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G = D4xM4(2)  order 128 = 27

Direct product of D4 and M4(2)

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

Aliases: D4xM4(2), C42.268C23, C8:15(C2xD4), (C8xD4):39C2, C4:C4o2M4(2), C8:9D4:33C2, C8:6D4:34C2, C4:C8:89C22, (C4xC8):58C22, (C4xD4).28C4, C4.152(C4xD4), C4:2(C2xM4(2)), C24.82(C2xC4), C22:C4o2M4(2), C22.67(C4xD4), C8:C4:61C22, (C4xM4(2)):33C2, C22:C8:78C22, C42.210(C2xC4), (C2xC8).406C23, (C2xC4).653C24, (C22xC8):53C22, (C22xD4).40C4, C4.199(C22xD4), C22:2(C2xM4(2)), C4:M4(2):35C2, C24.4C4:33C2, (C4xD4).360C22, C2.17(Q8oM4(2)), (C2xM4(2)):79C22, (C22xM4(2)):27C2, C22.180(C23xC4), C23.141(C22xC4), (C2xC42).760C22, (C23xC4).528C22, (C22xC4).920C23, C2.12(C22xM4(2)), C2.51(C2xC4xD4), (C2xC4xD4).72C2, C4:C4o(C2xM4(2)), (C2xC4:C4).72C4, (C4xD4)o(C2xM4(2)), (C2xD4)o(C2xM4(2)), C4:C4.247(C2xC4), C4.304(C2xC4oD4), C22:C4o(C2xM4(2)), (C2xD4).249(C2xC4), (C2xC4).1085(C2xD4), C22:C4.73(C2xC4), (C2xC22:C4).49C4, (C2xC4).831(C4oD4), (C22xC4).343(C2xC4), (C2xC4).294(C22xC4), SmallGroup(128,1666)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — D4xM4(2)
C1C2C4C2xC4C22xC4C2xM4(2)C22xM4(2) — D4xM4(2)
C1C22 — D4xM4(2)
C1C2xC4 — D4xM4(2)
C1C2C2C2xC4 — D4xM4(2)

Generators and relations for D4xM4(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, C2, C4, C4, C4, C22, C22, C22, C8, C8, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, M4(2), M4(2), C22xC4, C22xC4, C22xC4, C2xD4, C2xD4, C2xD4, C24, C4xC8, C8:C4, C22:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4xD4, C22xC8, C2xM4(2), C2xM4(2), C2xM4(2), C23xC4, C22xD4, C4xM4(2), C24.4C4, C4:M4(2), C8xD4, C8:9D4, C8:6D4, C2xC4xD4, C22xM4(2), D4xM4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, M4(2), C22xC4, C2xD4, C4oD4, C24, C4xD4, C2xM4(2), C23xC4, C22xD4, C2xC4oD4, C2xC4xD4, C22xM4(2), Q8oM4(2), D4xM4(2)

Smallest permutation representation of D4xM4(2)
On 32 points
Generators in S32
(1 10 23 26)(2 11 24 27)(3 12 17 28)(4 13 18 29)(5 14 19 30)(6 15 20 31)(7 16 21 32)(8 9 22 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 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 29)(10 26)(11 31)(12 28)(13 25)(14 30)(15 27)(16 32)

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

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

G=PermutationGroup([[(1,10,23,26),(2,11,24,27),(3,12,17,28),(4,13,18,29),(5,14,19,30),(6,15,20,31),(7,16,21,32),(8,9,22,25)], [(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)], [(1,23),(2,20),(3,17),(4,22),(5,19),(6,24),(7,21),(8,18),(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++++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4C4D4C4oD4M4(2)Q8oM4(2)
kernelD4xM4(2)C4xM4(2)C24.4C4C4:M4(2)C8xD4C8:9D4C8:6D4C2xC4xD4C22xM4(2)C2xC22:C4C2xC4:C4C4xD4C22xD4M4(2)C2xC4D4C2
# reps11212421242824482

Matrix representation of D4xM4(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] >;

D4xM4(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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