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G = C42.6D4order 128 = 27

6th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.6D4, C4.4(C4×D4), (C2×Q16)⋊4C4, (C2×SD16)⋊4C4, (C2×Q8).73D4, C4.9C423C2, C22.56(C4×D4), C22⋊C4.121D4, C23.130(C2×D4), Q8.7(C22⋊C4), C4.138(C4⋊D4), M4(2)⋊4C46C2, C22.32C22≀C2, (C22×C4).32C23, C23.38D424C2, C22.52(C4⋊D4), C42⋊C22.5C2, (C22×Q8).22C22, C42⋊C2.30C22, C23.32C232C2, C4.11(C22.D4), (C2×M4(2)).10C22, C23.C23.6C2, C2.44(C23.23D4), (C2×C8).7(C2×C4), (C2×D4).86(C2×C4), (C2×C4).241(C2×D4), C4.21(C2×C22⋊C4), (C2×Q8).74(C2×C4), (C2×C8.C22).3C2, (C2×C4).327(C4○D4), (C2×C4).192(C22×C4), (C2×C4○D4).26C22, SmallGroup(128,637)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.6D4
C1C2C4C2×C4C22×C4C22×Q8C23.32C23 — C42.6D4
C1C2C2×C4 — C42.6D4
C1C2C22×C4 — C42.6D4
C1C2C2C22×C4 — C42.6D4

Generators and relations for C42.6D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 300 in 155 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×10], C22 [×3], C22 [×3], C8 [×3], C2×C4 [×6], C2×C4 [×14], D4 [×4], Q8 [×4], Q8 [×8], C23, C23, C42 [×2], C42 [×6], C22⋊C4 [×2], C22⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×2], C2×C8, M4(2) [×4], SD16 [×4], Q16 [×4], C22×C4, C22×C4 [×2], C2×D4, C2×D4, C2×Q8, C2×Q8 [×6], C2×Q8 [×3], C4○D4 [×4], C23⋊C4 [×2], Q8⋊C4 [×2], C4≀C2 [×2], C42⋊C2 [×3], C42⋊C2 [×2], C4×Q8 [×4], C2×M4(2) [×2], C2×SD16 [×2], C2×Q16 [×2], C8.C22 [×4], C22×Q8, C2×C4○D4, C4.9C42, M4(2)⋊4C4, C23.C23, C23.38D4, C42⋊C22, C23.32C23, C2×C8.C22, C42.6D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42.6D4

