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G = D4×M4(2)  order 128 = 27

Direct product of D4 and M4(2)

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: D4×M4(2), C42.268C23, C815(C2×D4), (C8×D4)⋊39C2, C4⋊C42M4(2), C89D433C2, C86D434C2, C4⋊C889C22, (C4×C8)⋊58C22, (C4×D4).28C4, C4.152(C4×D4), C42(C2×M4(2)), C24.82(C2×C4), C22⋊C42M4(2), C22.67(C4×D4), C8⋊C461C22, (C4×M4(2))⋊33C2, C22⋊C878C22, C42.210(C2×C4), (C2×C8).406C23, (C2×C4).653C24, (C22×C8)⋊53C22, (C22×D4).40C4, C4.199(C22×D4), C222(C2×M4(2)), C4⋊M4(2)⋊35C2, C24.4C433C2, (C4×D4).360C22, C2.17(Q8○M4(2)), (C2×M4(2))⋊79C22, (C22×M4(2))⋊27C2, C22.180(C23×C4), C23.141(C22×C4), (C2×C42).760C22, (C23×C4).528C22, (C22×C4).920C23, C2.12(C22×M4(2)), C2.51(C2×C4×D4), (C2×C4×D4).72C2, C4⋊C4(C2×M4(2)), (C2×C4⋊C4).72C4, (C4×D4)(C2×M4(2)), (C2×D4)(C2×M4(2)), C4⋊C4.247(C2×C4), C4.304(C2×C4○D4), C22⋊C4(C2×M4(2)), (C2×D4).249(C2×C4), (C2×C4).1085(C2×D4), C22⋊C4.73(C2×C4), (C2×C22⋊C4).49C4, (C2×C4).831(C4○D4), (C22×C4).343(C2×C4), (C2×C4).294(C22×C4), SmallGroup(128,1666)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D4×M4(2)
C1C2C4C2×C4C22×C4C2×M4(2)C22×M4(2) — D4×M4(2)
C1C22 — D4×M4(2)
C1C2×C4 — D4×M4(2)
C1C2C2C2×C4 — D4×M4(2)

Generators and relations for D4×M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c5 >

Subgroups: 420 in 272 conjugacy classes, 150 normal (38 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×4], C4 [×7], C22, C22 [×6], C22 [×20], C8 [×4], C8 [×6], C2×C4 [×6], C2×C4 [×8], C2×C4 [×23], D4 [×4], D4 [×6], C23, C23 [×4], C23 [×10], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×2], C2×C8 [×8], C2×C8 [×8], M4(2) [×4], M4(2) [×14], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×2], C2×D4 [×2], C2×D4 [×4], C24 [×2], C4×C8 [×2], C8⋊C4 [×2], C22⋊C8 [×8], C4⋊C8 [×4], C2×C42, C2×C22⋊C4 [×2], C2×C4⋊C4, C4×D4 [×4], C4×D4 [×4], C22×C8 [×4], C2×M4(2) [×2], C2×M4(2) [×6], C2×M4(2) [×8], C23×C4 [×2], C22×D4, C4×M4(2), C24.4C4 [×2], C4⋊M4(2), C8×D4 [×2], C89D4 [×4], C86D4 [×2], C2×C4×D4, C22×M4(2) [×2], D4×M4(2)
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×4], C23 [×15], M4(2) [×4], C22×C4 [×14], C2×D4 [×6], C4○D4 [×2], C24, C4×D4 [×4], C2×M4(2) [×6], C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C22×M4(2), Q8○M4(2), D4×M4(2)

Smallest permutation representation of D4×M4(2)
On 32 points
Generators in S32
(1 10 17 26)(2 11 18 27)(3 12 19 28)(4 13 20 29)(5 14 21 30)(6 15 22 31)(7 16 23 32)(8 9 24 25)
(1 5)(2 6)(3 7)(4 8)(9 29)(10 30)(11 31)(12 32)(13 25)(14 26)(15 27)(16 28)(17 21)(18 22)(19 23)(20 24)
(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 17)(2 22)(3 19)(4 24)(5 21)(6 18)(7 23)(8 20)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D···2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H8I···8T
order12222···22244444···444448···88···8
size11112···24411112···244442···24···4

50 irreducible representations

dim11111111111112224
type++++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4D4C4○D4M4(2)Q8○M4(2)
kernelD4×M4(2)C4×M4(2)C24.4C4C4⋊M4(2)C8×D4C89D4C86D4C2×C4×D4C22×M4(2)C2×C22⋊C4C2×C4⋊C4C4×D4C22×D4M4(2)C2×C4D4C2
# reps11212421242824482

Matrix representation of D4×M4(2) in GL4(𝔽17) generated by

16200
16100
00160
00016
,
1000
11600
00160
00016
,
1000
0100
0001
00130
,
16000
01600
0010
00016
G:=sub<GL(4,GF(17))| [16,16,0,0,2,1,0,0,0,0,16,0,0,0,0,16],[1,1,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,0,13,0,0,1,0],[16,0,0,0,0,16,0,0,0,0,1,0,0,0,0,16] >;

D4×M4(2) in GAP, Magma, Sage, TeX

D_4\times M_4(2)
% in TeX

G:=Group("D4xM4(2)");
// GroupNames label

G:=SmallGroup(128,1666);
// by ID

G=gap.SmallGroup(128,1666);
# by ID

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

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

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