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G = C42.60D4order 128 = 27

42nd non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.60D4, C4⋊Q811C4, (C4×Q8)⋊6C4, C42.77(C2×C4), C425C4.2C2, C23.505(C2×D4), (C22×C4).217D4, C22⋊C8.136C22, C42.6C4.18C2, (C2×C42).180C22, (C22×C4).637C23, C23.31D4.6C2, C22⋊Q8.144C22, C22.26(C8.C22), C2.C42.5C22, C2.20(C42⋊C22), C2.10(C23.38D4), C23.37C23.7C2, C2.20(C23.C23), C4⋊C4.15(C2×C4), (C2×Q8).12(C2×C4), (C2×C4).1161(C2×D4), (C2×C4).94(C22⋊C4), (C2×C4).127(C22×C4), C22.191(C2×C22⋊C4), SmallGroup(128,247)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.60D4
C1C2C22C23C22×C4C2×C42C23.37C23 — C42.60D4
C1C22C2×C4 — C42.60D4
C1C22C2×C42 — C42.60D4
C1C2C22C22×C4 — C42.60D4

Generators and relations for C42.60D4
 G = < a,b,c,d | a4=b4=c4=1, d2=a2b, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 212 in 104 conjugacy classes, 44 normal (26 characteristic)
C1, C2 [×3], C2 [×2], C4 [×12], C22, C22 [×2], C22 [×2], C8 [×2], C2×C4 [×6], C2×C4 [×14], Q8 [×4], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C22×C4 [×3], C22×C4 [×2], C2×Q8 [×2], C2×Q8, C2.C42 [×2], C2.C42 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], C23.31D4 [×4], C425C4, C42.6C4, C23.37C23, C42.60D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8.C22 [×2], C23.C23, C23.38D4, C42⋊C22, C42.60D4

Character table of C42.60D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111-11-11-1-1-1-1-11-11-1111-11-1    linear of order 2
ρ31111111-11-11-1-1-111-11-11-1-11-11-1    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ51111111-11-11-1-1-1111-11-1-1-1-11-11    linear of order 2
ρ611111111111111-1-11111-1-1-1-1-1-1    linear of order 2
ρ71111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ81111111-11-11-1-1-1-1-1-11-1111-11-11    linear of order 2
ρ91111-1-1-11-111-11-1-11ii-i-i-11ii-i-i    linear of order 4
ρ101111-1-1-1-1-1-111-11-11-iii-i1-1i-i-ii    linear of order 4
ρ111111-1-1-11-111-11-11-1ii-i-i1-1-i-iii    linear of order 4
ρ121111-1-1-1-1-1-111-111-1-iii-i-11-iii-i    linear of order 4
ρ131111-1-1-1-1-1-111-111-1i-i-ii-11i-i-ii    linear of order 4
ρ141111-1-1-11-111-11-11-1-i-iii1-1ii-i-i    linear of order 4
ρ151111-1-1-1-1-1-111-11-11i-i-ii1-1-iii-i    linear of order 4
ρ161111-1-1-11-111-11-1-11-i-iii-11-i-iii    linear of order 4
ρ17222222-2-2-2-2-2-222000000000000    orthogonal lifted from D4
ρ18222222-22-22-22-2-2000000000000    orthogonal lifted from D4
ρ192222-2-22-22-2-222-2000000000000    orthogonal lifted from D4
ρ202222-2-22222-2-2-22000000000000    orthogonal lifted from D4
ρ214-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ224-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-44-4000-4i04i0000000000000000    complex lifted from C23.C23
ρ244-44-40004i0-4i0000000000000000    complex lifted from C23.C23
ρ2544-4-4004i0-4i00000000000000000    complex lifted from C42⋊C22
ρ2644-4-400-4i04i00000000000000000    complex lifted from C42⋊C22

Smallest permutation representation of C42.60D4
On 32 points
Generators in S32
(1 31 15 24)(2 28 16 21)(3 25 9 18)(4 30 10 23)(5 27 11 20)(6 32 12 17)(7 29 13 22)(8 26 14 19)
(1 9 5 13)(2 10 6 14)(3 11 7 15)(4 12 8 16)(17 26 21 30)(18 27 22 31)(19 28 23 32)(20 29 24 25)
(2 4 16 10)(3 13)(6 8 12 14)(7 9)(17 23 32 30)(18 25)(19 28 26 21)(20 24)(22 29)(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)

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

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

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

Matrix representation of C42.60D4 in GL8(𝔽17)

00100000
00010000
10000000
01000000
0000161500
00001100
00001191615
00002611
,
01000000
160000000
00010000
001600000
00004000
00000400
00000040
00000004
,
10000000
016000000
001600000
00010000
00001000
0000161600
000062130
000081444
,
314440000
331340000
13131430000
41314140000
000051190
0000161088
00001510611
000066013

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,1,11,2,0,0,0,0,15,1,9,6,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,16,6,8,0,0,0,0,0,16,2,14,0,0,0,0,0,0,13,4,0,0,0,0,0,0,0,4],[3,3,13,4,0,0,0,0,14,3,13,13,0,0,0,0,4,13,14,14,0,0,0,0,4,4,3,14,0,0,0,0,0,0,0,0,5,16,15,6,0,0,0,0,11,10,10,6,0,0,0,0,9,8,6,0,0,0,0,0,0,8,11,13] >;

C42.60D4 in GAP, Magma, Sage, TeX

C_4^2._{60}D_4
% in TeX

G:=Group("C4^2.60D4");
// GroupNames label

G:=SmallGroup(128,247);
// by ID

G=gap.SmallGroup(128,247);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,520,1123,1018,248,1971]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2*b,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^2,c*b*c^-1=a^2*b^-1,b*d=d*b,d*c*d^-1=b^-1*c^-1>;
// generators/relations

Export

Character table of C42.60D4 in TeX

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