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G = C42⋊3Q8  order 128 = 27

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

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

Aliases: C42⋊3Q8, M4(2)⋊2Q8, C4⋊C4.103D4, C4.25(C4⋊Q8), (C22×C4).86D4, C42⋊6C4.3C2, C23.594(C2×D4), C22.14(C4⋊Q8), C4.10(C22⋊Q8), C42⋊8C4.14C2, 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

Derived series
C1 — C22×C4 — C42⋊3Q8
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4⋊C4 — C42⋊8C4 — C42⋊3Q8
Lower central
C1 — C2 — C22×C4 — C42⋊3Q8
Upper central
C1 — C22 — C22×C4 — C42⋊3Q8
Jennings
C1 — C2 — C2 — C22×C4 — C42⋊3Q8

Generators and relations for C42⋊3Q8
 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], C42⋊6C4 [×2], C22.C42, C42⋊8C4, M4(2)⋊C4 [×2], C23.41C23, C42⋊3Q8
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, C42⋊3Q8

Character table of C42⋊3Q8

 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 C42⋊3Q8
►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 C42⋊3Q8 ►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] >;
 

C42⋊3Q8 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 C42⋊3Q8 in TeX

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