M4(2).9D4 - GroupNames

G = M₄(2).₉D₄ order 128 = 2⁷

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

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

Aliases: M₄(2).₉D₄, C₄.₇₁C₂²≀C₂, (C₂×D₄).₁₁₀D₄, (C₂×Q₈).₁₀₁D₄, C₄.₂₉(C₄⋊D₄), M₄(2).C₄⋊₅C₂, (C₂²×C₄).₄₂C₂³, M₄(2)⋊₄C₄⋊₁₁C₂, C₂³.₁₃₂(C₄○D₄), C₂³.₃₈D₄⋊₃₀C₂, C₂².₇₀(C₄⋊D₄), C₄²⋊C₂².₁C₂, (C₂²×Q₈).₇₀C₂², C₄²⋊C₂.₆₂C₂², C₄.₂₄(C₂².D₄), C₂².₁₁(C₄.₄D₄), C₂.₂₂(C₂³.₁₀D₄), (C₂×M₄(2)).₂₃₁C₂², C₂².₅₈(C₂².D₄), M₄(2).₈C₂².₈C₂, (C₂×C₄).₂₆₂(C₂×D₄), (C₂×C₄.₁₀D₄)⋊₆C₂, (C₂×C₈.C₂²).₇C₂, (C₂×C₄).₃₄₅(C₄○D₄), (C₂×C₄○D₄).₆₂C₂², SmallGroup(128,781)

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

Derived series

C₁ — C₂²×C₄ — M₄(2).₉D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×M₄(2) — M₄(2).₈C₂² — M₄(2).₉D₄

Lower central

C₁ — C₂ — C₂²×C₄ — M₄(2).₉D₄

Upper central

C₁ — C₂ — C₂²×C₄ — M₄(2).₉D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — M₄(2).₉D₄

Generators and relations for M₄(2).₉D₄
G = < a,b,c,d | a⁸=b²=1, c⁴=d²=a⁴, bab=a⁵, cac^-1=a^-1, dad^-1=a⁵b, cbc^-1=a⁴b, bd=db, dcd^-1=c³ >

Subgroups: 264 in 127 conjugacy classes, 40 normal (34 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄.D₄, C₄.₁₀D₄, Q₈⋊C₄, C₄≀C₂, C₈.C₄, C₄²⋊C₂, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×Q₈, C₂×C₄○D₄, M₄(2)⋊₄C₄, C₂×C₄.₁₀D₄, M₄(2).₈C₂², C₂³.₃₈D₄, C₄²⋊C₂², M₄(2).C₄, C₂×C₈.C₂², M₄(2).₉D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂³.₁₀D₄, M₄(2).₉D₄

Character table of M₄(2).₉D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H
size	1	1	2	2	2	8	2	2	2	2	8	8	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	0	2	-2	2	-2	0	0	0	0	0	0	2	0	0	0	0	-2	0	orthogonal lifted from D₄
ρ₁₀	2	2	-2	-2	2	0	-2	2	-2	2	0	2	0	-2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	-2	2	-2	2	2	-2	-2	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₂	2	2	-2	-2	2	0	-2	2	-2	2	0	-2	0	2	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	-2	-2	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	2	-2	0	0	orthogonal lifted from D₄
ρ₁₄	2	2	-2	-2	2	0	2	-2	2	-2	0	0	0	0	0	0	-2	0	0	0	0	2	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	-2	-2	0	-2	-2	2	2	0	0	0	0	0	0	0	0	0	-2	2	0	0	orthogonal lifted from D₄
ρ₁₆	2	2	-2	2	-2	-2	2	-2	-2	2	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₇	2	2	-2	2	-2	0	-2	2	2	-2	0	0	0	0	0	2i	0	0	0	0	0	0	-2i	complex lifted from C₄○D₄
ρ₁₈	2	2	2	-2	-2	0	2	2	-2	-2	-2i	0	0	0	2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	2	-2	2	-2	0	-2	2	2	-2	0	0	0	0	0	-2i	0	0	0	0	0	0	2i	complex lifted from C₄○D₄
ρ₂₀	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	-2i	2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	2i	-2i	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	2	2	-2	-2	0	2	2	-2	-2	2i	0	0	0	-2i	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	8	-8	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	symplectic faithful, Schur index 2

Smallest permutation representation of M₄(2).₉D₄
►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 30)(2 27)(3 32)(4 29)(5 26)(6 31)(7 28)(8 25)(9 20)(10 17)(11 22)(12 19)(13 24)(14 21)(15 18)(16 23)

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

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

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

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

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

Matrix representation of M₄(2).₉D₄ ►in GL₈(𝔽₁₇)

7	0	7	0	6	8	6	8
3	0	3	0	16	7	16	7
11	0	6	0	11	9	6	8
12	0	5	0	1	10	16	7
9	3	10	14	12	16	14	13
5	4	14	13	12	16	14	13
0	3	9	3	5	1	14	13
13	4	5	4	5	1	14	13

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
6	11	0	10	0	0	0	16
0	11	7	10	0	0	1	0
0	7	11	6	0	1	0	0
10	7	0	6	16	0	0	0

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
7	9	7	9	0	0	0	0
3	10	3	10	0	0	0	0
7	4	1	16	13	3	4	14
8	4	0	16	3	4	14	13
13	4	5	1	13	3	13	3
0	4	9	1	3	4	3	4

