M4(2).12D4 - GroupNames

G = M₄(2).₁₂D₄ order 128 = 2⁷

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

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

Aliases: M₄(2).₁₂D₄, C₄⋊C₄.₁₀₄D₄, (C₂×C₈).₁₆₀D₄, (C₂×D₄).₁₁Q₈, C₄.₁₅₁(C₄⋊D₄), M₄(2)⋊C₄⋊₇C₂, C₄.₁₀₀(C₂²⋊Q₈), C₂².C₄²⋊₂₃C₂, C₂².₅(C₂²⋊Q₈), C₂.₂₈(D₄.₃D₄), C₂.₂₁(D₄.₄D₄), C₂³.₂₈₀(C₄○D₄), (C₂²×C₈).₁₇₀C₂², (C₂²×C₄).₇₃₅C₂³, C₄².₆C₂²⋊₂C₂, C₂³.₃₇D₄.₉C₂, (C₂²×D₄).₈₈C₂², C₂.₄(C₂³.Q₈), C₂².₂₄₁(C₄⋊D₄), C₄²⋊C₂.₆₆C₂², C₂².₈(C₄²⋊₂C₂), (C₂×M₄(2)).₂₇C₂², (C₂×C₄).₂₀(C₂×Q₈), (C₂×C₈.C₄)⋊₁₆C₂, (C₂×C₄).₁₀₅₃(C₂×D₄), (C₂×C₄.D₄).₃C₂, (C₂×D₄⋊C₄).₂₇C₂, (C₂×C₄).₃₅₀(C₄○D₄), (C₂×C₄⋊C₄).₁₃₉C₂², SmallGroup(128,795)

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

Derived series

C₁ — C₂²×C₄ — M₄(2).₁₂D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂×D₄⋊C₄ — M₄(2).₁₂D₄

Lower central

C₁ — C₂ — C₂²×C₄ — M₄(2).₁₂D₄

Upper central

C₁ — C₂² — C₂²×C₄ — M₄(2).₁₂D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — M₄(2).₁₂D₄

Generators and relations for M₄(2).₁₂D₄
G = < a,b,c,d | a⁸=b²=c⁴=1, d²=a⁶b, bab=a⁵, cac^-1=a^-1, dad^-1=a^-1b, bc=cb, bd=db, dcd^-1=a⁶bc^-1 >

Subgroups: 280 in 114 conjugacy classes, 42 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₄.D₄, D₄⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₈.C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂²×D₄, C₂².C₄², C₂×C₄.D₄, C₂×D₄⋊C₄, C₂³.₃₇D₄, C₄².₆C₂², M₄(2)⋊C₄, C₂×C₈.C₄, M₄(2).₁₂D₄
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₄⋊D₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂³.Q₈, D₄.₃D₄, D₄.₄D₄, M₄(2).₁₂D₄

Character table of M₄(2).₁₂D₄

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	8	8	2	2	2	2	8	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	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	-2	2	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	-2	2	2	-2	0	0	2	-2	-2	2	0	0	0	0	0	0	0	0	0	2	-2	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	0	0	2	-2	2	-2	2	-2	0	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	2	-2	2	0	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	-2	2	-2	2	0	0	0	0	-2	2	-2	2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	2	2	-2	-2	0	0	2	-2	2	-2	-2	2	0	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	2	2	-2	-2	2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₆	2	-2	-2	2	2	-2	2	-2	-2	2	2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	-2	2	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	-2	2	0	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	complex lifted from C₄○D₄
ρ₁₉	2	2	2	2	2	2	0	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	-2	2	-2	2	0	0	2	2	-2	-2	0	0	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	-2	2	-2	2	0	0	2	2	-2	-2	0	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	-2	-2	-2	-2	0	0	0	0	0	0	0	0	-2i	0	0	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	-2√2	0	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	2√2	0	-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	0	-2√-2	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	0	2√-2	0	-2√-2	0	0	0	0	0	0	complex lifted from D₄.₃D₄

Smallest permutation representation of M₄(2).₁₂D₄
►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)

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

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

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

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

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

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

Matrix representation of M₄(2).₁₂D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	0	16	0
0	0	7	7	16	2
0	0	0	1	0	0
0	0	10	10	7	10

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	7	7	0	1

0	1	0	0	0	0
16	0	0	0	0	0
0	0	12	12	0	0
0	0	12	5	0	0
0	0	1	1	0	10
0	0	1	0	5	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	12	12	0	0
0	0	5	12	0	0
0	0	16	16	0	7
0	0	16	0	5	7

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,7,0,10,0,0,0,7,1,10,0,0,16,16,0,7,0,0,0,2,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,7,0,0,0,16,0,7,0,0,0,0,1,0,0,0,0,0,0,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,12,12,1,1,0,0,12,5,1,0,0,0,0,0,0,5,0,0,0,0,10,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,5,16,16,0,0,12,12,16,0,0,0,0,0,0,5,0,0,0,0,7,7] >;

M₄(2).₁₂D₄ in GAP, Magma, Sage, TeX

M_4(2)._{12}D_4

% in TeX

G:=Group("M4(2).12D4");

// GroupNames label

G:=SmallGroup(128,795);

// by ID

G=gap.SmallGroup(128,795);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,2019,521,248,2804,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=c^4=1,d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,b*c=c*b,b*d=d*b,d*c*d^-1=a^6*b*c^-1>;

// generators/relations

Export

Character table of M₄(2).₁₂D₄ in TeX

G = M4(2).12D4 order 128 = 27

12nd non-split extension by M4(2) of D4 acting via D4/C2=C22

G = M₄(2).₁₂D₄ order 128 = 2⁷

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