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G = M4(2)⋊19D4order 128 = 27

6th semidirect product of M4(2) and D4 acting via D4/C22=C2

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

Aliases: M4(2)⋊19D4, C4⋊Q814C4, C4.85(C4×D4), C41D410C4, (C2×D4).72D4, C4.4D411C4, Q8○M4(2)⋊6C2, (C22×C4).63D4, C426C421C2, C4.126C22≀C2, C42.147(C2×C4), C23.127(C2×D4), C22.2(C4⋊D4), C23.8(C22⋊C4), C42⋊C2212C2, C23.C234C2, (C22×C4).682C23, (C2×C42).283C22, C42⋊C2.19C22, C4.108(C22.D4), C2.23(C23.23D4), (C2×M4(2)).179C22, C22.26C24.19C2, (C2×D4).75(C2×C4), (C2×Q8).66(C2×C4), (C2×C4).54(C4○D4), (C2×C4).1003(C2×D4), (C2×C4).12(C22⋊C4), (C2×C4).184(C22×C4), (C2×C4○D4).19C22, C22.39(C2×C22⋊C4), SmallGroup(128,616)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — M4(2)⋊19D4
C1C2C22C23C22×C4C2×C4○D4Q8○M4(2) — M4(2)⋊19D4
C1C2C2×C4 — M4(2)⋊19D4
C1C4C22×C4 — M4(2)⋊19D4
C1C2C2C22×C4 — M4(2)⋊19D4

Generators and relations for M4(2)⋊19D4
 G = < a,b,c,d | a8=b2=c4=d2=1, bab=a5, cac-1=a3b, dad=a-1b, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 340 in 166 conjugacy classes, 54 normal (26 characteristic)
C1, C2, C2 [×6], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×13], D4 [×14], Q8 [×4], C23, C23 [×2], C23, C42 [×2], C42 [×3], C22⋊C4 [×6], C4⋊C4 [×4], C2×C8 [×6], M4(2) [×4], M4(2) [×6], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×D4 [×3], C2×Q8 [×2], C4○D4 [×8], C23⋊C4 [×2], C4≀C2 [×4], C2×C42, C42⋊C2 [×2], C4×D4 [×2], C4⋊D4 [×2], C4.4D4 [×2], C41D4, C4⋊Q8, C2×M4(2) [×2], C2×M4(2) [×2], C8○D4 [×4], C2×C4○D4 [×2], C426C4 [×2], C23.C23, C42⋊C22 [×2], C22.26C24, Q8○M4(2), M4(2)⋊19D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, M4(2)⋊19D4

Permutation representations of M4(2)⋊19D4
On 16 points - transitive group 16T251
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(2 6)(4 8)(10 14)(12 16)
(2 4 6 8)(9 15 13 11)
(1 10)(2 9)(3 12)(4 11)(5 14)(6 13)(7 16)(8 15)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (2,4,6,8)(9,15,13,11), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (2,4,6,8)(9,15,13,11), (1,10)(2,9)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(2,4,6,8),(9,15,13,11)], [(1,10),(2,9),(3,12),(4,11),(5,14),(6,13),(7,16),(8,15)])

G:=TransitiveGroup(16,251);

On 16 points - transitive group 16T299
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)
(1 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)
(1 14 5 10)(3 16 7 12)
(1 5)(2 15)(3 7)(4 9)(6 11)(8 13)

G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,14,5,10)(3,16,7,12), (1,5)(2,15)(3,7)(4,9)(6,11)(8,13)>;

G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,16)(2,13)(3,10)(4,15)(5,12)(6,9)(7,14)(8,11), (1,14,5,10)(3,16,7,12), (1,5)(2,15)(3,7)(4,9)(6,11)(8,13) );

G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,16),(2,13),(3,10),(4,15),(5,12),(6,9),(7,14),(8,11)], [(1,14,5,10),(3,16,7,12)], [(1,5),(2,15),(3,7),(4,9),(6,11),(8,13)])

G:=TransitiveGroup(16,299);

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F···4K4L···4P8A···8H
order12222222444444···44···48···8
size11222448112224···48···84···4

32 irreducible representations

dim11111111122224
type+++++++++
imageC1C2C2C2C2C2C4C4C4D4D4D4C4○D4M4(2)⋊19D4
kernelM4(2)⋊19D4C426C4C23.C23C42⋊C22C22.26C24Q8○M4(2)C4.4D4C41D4C4⋊Q8M4(2)C22×C4C2×D4C2×C4C1
# reps12121142242244

Matrix representation of M4(2)⋊19D4 in GL4(𝔽5) generated by

0002
0010
0300
4000
,
1000
0100
0040
0004
,
1000
0200
0010
0003
,
0020
0004
3000
0400
G:=sub<GL(4,GF(5))| [0,0,0,4,0,0,3,0,0,1,0,0,2,0,0,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,2,0,0,0,0,1,0,0,0,0,3],[0,0,3,0,0,0,0,4,2,0,0,0,0,4,0,0] >;

M4(2)⋊19D4 in GAP, Magma, Sage, TeX

M_4(2)\rtimes_{19}D_4
% in TeX

G:=Group("M4(2):19D4");
// GroupNames label

G:=SmallGroup(128,616);
// by ID

G=gap.SmallGroup(128,616);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,1018,521,248,2804,1411,2028,1027]);
// 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^3*b,d*a*d=a^-1*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations

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