M4(2).13D4 - GroupNames

G = M₄(2).₁₃D₄ order 128 = 2⁷

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

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

Aliases: M₄(2).₁₃D₄, (C₂×Q₈).₇Q₈, C₄⋊C₄.₁₀₅D₄, (C₂×C₈).₁₆₁D₄, C₄.₁₅₂(C₄⋊D₄), C₄.₁₀₁(C₂²⋊Q₈), C₂².₆(C₂²⋊Q₈), C₂.₂₉(D₄.₃D₄), C₂.₂₁(D₄.₅D₄), C₂³.₂₈₁(C₄○D₄), M₄(2)⋊C₄.₇C₂, (C₂²×C₈).₁₇₁C₂², (C₂²×C₄).₇₃₆C₂³, C₂³.₃₈D₄.₉C₂, C₂.₅(C₂³.Q₈), (C₂²×Q₈).₇₄C₂², C₂².₂₄₂(C₄⋊D₄), C₂².C₄².₁₅C₂, C₄²⋊C₂.₆₇C₂², C₂².₉(C₄²⋊₂C₂), (C₂×M₄(2)).₂₈C₂², C₄².₆C₂².₁C₂, (C₂×C₄).₂₁(C₂×Q₈), (C₂×C₄).₁₀₅₄(C₂×D₄), (C₂×C₈.C₄).₂₀C₂, (C₂×C₄).₃₅₁(C₄○D₄), (C₂×C₄⋊C₄).₁₄₀C₂², (C₂×Q₈⋊C₄).₂₅C₂, (C₂×C₄.₁₀D₄).₃C₂, SmallGroup(128,796)

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

Derived series

C₁ — C₂²×C₄ — M₄(2).₁₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂×Q₈⋊C₄ — M₄(2).₁₃D₄

Lower central

C₁ — C₂ — C₂²×C₄ — M₄(2).₁₃D₄

Upper central

C₁ — C₂² — C₂²×C₄ — M₄(2).₁₃D₄

Jennings

C₁ — C₂ — C₂ — C₂²×C₄ — M₄(2).₁₃D₄

Generators and relations for M₄(2).₁₃D₄
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=a²bc^-1 >

Subgroups: 200 in 104 conjugacy classes, 42 normal (38 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₄.₁₀D₄, Q₈⋊C₄, C₄⋊C₈, C₄.Q₈, C₂.D₈, C₈.C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂²×Q₈, C₂².C₄², C₂×C₄.₁₀D₄, C₂×Q₈⋊C₄, C₂³.₃₈D₄, C₄².₆C₂², M₄(2)⋊C₄, C₂×C₈.C₄, M₄(2).₁₃D₄
Quotients: C₁, C₂, C₂², D₄, Q₈, C₂³, C₂×D₄, C₂×Q₈, C₄○D₄, C₄⋊D₄, C₂²⋊Q₈, C₄²⋊₂C₂, C₂³.Q₈, D₄.₃D₄, D₄.₅D₄, M₄(2).₁₃D₄

Character table of M₄(2).₁₃D₄

class	1	2A	2B	2C	2D	2E	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	8A	8B	8C	8D	8E	8F	8G	8H	8I	8J
size	1	1	1	1	2	2	2	2	2	2	8	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	-2	2	2	-2	0	0	0	0	0	0	-2	2	2	-2	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₀	2	2	2	2	-2	-2	2	-2	-2	2	0	-2	0	0	2	0	0	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	0	0	0	0	0	0	0	0	0	0	0	-2	2	0	orthogonal lifted from D₄
ρ₁₂	2	2	2	2	-2	-2	-2	2	2	-2	0	0	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	2	-2	-2	2	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	0	orthogonal lifted from D₄
ρ₁₄	2	-2	-2	2	-2	2	2	2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	0	0	2	-2	0	orthogonal lifted from D₄
ρ₁₅	2	-2	-2	2	-2	2	-2	-2	2	2	0	0	-2	0	0	2	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	0	0	2	0	0	-2	0	0	0	0	0	0	0	0	0	0	symplectic lifted from Q₈, Schur index 2
ρ₁₇	2	-2	-2	2	2	-2	2	-2	2	-2	-2i	0	0	2i	0	0	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	-2	-2	2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	2i	0	-2i	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	-2	-2	2	2	-2	-2	2	-2	2	0	0	0	0	0	0	0	0	0	0	-2i	0	2i	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	-2i	0	0	0	2i	complex lifted from C₄○D₄
ρ₂₁	2	2	2	2	2	2	-2	-2	-2	-2	0	0	0	0	0	0	0	0	0	0	0	2i	0	0	0	-2i	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	2	-2	2	-2	2	-2	2i	0	0	-2i	0	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).₁₃D₄
►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 9)(2 14)(3 11)(4 16)(5 13)(6 10)(7 15)(8 12)(17 34)(18 39)(19 36)(20 33)(21 38)(22 35)(23 40)(24 37)(25 41)(26 46)(27 43)(28 48)(29 45)(30 42)(31 47)(32 44)(49 61)(50 58)(51 63)(52 60)(53 57)(54 62)(55 59)(56 64)

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

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

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

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

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

Matrix representation of M₄(2).₁₃D₄ ►in GL₆(𝔽₁₇)

13	0	0	0	0	0
0	4	0	0	0	0
0	0	3	2	15	0
0	0	5	16	0	15
0	0	1	10	14	15
0	0	14	14	12	1

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	3	2	16	0
0	0	5	16	0	16

0	1	0	0	0	0
16	0	0	0	0	0
0	0	14	3	0	0
0	0	3	3	0	0
0	0	12	6	3	3
0	0	11	10	3	14

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

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

M₄(2).₁₃D₄ in GAP, Magma, Sage, TeX

M_4(2)._{13}D_4

% in TeX

G:=Group("M4(2).13D4");

// GroupNames label

G:=SmallGroup(128,796);

// by ID

G=gap.SmallGroup(128,796);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,168,141,64,422,387,352,2019,521,248,2804,4037,1027,124]);

// 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^2*b*c^-1>;

// generators/relations

Export

Character table of M₄(2).₁₃D₄ in TeX

G = M4(2).13D4 order 128 = 27

13rd non-split extension by M4(2) of D4 acting via D4/C2=C22

G = M₄(2).₁₃D₄ order 128 = 2⁷

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