Copied to
clipboard

G = D4.C42  order 128 = 27

1st non-split extension by D4 of C42 acting via C42/C2×C4=C2

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

Aliases: D4.1C42, Q8.1C42, C42.380D4, C4≀C2⋊4C4, (C4×D4)⋊10C4, C42⋊7(C2×C4), (C4×Q8)⋊10C4, C4.162(C4×D4), C4.2(C2×C42), C42⋊4C4⋊6C2, C22.27(C4×D4), C42⋊6C4⋊11C2, M4(2)⋊13(C2×C4), (C4×M4(2))⋊22C2, (C22×C4).129D4, C23.542(C2×D4), C4.2(C42⋊C2), (C2×C42).234C22, (C22×C4).1306C23, C2.5(C42⋊C22), C42⋊C2.262C22, (C2×M4(2)).307C22, (C2×C4≀C2).5C2, (C4×C4○D4).6C2, C4⋊C4.188(C2×C4), C4○D4.14(C2×C4), C2.17(C4×C22⋊C4), (C2×D4).197(C2×C4), (C2×C4).1500(C2×D4), (C2×Q8).180(C2×C4), (C2×C4).537(C4○D4), (C2×C4).349(C22×C4), (C2×C4).326(C22⋊C4), (C2×C4○D4).251C22, C22.118(C2×C22⋊C4), SmallGroup(128,491)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C4 — D4.C42
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C42⋊4C4 — D4.C42
Lower central
C1 — C2 — C4 — D4.C42
Upper central
C1 — C2×C4 — C2×C42 — D4.C42
Jennings
C1 — C2 — C2 — C22×C4 — D4.C42

Generators and relations for D4.C42
 G = < a,b,c,d | a4=b2=c4=d4=1, bab=dad-1=a-1, dcd-1=ac=ca, cbc-1=ab, dbd-1=a2b >

Subgroups: 284 in 160 conjugacy classes, 76 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C4, C22, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, C23, C42, C42, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C4○D4, C2.C42, C4×C8, C8⋊C4, C4≀C2, C2×C42, C2×C42, C2×C42, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C2×M4(2), C2×C4○D4, C42⋊6C4, C42⋊4C4, C4×M4(2), C2×C4≀C2, C4×C4○D4, D4.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C42, C22⋊C4, C22×C4, C2×D4, C4○D4, C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4, C4×C22⋊C4, C42⋊C22, D4.C42

Smallest permutation representation of D4.C42
►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 31)(2 30)(3 29)(4 32)(5 12)(6 11)(7 10)(8 9)(13 28)(14 27)(15 26)(16 25)(17 22)(18 21)(19 24)(20 23)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4N4O···4AB8A···8H
order1222222244444···44···48···8
size1111224411112···24···44···4

44 irreducible representations

dim1111111112224
type++++++++
imageC1C2C2C2C2C2C4C4C4D4D4C4○D4C42⋊C22
kernelD4.C42C42⋊6C4C42⋊4C4C4×M4(2)C2×C4≀C2C4×C4○D4C4≀C2C4×D4C4×Q8C42C22×C4C2×C4C2
# reps12112116442244

Matrix representation of D4.C42 ►in GL6(𝔽17)

1600000
0160000
004000
000400
0000130
0000013
,
1620000
010000
000018
0000016
001800
0001600
,
100000
1160000
0041500
00161300
0000169
0000131
,
1380000
040000
0000160
0000016
001000
000100

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,8,16,0,0,1,0,0,0,0,0,8,16,0,0],[1,1,0,0,0,0,0,16,0,0,0,0,0,0,4,16,0,0,0,0,15,13,0,0,0,0,0,0,16,13,0,0,0,0,9,1],[13,0,0,0,0,0,8,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0] >;
 

D4.C42 in GAP, Magma, Sage, TeX 

D_4.C_4^2
 
% in TeX
 
G:=Group("D4.C4^2");
 
// GroupNames label
 
G:=SmallGroup(128,491);
 
// by ID
 
G=gap.SmallGroup(128,491);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,2,-2,112,141,232,723,100,2019,248,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=d*a*d^-1=a^-1,d*c*d^-1=a*c=c*a,c*b*c^-1=a*b,d*b*d^-1=a^2*b>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