,0,11,3,0,0,0,0,11,6,0,0] >;

M₄(2).₁₀D₄ in GAP, Magma, Sage, TeX

M_4(2)._{10}D_4

% in TeX

G:=Group("M4(2).10D4");

// GroupNames label

G:=SmallGroup(128,783);

// by ID

G=gap.SmallGroup(128,783);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=d^2=1,b*a*b=a^5,c*a*c^-1=a^5*b,d*a*d=a^-1,b*c=c*b,d*b*d=a^4*b,d*c*d=a^2*c^-1>;

// generators/relations

Export

Character table of M₄(2).₁₀D₄ in TeX

G = M4(2).10D4 order 128 = 27

10th non-split extension by M4(2) of D4 acting via D4/C2=C22

G = M₄(2).₁₀D₄ order 128 = 2⁷

10^th non-split extension by M₄(2) of D₄ acting via D₄/C₂=C₂²