Character table of C42.6D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D
 size 11222822224444444444448888888
ρ111111111111111111111111111111    trivial
ρ211111-111111-1-1-1-1111-1-1-1-1-1111-11-1    linear of order 2
ρ31111111111-11-1-1-1-1-1-1111-1111-1-1-1-1    linear of order 2
ρ411111-11111-1-1111-1-1-1-1-1-11-111-11-11    linear of order 2
ρ51111111111-1-1111-1-1-1-1-1-111-1-11-11-1    linear of order 2
ρ611111-11111-11-1-1-1-1-1-1111-1-1-1-11111    linear of order 2
ρ711111111111-1-1-1-1111-1-1-1-11-1-1-11-11    linear of order 2
ρ811111-11111111111111111-1-1-1-1-1-1-1    linear of order 2
ρ9111-1-1-111-1-1-ii1-11i-ii-ii-i-11i-i1i-1-i    linear of order 4
ρ10111-1-1-111-1-1i-i1-11-ii-ii-ii-11-ii1-i-1i    linear of order 4
ρ11111-1-1-111-1-1ii-11-1-ii-i-ii-i11i-i-1-i1i    linear of order 4
ρ12111-1-1-111-1-1-i-i-11-1i-iii-ii11-ii-1i1-i    linear of order 4
ρ13111-1-1111-1-1ii-11-1-ii-i-ii-i1-1-ii1i-1-i    linear of order 4
ρ14111-1-1111-1-1-i-i-11-1i-iii-ii1-1i-i1-i-1i    linear of order 4
ρ15111-1-1111-1-1-ii1-11i-ii-ii-i-1-1-ii-1-i1i    linear of order 4
ρ16111-1-1111-1-1i-i1-11-ii-ii-ii-1-1i-i-1i1-i    linear of order 4
ρ17222220-2-2-2-20-200000022-200000000    orthogonal lifted from D4
ρ1822-22-202-22-2002-2-200000020000000    orthogonal lifted from D4
ρ1922-22-202-22-200-222000000-20000000    orthogonal lifted from D4
ρ2022-2-220-222-2-20000-22200000000000    orthogonal lifted from D4
ρ21222220-2-2-2-202000000-2-2200000000    orthogonal lifted from D4
ρ2222-2-220-222-2200002-2-200000000000    orthogonal lifted from D4
ρ2322-2-2202-2-2200-2-2200000020000000    orthogonal lifted from D4
ρ2422-2-2202-2-220022-2000000-20000000    orthogonal lifted from D4
ρ25222-2-20-2-2220-2i000000-2i2i2i00000000    complex lifted from C4○D4
ρ26222-2-20-2-22202i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ2722-22-20-22-22-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ2822-22-20-22-222i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C42.6D4
On 32 points
Generators in S32
(1 32 27 2)(3 26 29 4)(5 28 31 6)(7 30 25 8)(9 22 17 10)(11 24 19 12)(13 18 21 14)(15 20 23 16)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(17 19 21 23)(18 20 22 24)(25 27 29 31)(26 28 30 32)
(1 13 31 21)(2 24 32 16)(3 11 25 19)(4 22 26 14)(5 9 27 17)(6 20 28 12)(7 15 29 23)(8 18 30 10)
(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)| (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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( (1,32,27,2)(3,26,29,4)(5,28,31,6)(7,30,25,8)(9,22,17,10)(11,24,19,12)(13,18,21,14)(15,20,23,16), (1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)(17,19,21,23)(18,20,22,24)(25,27,29,31)(26,28,30,32), (1,13,31,21)(2,24,32,16)(3,11,25,19)(4,22,26,14)(5,9,27,17)(6,20,28,12)(7,15,29,23)(8,18,30,10), (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([(1,32,27,2),(3,26,29,4),(5,28,31,6),(7,30,25,8),(9,22,17,10),(11,24,19,12),(13,18,21,14),(15,20,23,16)], [(1,3,5,7),(2,4,6,8),(9,11,13,15),(10,12,14,16),(17,19,21,23),(18,20,22,24),(25,27,29,31),(26,28,30,32)], [(1,13,31,21),(2,24,32,16),(3,11,25,19),(4,22,26,14),(5,9,27,17),(6,20,28,12),(7,15,29,23),(8,18,30,10)], [(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 C42.6D4 in GL8(𝔽17)

00001000
61116111500
519810150
3314310015
000160000
15251106116
1213111431689
151162314314
,
01000000
160000000
00010000
001600000
61116111500
6016011600
115141201601
416113116160
,
8511610070
615314710010
851160070
15610707010
0120120000
1229891699
02893161511
121251200013
,
61116111500
000016000
141431416002
519810150
10000000
149007111616
5154101299
131416103141414

G:=sub<GL(8,GF(17))| [0,6,5,3,0,15,12,15,0,11,1,3,0,2,13,11,0,16,9,14,0,5,11,6,0,1,8,3,16,11,14,2,1,1,1,1,0,0,3,3,0,15,0,0,0,6,16,14,0,0,15,0,0,1,8,3,0,0,0,15,0,16,9,14],[0,16,0,0,6,6,1,4,1,0,0,0,11,0,15,16,0,0,0,16,16,16,14,11,0,0,1,0,1,0,12,3,0,0,0,0,1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[8,6,8,15,0,12,0,12,5,15,5,6,12,2,2,12,11,3,11,10,0,9,8,5,6,14,6,7,12,8,9,12,10,7,0,0,0,9,3,0,0,10,0,7,0,16,16,0,7,0,7,0,0,9,15,0,0,10,0,10,0,9,11,13],[6,0,14,5,1,14,5,13,11,0,14,1,0,9,1,14,16,0,3,9,0,0,5,16,1,0,14,8,0,0,4,10,1,16,16,1,0,7,10,3,15,0,0,0,0,11,12,14,0,0,0,15,0,16,9,14,0,0,2,0,0,16,9,14] >;

C42.6D4 in GAP, Magma, Sage, TeX

C_4^2._6D_4
% in TeX

G:=Group("C4^2.6D4");
// GroupNames label

G:=SmallGroup(128,637);
// by ID

G=gap.SmallGroup(128,637);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,1411,718,172,1027,124]);
// Polycyclic

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

Export

Character table of C42.6D4 in TeX

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