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G = C42.16D4order 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, D4.10D41C2, (C2×Q8).2C23, C42.C42C2, C42.3C41C2, D4.8D4.2C2, C22.51C22≀C2, C4.10D4.1C22, C4.4D4.22C22, C22.57C243C2, (C2×C4).20(C2×D4), SmallGroup(128,935)

Series: Derived Chief Lower central Upper central Jennings

C1C2C2×Q8 — C42.16D4
C1C2C22C2×C4C2×Q8C4.4D4C22.57C24 — C42.16D4
C1C2C22C2×Q8 — C42.16D4
C1C2C22C2×Q8 — C42.16D4
C1C2C22C2×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 [×3], C4 [×9], C22, C22 [×4], C8 [×3], C2×C4, C2×C4 [×2], C2×C4 [×9], D4 [×5], Q8 [×6], C23, C42, C42 [×2], C22⋊C4 [×5], C4⋊C4 [×8], M4(2) [×3], D8, SD16 [×3], Q16 [×2], C22×C4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×4], C4.10D4, C4.10D4 [×2], C4≀C2 [×3], C22⋊Q8 [×2], C22.D4, C4.4D4, C42.C2, C422C2 [×2], C4⋊Q8 [×2], C8⋊C22, C8.C22 [×2], 2- 1+4, C42.C4, C42.3C4 [×2], D4.8D4, D4.10D4 [×2], C22.57C24, C42.16D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, C2×D4 [×3], 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 13)(3 21 7 17)(4 15)(6 9)(8 11)(10 27 14 31)(12 25 16 29)(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 20 13 24)(10 21 14 17)(11 18 15 22)(12 19 16 23)
(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 13)(2 16)(3 11)(4 14)(5 9)(6 12)(7 15)(8 10)(17 32)(18 27)(19 30)(20 25)(21 28)(22 31)(23 26)(24 29)

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

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

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