Copied to
clipboard

G = D4⋊M4(2)  order 128 = 27

1st semidirect product of D4 and M4(2) acting via M4(2)/C2×C4=C2

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

Aliases: D43M4(2), C42.48D4, C42.607C23, D4⋊C826C2, (C4×C8)⋊2C22, C4.80(C2×D8), C4⋊C844C22, (C4×D4).15C4, (C2×C4).127D8, C42.60(C2×C4), C4.93(C2×SD16), (C2×C4).98SD16, (C22×D4).24C4, (C22×C4).730D4, C4.18(C2×M4(2)), C4.34(D4⋊C4), C4⋊M4(2)⋊15C2, (C4×D4).264C22, C42.12C411C2, (C2×C42).163C22, C22.24(D4⋊C4), C23.169(C22⋊C4), C2.12(C24.4C4), C2.9(C42⋊C22), (C2×C4×D4).7C2, (C2×C4⋊C4).40C4, C4⋊C4.179(C2×C4), C2.5(C2×D4⋊C4), (C2×D4).192(C2×C4), (C2×C4).1449(C2×D4), (C2×C4).78(C22⋊C4), (C2×C4).312(C22×C4), (C22×C4).185(C2×C4), C22.162(C2×C22⋊C4), SmallGroup(128,218)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — D4⋊M4(2)
C1C2C22C2×C4C42C2×C42C2×C4×D4 — D4⋊M4(2)
C1C2C2×C4 — D4⋊M4(2)
C1C2×C4C2×C42 — 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 [×3], C2 [×6], C4 [×2], C4 [×4], C4 [×5], C22, C22 [×2], C22 [×18], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×15], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×4], M4(2) [×2], C22×C4 [×3], C22×C4 [×9], C2×D4 [×2], C2×D4 [×5], C24, C4×C8 [×2], C22⋊C8, C4⋊C8 [×2], C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4 [×4], C4×D4 [×2], C2×M4(2), C23×C4, C22×D4, D4⋊C8 [×4], C4⋊M4(2), C42.12C4, C2×C4×D4, D4⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], M4(2) [×4], D8 [×2], SD16 [×2], C22×C4, C2×D4 [×2], D4⋊C4 [×4], C2×C22⋊C4, C2×M4(2) [×2], C2×D8, C2×SD16, C24.4C4, C2×D4⋊C4, C42⋊C22, D4⋊M4(2)

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

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

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

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

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)C42.12C4C2×C4×D4C2×C4⋊C4C4×D4C22×D4C42C22×C4C2×C4C2×C4D4C2
# 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

׿
×
𝔽