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G = D4:M4(2)  order 128 = 27

1st semidirect product of D4 and M4(2) acting via M4(2)/C2xC4=C2

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

Aliases: D4:3M4(2), C42.48D4, C42.607C23, D4:C8:26C2, (C4xC8):2C22, C4.80(C2xD8), C4:C8:44C22, (C4xD4).15C4, (C2xC4).127D8, C42.60(C2xC4), C4.93(C2xSD16), (C2xC4).98SD16, (C22xD4).24C4, (C22xC4).730D4, C4.18(C2xM4(2)), C4.34(D4:C4), C4:M4(2):15C2, (C4xD4).264C22, C42.12C4:11C2, (C2xC42).163C22, C22.24(D4:C4), C23.169(C22:C4), C2.12(C24.4C4), C2.9(C42:C22), (C2xC4xD4).7C2, (C2xC4:C4).40C4, C4:C4.179(C2xC4), C2.5(C2xD4:C4), (C2xD4).192(C2xC4), (C2xC4).1449(C2xD4), (C2xC4).78(C22:C4), (C2xC4).312(C22xC4), (C22xC4).185(C2xC4), C22.162(C2xC22:C4), SmallGroup(128,218)

Series: Derived Chief Lower central Upper central Jennings

C1C2xC4 — D4:M4(2)
C1C2C22C2xC4C42C2xC42C2xC4xD4 — D4:M4(2)
C1C2C2xC4 — D4:M4(2)
C1C2xC4C2xC42 — D4:M4(2)
C1C22C22C42 — D4:M4(2)

Generators and relations for D4:M4(2)
 G = < a,b,c,d | a4=b2=c8=d2=1, bab=cac-1=a-1, ad=da, cbc-1=ab, bd=db, dcd=c5 >

Subgroups: 340 in 156 conjugacy classes, 58 normal (28 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2xC4, C2xC4, C2xC4, D4, D4, C23, C23, C42, C22:C4, C4:C4, C4:C4, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C24, C4xC8, C22:C8, C4:C8, C4:C8, C2xC42, C2xC22:C4, C2xC4:C4, C4xD4, C4xD4, C2xM4(2), C23xC4, C22xD4, D4:C8, C4:M4(2), C42.12C4, C2xC4xD4, D4:M4(2)
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22:C4, M4(2), D8, SD16, C22xC4, C2xD4, D4:C4, C2xC22:C4, C2xM4(2), C2xD8, C2xSD16, C24.4C4, C2xD4:C4, C42:C22, D4:M4(2)

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

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E···4J4K···4P8A···8H8I8J8K8L
order122222222244444···44···48···88888
size111122444411112···24···44···48888

38 irreducible representations

dim11111111222224
type++++++++
imageC1C2C2C2C2C4C4C4D4D4D8SD16M4(2)C42:C22
kernelD4:M4(2)D4:C8C4:M4(2)C42.12C4C2xC4xD4C2xC4:C4C4xD4C22xD4C42C22xC4C2xC4C2xC4D4C2
# reps14111242224482

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

0100
16000
00160
00016
,
01600
16000
00160
0001
,
14300
3300
0002
0020
,
16000
01600
00160
0001
G:=sub<GL(4,GF(17))| [0,16,0,0,1,0,0,0,0,0,16,0,0,0,0,16],[0,16,0,0,16,0,0,0,0,0,16,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,0,2,0,0,2,0],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1] >;

D4:M4(2) in GAP, Magma, Sage, TeX

D_4\rtimes M_4(2)
% in TeX

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

G:=SmallGroup(128,218);
// by ID

G=gap.SmallGroup(128,218);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,1123,570,136,172]);
// Polycyclic

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

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