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G = C24.16D4order 128 = 27

16th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.16D4, C23.8SD16, C4⋊C4.16D4, C23⋊C811C2, (C2×D4).19D4, (C22×C4).52D4, C2.9(D44D4), C22⋊C839C22, C233D4.3C2, C23.530(C2×D4), C23.4Q81C2, C2.8(C22⋊SD16), C22.SD1615C2, C4⋊D4.13C22, (C22×C4).19C23, C22.33(C2×SD16), C22.140C22≀C2, C23.46D425C2, C22.44(C8⋊C22), C2.C426C22, C2.10(C23.7D4), (C2×C4⋊C4)⋊2C22, (C2×C4).208(C2×D4), (C2×C22⋊C4).99C22, SmallGroup(128,345)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.16D4
C1C2C22C23C22×C4C2×C22⋊C4C233D4 — C24.16D4
C1C22C22×C4 — C24.16D4
C1C22C22×C4 — C24.16D4
C1C2C22C22×C4 — C24.16D4

Generators and relations for C24.16D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=c, f2=dc=cd, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, ebe-1=bd=db, bf=fb, ce=ec, cf=fc, de=ed, df=fd, fef-1=de3 >

Subgroups: 420 in 144 conjugacy classes, 34 normal (18 characteristic)
C1, C2 [×3], C2 [×6], C4 [×8], C22 [×3], C22 [×19], C8 [×2], C2×C4 [×2], C2×C4 [×14], D4 [×11], C23, C23 [×2], C23 [×10], C22⋊C4 [×8], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×9], C24, C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×2], C4.Q8 [×2], C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4, C22.D4 [×2], C22×D4, C23⋊C8, C22.SD16 [×2], C23.4Q8, C23.46D4 [×2], C233D4, C24.16D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×6], C23, SD16 [×2], C2×D4 [×3], C22≀C2, C2×SD16, C8⋊C22, C22⋊SD16, D44D4, C23.7D4, C24.16D4

Character table of C24.16D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I8A8B8C8D
 size 11112244884488888888888
ρ111111111111111111111111    trivial
ρ211111111-1-11111-11-111-1-1-1-1    linear of order 2
ρ311111111-1-111-11-1-1-1-1-11111    linear of order 2
ρ4111111111111-111-11-1-1-1-1-1-1    linear of order 2
ρ5111111-1-11-111-1-1-1111-11-11-1    linear of order 2
ρ6111111-1-1-1111-1-111-11-1-11-11    linear of order 2
ρ7111111-1-1-11111-11-1-1-111-11-1    linear of order 2
ρ8111111-1-11-1111-1-1-11-11-11-11    linear of order 2
ρ92222-2-200022-200-200000000    orthogonal lifted from D4
ρ102222-2-200-20-2200002000000    orthogonal lifted from D4
ρ11222222-2-200-2-202000000000    orthogonal lifted from D4
ρ122222-2-2000-22-200200000000    orthogonal lifted from D4
ρ132222-2-20020-220000-2000000    orthogonal lifted from D4
ρ142222222200-2-20-2000000000    orthogonal lifted from D4
ρ1522-2-2-222-200000000000-2-2--2--2    complex lifted from SD16
ρ1622-2-2-22-2200000000000--2-2-2--2    complex lifted from SD16
ρ1722-2-2-222-200000000000--2--2-2-2    complex lifted from SD16
ρ1822-2-2-22-2200000000000-2--2--2-2    complex lifted from SD16
ρ194-4-4400000000-20000020000    orthogonal lifted from D44D4
ρ204-4-4400000000200000-20000    orthogonal lifted from D44D4
ρ2144-4-44-400000000000000000    orthogonal lifted from C8⋊C22
ρ224-44-4000000000002i0-2i00000    complex lifted from C23.7D4
ρ234-44-400000000000-2i02i00000    complex lifted from C23.7D4

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

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

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

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

Matrix representation of C24.16D4 in GL6(𝔽17)

100000
010000
001000
000100
0000160
0000016
,
100000
010000
000100
001000
000001
000010
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
0016000
0001600
0000160
0000016
,
1250000
12120000
0022215
001515215
0021522
002151515
,
5120000
12120000
0022215
0022152
0021522
0015222

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[12,12,0,0,0,0,5,12,0,0,0,0,0,0,2,15,2,2,0,0,2,15,15,15,0,0,2,2,2,15,0,0,15,15,2,15],[5,12,0,0,0,0,12,12,0,0,0,0,0,0,2,2,2,15,0,0,2,2,15,2,0,0,2,15,2,2,0,0,15,2,2,2] >;

C24.16D4 in GAP, Magma, Sage, TeX

C_2^4._{16}D_4
% in TeX

G:=Group("C2^4.16D4");
// GroupNames label

G:=SmallGroup(128,345);
// by ID

G=gap.SmallGroup(128,345);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,672,141,422,1123,570,521,136,1411]);
// Polycyclic

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

Export

Character table of C24.16D4 in TeX

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