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G = C42.273D4order 128 = 27

255th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.273D4, C42.401C23, C4.1072+ 1+4, C4⋊C818C22, C8.2D49C2, C4⋊Q872C22, C8⋊C49C22, (C2×C8).63C23, (C4×Q8)⋊13C22, C4⋊C4.154C23, (C2×C4).413C24, Q8.D423C2, C22⋊Q1621C2, C22⋊SD16.3C2, (C2×Q16)⋊25C22, C23.694(C2×D4), (C22×C4).502D4, C42.6C411C2, Q8⋊C434C22, (C2×D4).162C23, C22⋊C8.48C22, (C2×Q8).150C23, D4⋊C4.44C22, (C2×C42).880C22, (C2×SD16).33C22, C22.673(C22×D4), C22⋊Q8.196C22, C2.58(D8⋊C22), (C22×C4).1084C23, C42.28C223C2, C4.4D4.154C22, (C22×D4).390C22, C22.40(C8.C22), (C22×Q8).323C22, C23.37C2318C2, C2.84(C22.29C24), (C2×C4).542(C2×D4), C2.56(C2×C8.C22), (C2×C4.4D4).42C2, SmallGroup(128,1947)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.273D4
C1C2C4C2×C4C22×C4C22×D4C2×C4.4D4 — C42.273D4
C1C2C2×C4 — C42.273D4
C1C22C2×C42 — C42.273D4
C1C2C2C2×C4 — C42.273D4

Generators and relations for C42.273D4
 G = < a,b,c,d | a4=b4=1, c4=d2=b2, ab=ba, cac-1=ab2, dad-1=a-1, cbc-1=a2b-1, dbd-1=a2b, dcd-1=c3 >

Subgroups: 428 in 200 conjugacy classes, 86 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×12], C8 [×4], C2×C4 [×6], C2×C4 [×13], D4 [×6], Q8 [×12], C23, C23 [×8], C42 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4, C2×D4 [×2], C2×D4 [×3], C2×Q8 [×4], C2×Q8 [×4], C24, C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4 [×2], C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C4.4D4 [×4], C4.4D4 [×2], C42.C2, C4⋊Q8 [×2], C2×SD16 [×4], C2×Q16 [×4], C22×D4, C22×Q8, C42.6C4, C22⋊SD16 [×2], C22⋊Q16 [×2], Q8.D4 [×4], C42.28C22 [×2], C8.2D4 [×2], C2×C4.4D4, C23.37C23, C42.273D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×2], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8.C22, D8⋊C22, C42.273D4

Character table of C42.273D4

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D
 size 11112288222244448888888888
ρ111111111111111111111111111    trivial
ρ21111-1-1-111111-1-11-11-11-1-11-111-1    linear of order 2
ρ3111111-1-1-11-11-1-1-111111-1-11-11-1    linear of order 2
ρ41111-1-11-1-11-1111-1-11-11-11-1-1-111    linear of order 2
ρ5111111-1-1-11-11-1-1-1111-1-111-11-11    linear of order 2
ρ61111-1-11-1-11-1111-1-11-1-11-1111-1-1    linear of order 2
ρ7111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-1-111111-1-11-11-1-111-11-1-11    linear of order 2
ρ91111-1-1-11-11-1111-1-1-11-11-11-1-111    linear of order 2
ρ1011111111-11-11-1-1-11-1-1-1-1111-11-1    linear of order 2
ρ111111-1-11-11111-1-11-1-11-111-1-111-1    linear of order 2
ρ12111111-1-111111111-1-1-1-1-1-11111    linear of order 2
ρ131111-1-11-11111-1-11-1-111-1-111-1-11    linear of order 2
ρ14111111-1-111111111-1-11111-1-1-1-1    linear of order 2
ρ151111-1-1-11-11-1111-1-1-111-11-111-1-1    linear of order 2
ρ1611111111-11-11-1-1-11-1-111-1-1-11-11    linear of order 2
ρ17222222002-22-22-2-2-20000000000    orthogonal lifted from D4
ρ1822222200-2-2-2-2-222-20000000000    orthogonal lifted from D4
ρ192222-2-200-2-2-2-22-2220000000000    orthogonal lifted from D4
ρ202222-2-2002-22-2-22-220000000000    orthogonal lifted from D4
ρ214-44-40000040-400000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000-40400000000000000    orthogonal lifted from 2+ 1+4
ρ234-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2544-4-400004i0-4i000000000000000    complex lifted from D8⋊C22
ρ2644-4-40000-4i04i000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.273D4
On 32 points
Generators in S32
(1 9 25 17)(2 14 26 22)(3 11 27 19)(4 16 28 24)(5 13 29 21)(6 10 30 18)(7 15 31 23)(8 12 32 20)
(1 3 5 7)(2 32 6 28)(4 26 8 30)(9 11 13 15)(10 24 14 20)(12 18 16 22)(17 19 21 23)(25 27 29 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)
(1 2 5 6)(3 8 7 4)(9 22 13 18)(10 17 14 21)(11 20 15 24)(12 23 16 19)(25 26 29 30)(27 32 31 28)

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

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

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

Matrix representation of C42.273D4 in GL8(𝔽17)

12000000
1616000000
0016150000
00110000
00004000
00000400
000000130
000000013
,
160000000
016000000
00100000
00010000
0000161500
00001100
00000012
0000001616
,
001600000
00110000
10000000
1616000000
0000001615
00000001
000016000
00001100
,
00100000
00010000
10000000
01000000
00000010
00000001
000016000
000001600

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

C42.273D4 in GAP, Magma, Sage, TeX

C_4^2._{273}D_4
% in TeX

G:=Group("C4^2.273D4");
// GroupNames label

G:=SmallGroup(128,1947);
// by ID

G=gap.SmallGroup(128,1947);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,891,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^3>;
// generators/relations

Export

Character table of C42.273D4 in TeX

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