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G = M4(2)○D8order 128 = 27

Central product of M4(2) and D8

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

Aliases: M4(2)D8, M4(2)Q16, M4(2)SD16, M4(2).52D4, C42.285C23, M4(2).34C23, C8○D87C2, C4○D89C4, (C2×D8)⋊20C4, C8.89(C2×D4), C8.26D43C2, (C4×C8)⋊24C22, (C2×Q16)⋊20C4, D8.14(C2×C4), C4.104(C4×D4), C4≀C219C22, (C2×SD16)⋊12C4, C8○D410C22, M4(2)(C4○D8), C4.34(C23×C4), C8.26(C22×C4), Q16.15(C2×C4), SD16.2(C2×C4), C22.23(C4×D4), C8⋊C443C22, Q8○M4(2)⋊15C2, (C2×C8).422C23, (C2×C4).214C24, C4○D8.26C22, C4○D4.26C23, D4.16(C22×C4), C4.205(C22×D4), Q8.16(C22×C4), C82M4(2)⋊11C2, C8.C414C22, M4(2)(C8.C4), C42⋊C2220C2, C23.111(C4○D4), (C22×C8).254C22, (C22×C4).933C23, C42⋊C2.302C22, (C2×M4(2)).245C22, C2.74(C2×C4×D4), (C2×C8).99(C2×C4), (C2×C4○D8).18C2, C4○D4.12(C2×C4), (C2×C8.C4)⋊24C2, C22.5(C2×C4○D4), (C2×D4).140(C2×C4), (C2×C4).1088(C2×D4), (C2×Q8).117(C2×C4), (C2×C4).270(C4○D4), (C2×C4).269(C22×C4), (C2×C4○D4).92C22, SmallGroup(128,1689)

Series: Derived Chief Lower central Upper central Jennings

C1C4 — M4(2)○D8
C1C2C4C2×C4C22×C4C2×M4(2)Q8○M4(2) — M4(2)○D8
C1C2C4 — M4(2)○D8
C1C4C2×M4(2) — M4(2)○D8
C1C2C2C2×C4 — M4(2)○D8

Generators and relations for M4(2)○D8
 G = < a,b,c,d | a8=b2=d2=1, c4=a4, bab=a5, ac=ca, ad=da, bc=cb, bd=db, dcd=a4c3 >

Subgroups: 340 in 226 conjugacy classes, 138 normal (36 characteristic)
C1, C2, C2 [×7], C4 [×4], C4 [×6], C22 [×3], C22 [×7], C8 [×4], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×12], D4 [×4], D4 [×10], Q8 [×4], Q8 [×2], C23, C23 [×2], C42 [×2], C22⋊C4, C4⋊C4, C2×C8 [×6], C2×C8 [×2], C2×C8 [×10], M4(2) [×8], M4(2) [×10], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×8], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], C4≀C2 [×8], C8.C4 [×4], C42⋊C2, C22×C8, C2×M4(2), C2×M4(2) [×2], C2×M4(2) [×4], C8○D4 [×8], C8○D4 [×4], C2×D8, C2×SD16 [×2], C2×Q16, C4○D8 [×8], C2×C4○D4 [×2], C82M4(2), C42⋊C22 [×2], C2×C8.C4, C8○D8 [×4], C8.26D4 [×4], Q8○M4(2) [×2], C2×C4○D8, M4(2)○D8
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, M4(2)○D8

Smallest permutation representation of M4(2)○D8
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 5)(3 7)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)
(1 11 18 31 5 15 22 27)(2 12 19 32 6 16 23 28)(3 13 20 25 7 9 24 29)(4 14 21 26 8 10 17 30)
(1 22)(2 23)(3 24)(4 17)(5 18)(6 19)(7 20)(8 21)(9 13)(10 14)(11 15)(12 16)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A4B4C4D4E4F···4M8A···8L8M···8V
order122222222444444···48···88···8
size112224444112224···42···24···4

44 irreducible representations

dim1111111111112224
type+++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4D4C4○D4C4○D4M4(2)○D8
kernelM4(2)○D8C82M4(2)C42⋊C22C2×C8.C4C8○D8C8.26D4Q8○M4(2)C2×C4○D8C2×D8C2×SD16C2×Q16C4○D8M4(2)C2×C4C23C1
# reps1121442124284224

Matrix representation of M4(2)○D8 in GL4(𝔽17) generated by

00160
00016
4000
0400
,
1000
0100
00160
00016
,
141400
31400
001414
00314
,
1000
01600
0010
00016
G:=sub<GL(4,GF(17))| [0,0,4,0,0,0,0,4,16,0,0,0,0,16,0,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[14,3,0,0,14,14,0,0,0,0,14,3,0,0,14,14],[1,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

M4(2)○D8 in GAP, Magma, Sage, TeX

M_4(2)\circ D_8
% in TeX

G:=Group("M4(2)oD8");
// GroupNames label

G:=SmallGroup(128,1689);
// by ID

G=gap.SmallGroup(128,1689);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,184,2019,521,2804,1411,172,124]);
// Polycyclic

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

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