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G = C24.6D4order 128 = 27

6th non-split extension by C24 of D4 acting faithfully

p-group, metabelian, nilpotent (class 4), monomial

Aliases: C24.6D4, C23⋊C43C4, (C2×C42)⋊3C4, (C2×D4).3Q8, (C2×D4).41D4, (C2×C4).1C42, C23.2(C4⋊C4), C2.3(C42⋊C4), C2.3(C423C4), C23.3(C22⋊C4), (C22×D4).2C22, C22.14(C23⋊C4), C2.15(C23.9D4), C24.3C22.3C2, C22.4(C2.C42), (C2×C4⋊C4)⋊4C4, (C2×C4).9(C4⋊C4), (C2×D4).46(C2×C4), (C2×C23⋊C4).2C2, (C22×C4).66(C2×C4), SmallGroup(128,125)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.6D4
C1C2C22C23C24C22×D4C24.3C22 — C24.6D4
C1C2C22C2×C4 — C24.6D4
C1C22C23C22×D4 — C24.6D4
C1C2C22C22×D4 — C24.6D4

Generators and relations for C24.6D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=a, ab=ba, eae-1=ac=ca, ad=da, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=abe-1 >

Subgroups: 352 in 116 conjugacy classes, 32 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22 [×3], C22 [×16], C2×C4 [×2], C2×C4 [×17], D4 [×4], C23, C23 [×4], C23 [×8], C42, C22⋊C4 [×10], C4⋊C4, C22×C4, C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×D4 [×2], C24 [×2], C23⋊C4 [×4], C23⋊C4 [×2], C2×C42, C2×C22⋊C4 [×4], C2×C4⋊C4, C22×D4, C24.3C22, C2×C23⋊C4 [×2], C24.6D4
Quotients: C1, C2 [×3], C4 [×6], C22, C2×C4 [×3], D4 [×3], Q8, C42, C22⋊C4 [×3], C4⋊C4 [×3], C2.C42, C23⋊C4 [×2], C23.9D4, C42⋊C4, C423C4, C24.6D4

Character table of C24.6D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ21111111111111111-1-1-1-111-1-1-1-1    linear of order 2
ρ311111111111-11-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ411111111111-11-1-1-11111-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-11-11-1-1-1ii-i-i11-i-iii    linear of order 4
ρ6111111-1-1-1-1111111-i-iii-1-1-i-iii    linear of order 4
ρ71-11-11-11-1-11-1i1-i-ii-111-1i-i-iii-i    linear of order 4
ρ81-11-11-1-111-1-1i1-i-ii-ii-ii-ii1-11-1    linear of order 4
ρ91-11-11-11-1-11-1i1-i-ii1-1-11i-ii-i-ii    linear of order 4
ρ101-11-11-1-111-1-1-i1ii-ii-ii-ii-i1-11-1    linear of order 4
ρ111-11-11-1-111-1-1-i1ii-i-ii-iii-i-11-11    linear of order 4
ρ121-11-11-11-1-11-1-i1ii-i1-1-11-ii-iii-i    linear of order 4
ρ131-11-11-1-111-1-1i1-i-iii-ii-i-ii-11-11    linear of order 4
ρ14111111-1-1-1-1111111ii-i-i-1-1ii-i-i    linear of order 4
ρ151-11-11-11-1-11-1-i1ii-i-111-1-iii-i-ii    linear of order 4
ρ16111111-1-1-1-11-11-1-1-1-i-iii11ii-i-i    linear of order 4
ρ1722222222-2-2-20-20000000000000    orthogonal lifted from D4
ρ182-22-22-22-22-220-20000000000000    orthogonal lifted from D4
ρ19222222-2-222-20-20000000000000    orthogonal lifted from D4
ρ202-22-22-2-22-2220-20000000000000    symplectic lifted from Q8, Schur index 2
ρ2144-4-40000000202-2-20000000000    orthogonal lifted from C42⋊C4
ρ224-44-4-4400000000000000000000    orthogonal lifted from C23⋊C4
ρ234444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2444-4-40000000-20-2220000000000    orthogonal lifted from C42⋊C4
ρ254-4-440000000-2i02i-2i2i0000000000    complex lifted from C423C4
ρ264-4-4400000002i0-2i2i-2i0000000000    complex lifted from C423C4

Smallest permutation representation of C24.6D4
On 32 points
Generators in S32
(1 11)(2 16)(3 12)(4 15)(5 9)(6 14)(7 13)(8 10)(17 21)(18 24)(19 20)(22 23)(25 32)(26 27)(28 29)(30 31)
(1 3)(2 7)(4 6)(5 8)(9 10)(11 12)(13 16)(14 15)(17 30)(18 27)(19 32)(20 25)(21 31)(22 28)(23 29)(24 26)
(1 4)(2 8)(3 6)(5 7)(9 13)(10 16)(11 15)(12 14)(17 22)(18 20)(19 24)(21 23)(25 27)(26 32)(28 30)(29 31)
(1 5)(2 6)(3 8)(4 7)(9 11)(10 12)(13 15)(14 16)(17 24)(18 21)(19 22)(20 23)(25 29)(26 30)(27 31)(28 32)
(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 30 11 31)(2 19 16 20)(3 24 12 18)(4 32 15 25)(5 26 9 27)(6 22 14 23)(7 28 13 29)(8 17 10 21)

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

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

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

Matrix representation of C24.6D4 in GL8(ℤ)

01000000
10000000
000-10000
00-100000
00000010
00000001
00001000
00000100
,
10000000
01000000
00100000
00010000
00001200
00000-100
00000012
0000000-1
,
-10000000
0-1000000
00-100000
000-10000
0000-1-200
00000100
000000-1-2
00000001
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1
,
10000000
0-1000000
000-10000
00100000
0000-1-200
00001100
00000010
000000-1-1
,
000-10000
00100000
10000000
0-1000000
000001-1-1
00000011
0000-1-101
00001100

G:=sub<GL(8,Integers())| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,2,-1],[-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-2,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,0,0,-2,1,0,0,0,0,0,0,0,0,1,-1,0,0,0,0,0,0,0,-1],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,-1,1,0,0,0,0,1,0,-1,1,0,0,0,0,-1,1,0,0,0,0,0,0,-1,1,1,0] >;

C24.6D4 in GAP, Magma, Sage, TeX

C_2^4._6D_4
% in TeX

G:=Group("C2^4.6D4");
// GroupNames label

G:=SmallGroup(128,125);
// by ID

G=gap.SmallGroup(128,125);
# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,352,1466,521,2804,4037]);
// Polycyclic

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

Export

Character table of C24.6D4 in TeX

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