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G = C42.59D4order 128 = 27

41st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.59D4, (C4×D4)⋊6C4, C4⋊Q810C4, C41D48C4, C425C41C2, C42.76(C2×C4), C23.504(C2×D4), (C22×C4).216D4, C42.6C432C2, C22.SD1622C2, C4⋊D4.139C22, C22⋊C8.135C22, C22.37(C8⋊C22), (C22×C4).636C23, (C2×C42).179C22, C22.26C24.7C2, C2.C42.4C22, C2.19(C42⋊C22), C2.10(C23.37D4), C2.19(C23.C23), C4⋊C4.14(C2×C4), (C2×D4).12(C2×C4), (C2×C4).1160(C2×D4), (C2×C4).93(C22⋊C4), (C2×C4).126(C22×C4), C22.190(C2×C22⋊C4), SmallGroup(128,246)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.59D4
C1C2C22C23C22×C4C2×C42C22.26C24 — C42.59D4
C1C22C2×C4 — C42.59D4
C1C22C2×C42 — C42.59D4
C1C2C22C22×C4 — C42.59D4

Generators and relations for C42.59D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b-1, ab=ba, cac-1=dad-1=ab2, cbc-1=a2b-1, bd=db, dcd-1=b-1c-1 >

Subgroups: 308 in 126 conjugacy classes, 44 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×10], C22, C22 [×2], C22 [×8], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×10], Q8 [×2], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4, C2×C8 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C2.C42 [×2], C2.C42 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8, C2×C42, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C2×C4○D4, C22.SD16 [×4], C425C4, C42.6C4, C22.26C24, C42.59D4
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, C23.37D4, C42⋊C22, C42.59D4

Character table of C42.59D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ2111111-1-11111111111-1-111-1-1-1-1    linear of order 2
ρ3111111-111-1-11-1-11-11-1-111-11-11-1    linear of order 2
ρ41111111-11-1-11-1-11-11-11-11-1-11-11    linear of order 2
ρ51111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ6111111-1-111111111-1-1-1-1-1-11111    linear of order 2
ρ7111111-111-1-11-1-11-1-11-11-11-11-11    linear of order 2
ρ81111111-11-1-11-1-11-1-111-1-111-11-1    linear of order 2
ρ91111-1-1-11-111-11-11-1-i-i1-1ii-i-iii    linear of order 4
ρ101111-1-1-11-111-11-11-1ii1-1-i-iii-i-i    linear of order 4
ρ111111-1-11-1-111-11-11-1-i-i-11iiii-i-i    linear of order 4
ρ121111-1-11-1-111-11-11-1ii-11-i-i-i-iii    linear of order 4
ρ131111-1-111-1-1-1-1-1111i-i-1-1-iii-i-ii    linear of order 4
ρ141111-1-111-1-1-1-1-1111-ii-1-1i-i-iii-i    linear of order 4
ρ151111-1-1-1-1-1-1-1-1-1111i-i11-ii-iii-i    linear of order 4
ρ161111-1-1-1-1-1-1-1-1-1111-ii11i-ii-i-ii    linear of order 4
ρ172222-2-2002-2-222-2-220000000000    orthogonal lifted from D4
ρ182222-2-2002222-22-2-20000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-222-2-20000000000    orthogonal lifted from D4
ρ2022222200-222-2-2-2-220000000000    orthogonal lifted from D4
ρ214-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ224-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-40000-4i004i00000000000000    complex lifted from C42⋊C22
ρ2444-4-400000-4i4i000000000000000    complex lifted from C23.C23
ρ254-44-400004i00-4i00000000000000    complex lifted from C42⋊C22
ρ2644-4-4000004i-4i000000000000000    complex lifted from C23.C23

Smallest permutation representation of C42.59D4
On 32 points
Generators in S32
(1 16 29 22)(2 13 30 19)(3 10 31 24)(4 15 32 21)(5 12 25 18)(6 9 26 23)(7 14 27 20)(8 11 28 17)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)
(2 28 30 8)(3 27)(4 6 32 26)(7 31)(9 17 23 11)(10 24)(12 16)(13 21 19 15)(14 20)(18 22)
(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)| (1,16,29,22)(2,13,30,19)(3,10,31,24)(4,15,32,21)(5,12,25,18)(6,9,26,23)(7,14,27,20)(8,11,28,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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( (1,16,29,22)(2,13,30,19)(3,10,31,24)(4,15,32,21)(5,12,25,18)(6,9,26,23)(7,14,27,20)(8,11,28,17), (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,23,21,19)(18,24,22,20)(25,31,29,27)(26,32,30,28), (2,28,30,8)(3,27)(4,6,32,26)(7,31)(9,17,23,11)(10,24)(12,16)(13,21,19,15)(14,20)(18,22), (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([(1,16,29,22),(2,13,30,19),(3,10,31,24),(4,15,32,21),(5,12,25,18),(6,9,26,23),(7,14,27,20),(8,11,28,17)], [(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,23,21,19),(18,24,22,20),(25,31,29,27),(26,32,30,28)], [(2,28,30,8),(3,27),(4,6,32,26),(7,31),(9,17,23,11),(10,24),(12,16),(13,21,19,15),(14,20),(18,22)], [(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 C42.59D4 in GL8(𝔽17)

40000000
04000000
00400000
00040000
0000011616
000016010
0000001616
00000021
,
115000000
016000000
001620000
00010000
00004000
00000400
00000040
00000004
,
10000000
116000000
001620000
001610000
00001007
0000016011
0000001313
00000004
,
001620000
001610000
10000000
116000000
0000016137
00000106
000011601
000002016

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

C42.59D4 in GAP, Magma, Sage, TeX

C_4^2._{59}D_4
% in TeX

G:=Group("C4^2.59D4");
// GroupNames label

G:=SmallGroup(128,246);
// by ID

G=gap.SmallGroup(128,246);
# by ID

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

G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=b^-1,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 C42.59D4 in TeX

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