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G = C24.127D4order 128 = 27

82nd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.127D4, C4.92+ 1+4, C8⋊D413C2, C82D410C2, C22⋊D818C2, D4⋊D421C2, C22⋊SD168C2, (C2×D8)⋊23C22, (C2×C8).57C23, C4.Q816C22, C2.D828C22, C4⋊C4.133C23, C4⋊D459C22, C22⋊C815C22, (C2×C4).392C24, C23.276(C2×D4), (C22×C4).489D4, C22⋊Q871C22, D4⋊C426C22, C24.4C412C2, Q8⋊C428C22, (C2×SD16)⋊19C22, (C2×D4).144C23, C23.47D48C2, C22.D820C2, (C2×Q8).131C23, C22.19C2411C2, C42⋊C217C22, C23.19D422C2, C2.73(C233D4), C22.31(C8⋊C22), (C2×M4(2))⋊15C22, (C23×C4).572C22, C22.652(C22×D4), C2.51(D8⋊C22), (C22×C4).1070C23, (C22×D4).383C22, (C2×C4⋊D4)⋊52C2, (C2×C4).530(C2×D4), (C2×C4○D4)⋊9C22, C2.52(C2×C8⋊C22), (C2×C4⋊C4).639C22, SmallGroup(128,1926)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.127D4
C1C2C4C2×C4C22×C4C22×D4C2×C4⋊D4 — C24.127D4
C1C2C2×C4 — C24.127D4
C1C22C23×C4 — C24.127D4
C1C2C2C2×C4 — C24.127D4

Generators and relations for C24.127D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, ab=ba, ac=ca, eae-1=ad=da, faf=acd, ebe-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de3 >

Subgroups: 556 in 233 conjugacy classes, 86 normal (42 characteristic)
C1, C2 [×3], C2 [×7], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×25], C8 [×4], C2×C4 [×4], C2×C4 [×18], D4 [×22], Q8 [×2], C23 [×3], C23 [×13], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×3], C2×C8 [×4], M4(2) [×2], D8 [×2], SD16 [×2], C22×C4 [×6], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×14], C2×Q8, C4○D4 [×2], C24, C24, C22⋊C8 [×2], C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C4⋊D4 [×4], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C23×C4, C22×D4, C22×D4, C2×C4○D4, C24.4C4, C22⋊D8, D4⋊D4 [×2], C22⋊SD16, C8⋊D4 [×2], C82D4 [×2], C22.D8, C23.19D4 [×2], C23.47D4, C2×C4⋊D4, C22.19C24, C24.127D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C22×D4, 2+ 1+4 [×2], C233D4, C2×C8⋊C22, D8⋊C22, C24.127D4

Character table of C24.127D4

 class 12A2B2C2D2E2F2G2H2I2J4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244888222244888888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-1-1-11-111-11-1-11-111-11-11    linear of order 2
ρ31111-1-11-1-11-1-111-11-111-11-11-11-1    linear of order 2
ρ4111111111-1-1111111-1111-1-1-1-1-1    linear of order 2
ρ5111111-1-11-1-1-111-1-1111-1-11-111-1    linear of order 2
ρ61111-1-1-11-11-11111-1-1-111-1111-1-1    linear of order 2
ρ71111-1-1-11-1-111111-1-1111-1-1-1-111    linear of order 2
ρ8111111-1-1111-111-1-11-11-1-1-11-1-11    linear of order 2
ρ91111-1-11-11-11-111-11-1-1-11-111-11-1    linear of order 2
ρ1011111111-1111111111-1-1-11-1-1-1-1    linear of order 2
ρ1111111111-1-1-1111111-1-1-1-1-11111    linear of order 2
ρ121111-1-11-111-1-111-11-11-11-1-1-11-11    linear of order 2
ρ131111-1-1-1111-11111-1-1-1-1-111-1-111    linear of order 2
ρ14111111-1-1-1-1-1-111-1-111-11111-1-11    linear of order 2
ρ15111111-1-1-111-111-1-11-1-111-1-111-1    linear of order 2
ρ161111-1-1-111-111111-1-11-1-11-111-1-1    linear of order 2
ρ172222-2-22-20002-2-22-22000000000    orthogonal lifted from D4
ρ182222-2-2-22000-2-2-2-222000000000    orthogonal lifted from D4
ρ1922222222000-2-2-2-2-2-2000000000    orthogonal lifted from D4
ρ20222222-2-20002-2-222-2000000000    orthogonal lifted from D4
ρ214-44-400000000-44000000000000    orthogonal lifted from 2+ 1+4
ρ224-4-444-400000000000000000000    orthogonal lifted from C8⋊C22
ρ234-44-4000000004-4000000000000    orthogonal lifted from 2+ 1+4
ρ244-4-44-4400000000000000000000    orthogonal lifted from C8⋊C22
ρ2544-4-400000004i00-4i00000000000    complex lifted from D8⋊C22
ρ2644-4-40000000-4i004i00000000000    complex lifted from D8⋊C22

Smallest permutation representation of C24.127D4
On 32 points
Generators in S32
(1 20)(2 17)(3 22)(4 19)(5 24)(6 21)(7 18)(8 23)(9 25)(10 30)(11 27)(12 32)(13 29)(14 26)(15 31)(16 28)
(2 30)(4 32)(6 26)(8 28)(10 17)(12 19)(14 21)(16 23)
(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 24)(10 17)(11 18)(12 19)(13 20)(14 21)(15 22)(16 23)
(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 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)(3 8)(4 7)(5 6)(9 17)(10 24)(11 23)(12 22)(13 21)(14 20)(15 19)(16 18)(25 26)(27 32)(28 31)(29 30)

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

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

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

Matrix representation of C24.127D4 in GL8(𝔽17)

162000000
01000000
001150000
000160000
000001300
00004000
000000013
00000040
,
10000000
01000000
001600000
000160000
00001000
00000100
000000160
000000016
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
001600000
001610000
10000000
116000000
00000001
00000010
00001000
000001600
,
00100000
00010000
10000000
01000000
00000010
00000001
00001000
00000100

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

C24.127D4 in GAP, Magma, Sage, TeX

C_2^4._{127}D_4
% in TeX

G:=Group("C2^4.127D4");
// GroupNames label

G:=SmallGroup(128,1926);
// by ID

G=gap.SmallGroup(128,1926);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,891,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C24.127D4 in TeX

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