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
10	8	10	8	0	0	0	0
14	7	14	7	0	0	0	0
16	1	10	13	13	3	4	14
0	1	9	13	3	4	14	13
12	16	4	13	4	14	4	14
8	16	0	13	14	13	14	13

G:=sub<GL(8,GF(17))| [7,3,11,12,9,5,0,13,0,0,0,0,3,4,3,4,7,3,6,5,10,14,9,5,0,0,0,0,14,13,3,4,6,16,11,1,12,12,5,5,8,7,9,10,16,16,1,1,6,16,6,16,14,14,14,14,8,7,8,7,13,13,13,13],[0,0,1,1,6,0,0,10,0,0,15,16,11,11,7,7,16,16,0,0,0,7,11,0,2,1,0,0,10,10,6,6,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0],[7,3,7,3,7,8,13,0,9,10,9,10,4,4,4,4,10,14,7,3,1,0,5,9,8,7,9,10,16,16,1,1,0,0,0,0,13,3,13,3,0,0,0,0,3,4,3,4,0,0,0,0,4,14,13,3,0,0,0,0,14,13,3,4],[7,3,10,14,16,0,12,8,9,10,8,7,1,1,16,16,10,14,10,14,10,9,4,0,8,7,8,7,13,13,13,13,0,0,0,0,13,3,4,14,0,0,0,0,3,4,14,13,0,0,0,0,4,14,4,14,0,0,0,0,14,13,14,13] >;

M₄(2).₉D₄ in GAP, Magma, Sage, TeX

M_4(2)._9D_4

% in TeX

G:=Group("M4(2).9D4");

// GroupNames label

G:=SmallGroup(128,781);

// by ID

G=gap.SmallGroup(128,781);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,58,2019,1018,248,2804,172,1027]);

// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^4=d^2=a^4,b*a*b=a^5,c*a*c^-1=a^-1,d*a*d^-1=a^5*b,c*b*c^-1=a^4*b,b*d=d*b,d*c*d^-1=c^3>;

// generators/relations

Export

Character table of M₄(2).₉D₄ in TeX

7	0	7	0	6	8	6	8
3	0	3	0	16	7	16	7
11	0	6	0	11	9	6	8
12	0	5	0	1	10	16	7
9	3	10	14	12	16	14	13
5	4	14	13	12	16	14	13
0	3	9	3	5	1	14	13
13	4	5	4	5	1	14	13

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
6	11	0	10	0	0	0	16
0	11	7	10	0	0	1	0
0	7	11	6	0	1	0	0
10	7	0	6	16	0	0	0

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
7	9	7	9	0	0	0	0
3	10	3	10	0	0	0	0
7	4	1	16	13	3	4	14
8	4	0	16	3	4	14	13
13	4	5	1	13	3	13	3
0	4	9	1	3	4	3	4

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
10	8	10	8	0	0	0	0
14	7	14	7	0	0	0	0
16	1	10	13	13	3	4	14
0	1	9	13	3	4	14	13
12	16	4	13	4	14	4	14
8	16	0	13	14	13	14	13

7	0	7	0	6	8	6	8
3	0	3	0	16	7	16	7
11	0	6	0	11	9	6	8
12	0	5	0	1	10	16	7
9	3	10	14	12	16	14	13
5	4	14	13	12	16	14	13
0	3	9	3	5	1	14	13
13	4	5	4	5	1	14	13

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
6	11	0	10	0	0	0	16
0	11	7	10	0	0	1	0
0	7	11	6	0	1	0	0
10	7	0	6	16	0	0	0

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
7	9	7	9	0	0	0	0
3	10	3	10	0	0	0	0
7	4	1	16	13	3	4	14
8	4	0	16	3	4	14	13
13	4	5	1	13	3	13	3
0	4	9	1	3	4	3	4

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
10	8	10	8	0	0	0	0
14	7	14	7	0	0	0	0
16	1	10	13	13	3	4	14
0	1	9	13	3	4	14	13
12	16	4	13	4	14	4	14
8	16	0	13	14	13	14	13

G = M4(2).9D4 order 128 = 27

9th non-split extension by M4(2) of D4 acting via D4/C2=C22

G = M₄(2).₉D₄ order 128 = 2⁷

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

7	0	7	0	6	8	6	8
3	0	3	0	16	7	16	7
11	0	6	0	11	9	6	8
12	0	5	0	1	10	16	7
9	3	10	14	12	16	14	13
5	4	14	13	12	16	14	13
0	3	9	3	5	1	14	13
13	4	5	4	5	1	14	13

0	0	16	2	0	0	0	0
0	0	16	1	0	0	0	0
1	15	0	0	0	0	0	0
1	16	0	0	0	0	0	0
6	11	0	10	0	0	0	16
0	11	7	10	0	0	1	0
0	7	11	6	0	1	0	0
10	7	0	6	16	0	0	0

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
7	9	7	9	0	0	0	0
3	10	3	10	0	0	0	0
7	4	1	16	13	3	4	14
8	4	0	16	3	4	14	13
13	4	5	1	13	3	13	3
0	4	9	1	3	4	3	4

7	9	10	8	0	0	0	0
3	10	14	7	0	0	0	0
10	8	10	8	0	0	0	0
14	7	14	7	0	0	0	0
16	1	10	13	13	3	4	14
0	1	9	13	3	4	14	13
12	16	4	13	4	14	4	14
8	16	0	13	14	13	14	13