Copied to
clipboard

G = (C2×D4).D4  order 128 = 27

4th non-split extension by C2×D4 of D4 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: C4⋊Q8⋊1C4, C8⋊C4⋊2C4, (C2×D4).4D4, (C2×Q8).4D4, C42.4(C2×C4), C8.2D4.1C2, C42⋊3C4.1C2, C2.7(C42⋊C4), C42.C4.1C2, C4.4D4.4C22, C22.17(C23⋊C4), (C2×C4).33(C22⋊C4), SmallGroup(128,139)

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

Derived series
C1 — C42 — (C2×D4).D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×D4 — C4.4D4 — C8.2D4 — (C2×D4).D4
Lower central
C1 — C2 — C22 — C2×C4 — C42 — (C2×D4).D4
Upper central
C1 — C2 — C22 — C2×C4 — C4.4D4 — (C2×D4).D4
Jennings
C1 — C2 — C2 — C22 — C2×C4 — C4.4D4 — (C2×D4).D4

Generators and relations for (C2×D4).D4
 G = < a,b,c,d,e | a2=b4=c2=1, d4=b2, e2=c, ebe-1=ab=ba, ac=ca, dad-1=eae-1=ab2, cbc=b-1, dbd-1=ab-1, dcd-1=ab-1c, ce=ec, ede-1=cd3 >

C1
C2
2C2
8C2
C22
2C4
4C4
4C4
4C22
8C4
8C22
16C4
C2×C4
2C23
2C8
2C2×C4
2C2×C4
2C8
4D4
4C2×C4
4Q8
4Q8
4Q8
8C8
8C2×C4
C42
C2×D4
C2×Q8
2C2×Q8
2C2×C8
4M4(2)
4C22⋊C4
4SD16
4Q16
4C4⋊C4
4SD16
4C22⋊C4
4Q16
C4.4D4
C8⋊C4
C4⋊Q8
2C2×SD16
2C4.10D4
2C23⋊C4
2C2×Q16
C42.C4
C8.2D4
C42⋊3C4
(C2×D4).D4

Character table of (C2×D4).D4

 class 12A2B2C4A4B4C4D4E4F8A8B8C8D
 size 1128488161616881616
ρ111111111111111    trivial
ρ2111111111-1-1-1-1-1    linear of order 2
ρ31111111-1-1-1-1-111    linear of order 2
ρ41111111-1-1111-1-1    linear of order 2
ρ5111-11-11-ii-111i-i    linear of order 4
ρ6111-11-11-ii1-1-1-ii    linear of order 4
ρ7111-11-11i-i1-1-1i-i    linear of order 4
ρ8111-11-11i-i-111-ii    linear of order 4
ρ922222-2-20000000    orthogonal lifted from D4
ρ10222-222-20000000    orthogonal lifted from D4
ρ114440-4000000000    orthogonal lifted from C23⋊C4
ρ1244-400000002-200    orthogonal lifted from C42⋊C4
ρ1344-40000000-2200    orthogonal lifted from C42⋊C4
ρ148-8000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of (C2×D4).D4
►On 32 points
Generators in S32
(1 23)(2 20)(3 17)(4 22)(5 19)(6 24)(7 21)(8 18)(9 31)(10 28)(11 25)(12 30)(13 27)(14 32)(15 29)(16 26)

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

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

(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)

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

Matrix representation of (C2×D4).D4 ►in GL8(𝔽17)

115000000
016000000
001150000
000160000
000011500
000001600
0160160101
0160160110
,
001620000
00010000
115000000
016000000
1601601002
000000161
11601161016
11601160016
,
10000000
01000000
001600000
000160000
000016200
00000100
0016011601
0016011610
,
7167161116110
7137131131414
107110016167
1041130333
161161081671
1614161481344
0313103143
168010161016
,
10000000
116000000
001620000
001610000
1010160015
10101601616
001610001
0016111601

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,15,16,0,0,0,0,16,16,0,0,1,0,0,0,0,0,0,0,15,16,0,0,16,16,0,0,0,0,1,0,0,0,0,0,0,0,15,16,1,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,1,0,16,0,1,1,0,0,15,16,0,0,16,16,16,0,0,0,16,0,0,0,2,1,0,0,0,0,1,1,0,0,0,0,1,0,16,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,2,1,16,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,16,16,0,0,0,16,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,2,1,16,16,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[7,7,10,10,16,16,0,1,16,13,7,4,1,14,3,6,7,7,1,1,16,16,1,8,16,13,10,13,10,14,3,0,11,11,0,0,8,8,10,10,16,3,16,3,16,13,3,16,1,14,16,3,7,4,14,10,10,14,7,3,1,4,3,16],[1,1,0,0,1,1,0,0,0,16,0,0,0,0,0,0,0,0,16,16,1,1,16,16,0,0,2,1,0,0,1,1,0,0,0,0,16,16,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,0,15,16,1,1] >;
 

(C2×D4).D4 in GAP, Magma, Sage, TeX 

(C_2\times D_4).D_4
 
% in TeX
 
G:=Group("(C2xD4).D4");
 
// GroupNames label
 
G:=SmallGroup(128,139);
 
// by ID
 
G=gap.SmallGroup(128,139);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,456,422,387,184,1690,745,1684,1411,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=1,d^4=b^2,e^2=c,e*b*e^-1=a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a*b^2,c*b*c=b^-1,d*b*d^-1=a*b^-1,d*c*d^-1=a*b^-1*c,c*e=e*c,e*d*e^-1=c*d^3>;
 
// generators/relations
 

Export

Subgroup lattice of (C2×D4).D4 in TeX
Character table of (C2×D4).D4 in TeX

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