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G = C42.16D4  order 128 = 27

16th non-split extension by C42 of D4 acting faithfully

p-group, non-abelian, nilpotent (class 4), monomial, rational

Aliases: C42.16D4, 2- 1+4.1C22, (C2×D4).37D4, (C2×Q8).36D4, C2.27C2≀C22, C4⋊Q8.97C22, D4.10D4⋊1C2, (C2×Q8).2C23, C42.C4⋊2C2, C42.3C4⋊1C2, D4.8D4.2C2, C22.51C22≀C2, C4.10D4.1C22, C4.4D4.22C22, C22.57C24⋊3C2, (C2×C4).20(C2×D4), SmallGroup(128,935)

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

Derived series
C1 — C2 — C2×Q8 — C42.16D4
Chief series
C1 — C2 — C22 — C2×C4 — C2×Q8 — C4.4D4 — C22.57C24 — C42.16D4
Lower central
C1 — C2 — C22 — C2×Q8 — C42.16D4
Upper central
C1 — C2 — C22 — C2×Q8 — C42.16D4
Jennings
C1 — C2 — C22 — C2×Q8 — C42.16D4

Generators and relations for C42.16D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=a-1b-1, dad=a-1b, cbc-1=a2b, bd=db, dcd=c3 >

Subgroups: 264 in 108 conjugacy classes, 28 normal (14 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C42, C42, C22⋊C4, C4⋊C4, M4(2), D8, SD16, Q16, C22×C4, C2×D4, C2×Q8, C2×Q8, C2×Q8, C4○D4, C4.10D4, C4.10D4, C4≀C2, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C42⋊2C2, C4⋊Q8, C8⋊C22, C8.C22, 2- 1+4, C42.C4, C42.3C4, D4.8D4, D4.10D4, C22.57C24, C42.16D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C22≀C2, C2≀C22, C42.16D4

Character table of C42.16D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I8A8B8C
 size 11288444888888161616
ρ111111111111111111    trivial
ρ2111-1111111111-1-1-1-1    linear of order 2
ρ3111-11111-1-11-1-1-11-11    linear of order 2
ρ411111111-1-11-1-11-11-1    linear of order 2
ρ5111-1-11111-1-1-11-111-1    linear of order 2
ρ61111-11111-1-1-111-1-11    linear of order 2
ρ71111-1111-11-11-111-1-1    linear of order 2
ρ8111-1-1111-11-11-1-1-111    linear of order 2
ρ92220-2-22-2002000000    orthogonal lifted from D4
ρ1022200-2-222000-20000    orthogonal lifted from D4
ρ11222002-2-2020-200000    orthogonal lifted from D4
ρ12222002-2-20-20200000    orthogonal lifted from D4
ρ1322200-2-22-200020000    orthogonal lifted from D4
ρ1422202-22-200-2000000    orthogonal lifted from D4
ρ1544-42000000000-2000    orthogonal lifted from C2≀C22
ρ1644-4-20000000002000    orthogonal lifted from C2≀C22
ρ178-8000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C42.16D4
►On 32 points
Generators in S32
(1 23 5 19)(2 11)(3 21 7 17)(4 13)(6 15)(8 9)(10 25 14 29)(12 31 16 27)(18 28)(20 30)(22 32)(24 26)

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

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

Matrix representation of C42.16D4 ►in GL8(𝔽17)

55000000
512000000
5012120000
051250000
12120070710
120005007
0500105512
0000125125
,
100150000
016200000
016100000
100160000
000010015
160161160161
011610101
000010016
,
1212007700
1212000700
0000512512
12120012121212
5121070000
0501051000
5512551000
0125120000
,
00001000
161600161500
000001610
00000001
10000000
00000100
00100100
00010000

G:=sub<GL(8,GF(17))| [5,5,5,0,12,12,0,0,5,12,0,5,12,0,5,0,0,0,12,12,0,0,0,0,0,0,12,5,0,0,0,0,0,0,0,0,7,5,10,12,0,0,0,0,0,0,5,5,0,0,0,0,7,0,5,12,0,0,0,0,10,7,12,5],[1,0,0,1,0,16,0,0,0,16,16,0,0,0,1,0,0,2,1,0,0,16,16,0,15,0,0,16,0,1,1,0,0,0,0,0,1,16,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,15,1,1,16],[12,12,0,12,5,0,5,0,12,12,0,12,12,5,5,12,0,0,0,0,10,0,12,5,0,0,0,0,7,10,5,12,7,0,5,12,0,5,5,0,7,7,12,12,0,10,10,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0],[0,16,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,16,0,0,0,0,0,0,0,15,16,0,0,1,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
 

C42.16D4 in GAP, Magma, Sage, TeX 

C_4^2._{16}D_4
 
% in TeX
 
G:=Group("C4^2.16D4");
 
// GroupNames label
 
G:=SmallGroup(128,935);
 
// by ID
 
G=gap.SmallGroup(128,935);
 
# by ID
 
G:=PCGroup([7,-2,2,2,-2,2,-2,-2,141,456,422,2019,1018,297,248,2804,1971,718,375,172,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a^-1*b^-1,d*a*d=a^-1*b,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=c^3>;
 
// generators/relations
 

Export

Character table of C42.16D4 in TeX

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