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G = C423Q8order 128 = 27

3rd semidirect product of C42 and Q8 acting via Q8/C2=C22

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

Aliases: C423Q8, M4(2)⋊2Q8, C4⋊C4.103D4, C4.25(C4⋊Q8), (C22×C4).86D4, C426C4.3C2, C23.594(C2×D4), C22.14(C4⋊Q8), C4.10(C22⋊Q8), C428C4.14C2, C22.231C22≀C2, C2.29(D4.9D4), C2.31(D4.8D4), M4(2)⋊C4.6C2, C22.C42.4C2, (C2×C42).375C22, (C22×C4).734C23, C22.36(C22⋊Q8), C42⋊C2.65C22, (C2×M4(2)).26C22, C23.41C23.6C2, C2.8(C23.78C23), (C2×C4).19(C2×Q8), (C2×C4).1052(C2×D4), (C2×C4).349(C4○D4), (C2×C4⋊C4).138C22, SmallGroup(128,793)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C423Q8
C1C2C22C2×C4C22×C4C2×C4⋊C4C428C4 — C423Q8
C1C2C22×C4 — C423Q8
C1C22C22×C4 — C423Q8
C1C2C2C22×C4 — C423Q8

Generators and relations for C423Q8
 G = < a,b,c,d | a4=b4=c4=1, d2=c2, dad-1=ab=ba, cac-1=a-1b2, cbc-1=dbd-1=b-1, dcd-1=c-1 >

Subgroups: 232 in 115 conjugacy classes, 46 normal (18 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×9], C22 [×3], C22 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×15], Q8 [×2], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×2], M4(2) [×4], M4(2) [×2], C22×C4, C22×C4 [×2], C22×C4 [×2], C2×Q8 [×2], C2.C42 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C4⋊C4 [×2], C42⋊C2 [×2], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C426C4 [×2], C22.C42, C428C4, M4(2)⋊C4 [×2], C23.41C23, C423Q8
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], Q8 [×6], C23, C2×D4 [×3], C2×Q8 [×3], C4○D4, C22≀C2, C22⋊Q8 [×3], C4⋊Q8 [×3], C23.78C23, D4.8D4, D4.9D4, C423Q8

Character table of C423Q8

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D
 size 11112222224444888888888888
ρ111111111111111111111111111    trivial
ρ21111111111-1-1-1-1-1-11-1-1-1-111111    linear of order 2
ρ31111111111-1-1-1-111-1-1-111-1-1-111    linear of order 2
ρ411111111111111-1-1-111-1-1-1-1-111    linear of order 2
ρ51111111111-1-1-1-11-11111-11-1-1-1-1    linear of order 2
ρ611111111111111-111-1-1-111-1-1-1-1    linear of order 2
ρ7111111111111111-1-1-1-11-1-111-1-1    linear of order 2
ρ81111111111-1-1-1-1-11-111-11-111-1-1    linear of order 2
ρ92222-2-22-2-220000000-220000000    orthogonal lifted from D4
ρ10222222-2-2-2-2000000-2000020000    orthogonal lifted from D4
ρ112222-2-2-222-20000-200002000000    orthogonal lifted from D4
ρ12222222-2-2-2-200000020000-20000    orthogonal lifted from D4
ρ132222-2-22-2-2200000002-20000000    orthogonal lifted from D4
ρ142222-2-2-222-2000020000-2000000    orthogonal lifted from D4
ρ152-2-22-22-22-22000000000000-2200    symplectic lifted from Q8, Schur index 2
ρ162-2-22-22-22-220000000000002-200    symplectic lifted from Q8, Schur index 2
ρ172-2-222-222-2-2-22-22000000000000    symplectic lifted from Q8, Schur index 2
ρ182-2-22-222-22-2000000000000002-2    symplectic lifted from Q8, Schur index 2
ρ192-2-22-222-22-200000000000000-22    symplectic lifted from Q8, Schur index 2
ρ202-2-222-222-2-22-22-2000000000000    symplectic lifted from Q8, Schur index 2
ρ212-2-222-2-2-22200000-2i00002i00000    complex lifted from C4○D4
ρ222-2-222-2-2-222000002i0000-2i00000    complex lifted from C4○D4
ρ234-44-40000002i2i-2i-2i000000000000    complex lifted from D4.9D4
ρ2444-4-40000002i-2i-2i2i000000000000    complex lifted from D4.8D4
ρ254-44-4000000-2i-2i2i2i000000000000    complex lifted from D4.9D4
ρ2644-4-4000000-2i2i2i-2i000000000000    complex lifted from D4.8D4

Smallest permutation representation of C423Q8
On 32 points
Generators in S32
(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 8 2 7)(3 5 4 6)(9 15 10 16)(11 14 12 13)(17 18 19 20)(21 24 23 22)(25 26 27 28)(29 32 31 30)
(1 11 5 15)(2 12 6 16)(3 10 7 14)(4 9 8 13)(17 32 26 22)(18 29 27 23)(19 30 28 24)(20 31 25 21)
(1 21 5 31)(2 23 6 29)(3 30 7 24)(4 32 8 22)(9 17 13 26)(10 19 14 28)(11 25 15 20)(12 27 16 18)

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

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

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

Matrix representation of C423Q8 in GL6(𝔽17)

0160000
100000
001500
0001600
000040
000004
,
1600000
0160000
004300
0001300
00001314
000004
,
040000
400000
00141600
008300
000031
0000914
,
1300000
040000
000043
0000013
00131400
000400

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

C423Q8 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3Q_8
% in TeX

G:=Group("C4^2:3Q8");
// GroupNames label

G:=SmallGroup(128,793);
// by ID

G=gap.SmallGroup(128,793);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,56,141,64,422,387,394,2804,1411,718,172,4037]);
// Polycyclic

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

Export

Character table of C423Q8 in TeX

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