M4(2).11D4 - GroupNames

G = M₄(2).₁₁D₄ order 128 = 2⁷

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

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

Aliases: M₄(2).₁₁D₄, (C₂×C₈).₄₈D₄, C₄.₇₄C₂²≀C₂, (C₂×D₄).₁₁₃D₄, (C₂×Q₈).₁₀₄D₄, C₄.₁₅₀(C₄⋊D₄), C₂².C₄²⋊₁₀C₂, C₂.₂₀(D₄.₅D₄), C₂.₂₇(D₄.₃D₄), C₂³.₂₇₉(C₄○D₄), (C₂²×C₄).₇₃₁C₂³, (C₂²×C₈).₁₆₉C₂², (C₂²×Q₈).₇₁C₂², C₂².₂₃₇(C₄⋊D₄), C₂³.₃₆D₄.₁₁C₂, C₄.₇₅(C₂².D₄), C₂².₁₃(C₄.₄D₄), C₂.₂₅(C₂³.₁₀D₄), (C₂×M₄(2)).₂₃₄C₂², C₂².₁₄(C₂².D₄), (C₂×C₈.C₄)⋊₁₅C₂, (C₂×Q₈⋊C₄)⋊₂₇C₂, (C₂×C₄).₁₃₇₂(C₂×D₄), (C₂×C₈.C₂²).₈C₂, (C₂×C₄).₃₄₇(C₄○D₄), (C₂×C₄.₁₀D₄)⋊₂₅C₂, (C₂×C₄⋊C₄).₁₃₀C₂², (C₂²×C₈)⋊C₂.₁C₂, (C₂×C₄○D₄).₆₅C₂², SmallGroup(128,784)

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

Derived series

C₁ — C₂²×C₄ — M₄(2).₁₁D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×M₄(2) — C₂×C₄.₁₀D₄ — M₄(2).₁₁D₄

Lower central

C₁ — C₂ — C₂²×C₄ — M₄(2).₁₁D₄

Upper central

C₁ — C₂² — C₂²×C₄ — M₄(2).₁₁D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — M₄(2).₁₁D₄

Generators and relations for M₄(2).₁₁D₄
G = < a,b,c,d | a⁸=b²=c⁴=d²=1, bab=a⁵, cac^-1=ab, dad=a³, bc=cb, dbd=a⁴b, dcd=a⁶c^-1 >

Subgroups: 272 in 131 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), SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂²⋊C₈, C₄.₁₀D₄, D₄⋊C₄, Q₈⋊C₄, C₈.C₄, C₂×C₄⋊C₄, C₂²×C₈, C₂×M₄(2), C₂×SD₁₆, C₂×Q₁₆, C₈.C₂², C₂²×Q₈, C₂×C₄○D₄, C₂².C₄², (C₂²×C₈)⋊C₂, C₂×C₄.₁₀D₄, C₂×Q₈⋊C₄, C₂³.₃₆D₄, C₂×C₈.C₄, C₂×C₈.C₂², M₄(2).₁₁D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂³.₁₀D₄, D₄.₃D₄, D₄.₅D₄, M₄(2).₁₁D₄

Character table of M₄(2).₁₁D₄

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	4E	4F	4G	4H	4I	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	8	2	2	2	2	8	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	-2	2	-2	2	0	0	0	0	0	0	0	0	0	-2	0	0	2	0	0	orthogonal lifted from D₄
ρ₁₀	2	-2	2	-2	2	-2	0	-2	2	-2	2	0	0	0	0	0	0	0	0	0	2	0	0	-2	0	0	orthogonal lifted from D₄
ρ₁₁	2	2	2	2	-2	-2	2	-2	2	2	-2	0	0	-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	2	2	-2	-2	0	-2	0	0	2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₃	2	2	2	2	-2	-2	0	2	-2	-2	2	0	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	2	2	-2	-2	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₅	2	2	2	2	-2	-2	0	2	-2	-2	2	0	0	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	-2	2	2	-2	0	0	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	-2	-2	2	2	0	0	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	-2	-2	2	2	0	0	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	complex lifted from C₄○D₄
ρ₁₉	2	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	-2	2	-2	0	2	-2	2	-2	2i	0	0	-2i	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	2	2	2	2	2	0	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	2	-2	2	-2	0	2	-2	2	-2	-2i	0	0	2i	0	0	0	0	0	0	0	0	0	0	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	0	2√2	0	0	0	0	0	0	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₄	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	symplectic lifted from D₄.₅D₄, Schur index 2
ρ₂₅	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	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	-2√-2	0	0	0	0	0	0	0	complex lifted from D₄.₃D₄

Smallest permutation representation of M₄(2).₁₁D₄
►On 64 points

Generators in S₆₄

(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)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)

(1 27)(2 32)(3 29)(4 26)(5 31)(6 28)(7 25)(8 30)(9 64)(10 61)(11 58)(12 63)(13 60)(14 57)(15 62)(16 59)(17 50)(18 55)(19 52)(20 49)(21 54)(22 51)(23 56)(24 53)(33 45)(34 42)(35 47)(36 44)(37 41)(38 46)(39 43)(40 48)

(1 57 22 43)(2 11 23 36)(3 63 24 41)(4 9 17 34)(5 61 18 47)(6 15 19 40)(7 59 20 45)(8 13 21 38)(10 55 35 31)(12 53 37 29)(14 51 39 27)(16 49 33 25)(26 64 50 42)(28 62 52 48)(30 60 54 46)(32 58 56 44)

(2 4)(3 7)(6 8)(9 38)(10 33)(11 36)(12 39)(13 34)(14 37)(15 40)(16 35)(17 23)(19 21)(20 24)(26 28)(27 31)(30 32)(41 61)(42 64)(43 59)(44 62)(45 57)(46 60)(47 63)(48 58)(50 52)(51 55)(54 56)

G:=sub<Sym(64)| (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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,64)(10,61)(11,58)(12,63)(13,60)(14,57)(15,62)(16,59)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,57,22,43)(2,11,23,36)(3,63,24,41)(4,9,17,34)(5,61,18,47)(6,15,19,40)(7,59,20,45)(8,13,21,38)(10,55,35,31)(12,53,37,29)(14,51,39,27)(16,49,33,25)(26,64,50,42)(28,62,52,48)(30,60,54,46)(32,58,56,44), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56)>;

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)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64), (1,27)(2,32)(3,29)(4,26)(5,31)(6,28)(7,25)(8,30)(9,64)(10,61)(11,58)(12,63)(13,60)(14,57)(15,62)(16,59)(17,50)(18,55)(19,52)(20,49)(21,54)(22,51)(23,56)(24,53)(33,45)(34,42)(35,47)(36,44)(37,41)(38,46)(39,43)(40,48), (1,57,22,43)(2,11,23,36)(3,63,24,41)(4,9,17,34)(5,61,18,47)(6,15,19,40)(7,59,20,45)(8,13,21,38)(10,55,35,31)(12,53,37,29)(14,51,39,27)(16,49,33,25)(26,64,50,42)(28,62,52,48)(30,60,54,46)(32,58,56,44), (2,4)(3,7)(6,8)(9,38)(10,33)(11,36)(12,39)(13,34)(14,37)(15,40)(16,35)(17,23)(19,21)(20,24)(26,28)(27,31)(30,32)(41,61)(42,64)(43,59)(44,62)(45,57)(46,60)(47,63)(48,58)(50,52)(51,55)(54,56) );

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),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64)], [(1,27),(2,32),(3,29),(4,26),(5,31),(6,28),(7,25),(8,30),(9,64),(10,61),(11,58),(12,63),(13,60),(14,57),(15,62),(16,59),(17,50),(18,55),(19,52),(20,49),(21,54),(22,51),(23,56),(24,53),(33,45),(34,42),(35,47),(36,44),(37,41),(38,46),(39,43),(40,48)], [(1,57,22,43),(2,11,23,36),(3,63,24,41),(4,9,17,34),(5,61,18,47),(6,15,19,40),(7,59,20,45),(8,13,21,38),(10,55,35,31),(12,53,37,29),(14,51,39,27),(16,49,33,25),(26,64,50,42),(28,62,52,48),(30,60,54,46),(32,58,56,44)], [(2,4),(3,7),(6,8),(9,38),(10,33),(11,36),(12,39),(13,34),(14,37),(15,40),(16,35),(17,23),(19,21),(20,24),(26,28),(27,31),(30,32),(41,61),(42,64),(43,59),(44,62),(45,57),(46,60),(47,63),(48,58),(50,52),(51,55),(54,56)]])

Matrix representation of M₄(2).₁₁D₄ ►in GL₆(𝔽₁₇)

16	15	0	0	0	0
1	1	0	0	0	0
0	0	4	0	4	13
0	0	0	13	4	13
0	0	15	2	0	4
0	0	15	2	13	0

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

4	0	0	0	0	0
13	13	0	0	0	0
0	0	0	6	0	6
0	0	6	0	6	0
0	0	0	14	0	11
0	0	14	0	11	0

1	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	1	0	1	0
0	0	0	16	0	16

G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,4,0,15,15,0,0,0,13,2,2,0,0,4,4,0,13,0,0,13,13,4,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,6,0,14,0,0,6,0,14,0,0,0,0,6,0,11,0,0,6,0,11,0],[1,16,0,0,0,0,0,16,0,0,0,0,0,0,16,0,1,0,0,0,0,1,0,16,0,0,0,0,1,0,0,0,0,0,0,16] >;

M₄(2).₁₁D₄ in GAP, Magma, Sage, TeX

M_4(2)._{11}D_4

% in TeX

G:=Group("M4(2).11D4");

// GroupNames label

G:=SmallGroup(128,784);

// by ID

G=gap.SmallGroup(128,784);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,456,422,387,58,2019,1018,248,2804,1411,172,4037,1027]);

// 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*b,d*a*d=a^3,b*c=c*b,d*b*d=a^4*b,d*c*d=a^6*c^-1>;

// generators/relations

Export

Character table of M₄(2).₁₁D₄ in TeX

G = M4(2).11D4 order 128 = 27

11st non-split extension by M4(2) of D4 acting via D4/C2=C22

G = M₄(2).₁₁D₄ order 128 = 2⁷

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