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G = C24.128D4order 128 = 27

83rd non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.128D4, C4.102+ 1+4, C8.D49C2, (C2×C8).58C23, C4.Q817C22, C4⋊C4.134C23, (C2×C4).393C24, C22⋊Q1617C2, (C2×Q16)⋊22C22, (C22×C4).490D4, C23.277(C2×D4), Q8⋊C429C22, C23.47D49C2, C22⋊C8.39C22, C24.4C4.3C2, (C2×Q8).132C23, C2.74(C233D4), (C23×C4).573C22, C22.653(C22×D4), C22⋊Q8.188C22, (C22×C4).1071C23, C22.39(C8.C22), (C2×M4(2)).80C22, (C22×Q8).315C22, (C2×C4).531(C2×D4), C2.51(C2×C8.C22), (C2×C22⋊Q8).59C2, (C2×C4⋊C4).640C22, SmallGroup(128,1927)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.128D4
C1C2C4C2×C4C22×C4C22×Q8C2×C22⋊Q8 — C24.128D4
C1C2C2×C4 — C24.128D4
C1C22C23×C4 — C24.128D4
C1C2C2C2×C4 — C24.128D4

Generators and relations for C24.128D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, ab=ba, faf-1=ac=ca, eae-1=ad=da, ebe-1=fbf-1=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=de3 >

Subgroups: 404 in 205 conjugacy classes, 88 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×11], C22, C22 [×4], C22 [×11], C8 [×4], C2×C4 [×2], C2×C4 [×2], C2×C4 [×25], Q8 [×12], C23, C23 [×2], C23 [×5], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], M4(2) [×2], Q16 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×6], C2×Q8 [×4], C2×Q8 [×6], C24, C22⋊C8 [×4], Q8⋊C4 [×8], C4.Q8 [×4], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C22⋊Q8 [×8], C22⋊Q8 [×4], C2×M4(2) [×2], C2×Q16 [×4], C23×C4, C22×Q8 [×2], C24.4C4, C22⋊Q16 [×4], C8.D4 [×4], C23.47D4 [×4], C2×C22⋊Q8 [×2], C24.128D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×4], C22×D4, 2+ 1+4 [×2], C233D4, C2×C8.C22 [×2], C24.128D4

Character table of C24.128D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D
 size 11112222422444888888888888
ρ111111111111111111111111111    trivial
ρ2111111-1-1-1111-1-11-1-11-11-111-1-11    linear of order 2
ρ3111111-1-1-1111-1-1-111-11-11-11-1-11    linear of order 2
ρ411111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ51111-1-1-1-1111-1-11-11-1111-1-1-11-11    linear of order 2
ρ61111-1-111-111-11-1-1-111-111-1-1-111    linear of order 2
ρ71111-1-111-111-11-111-1-11-1-11-1-111    linear of order 2
ρ81111-1-1-1-1111-1-111-11-1-1-111-11-11    linear of order 2
ρ91111-1-1-1-1111-1-1111-11-1-11-11-11-1    linear of order 2
ρ101111-1-111-111-11-11-1111-1-1-111-1-1    linear of order 2
ρ111111-1-111-111-11-1-11-1-1-111111-1-1    linear of order 2
ρ121111-1-1-1-1111-1-11-1-11-111-111-11-1    linear of order 2
ρ1311111111111111-1111-1-1-11-1-1-1-1    linear of order 2
ρ14111111-1-1-1111-1-1-1-1-111-111-111-1    linear of order 2
ρ15111111-1-1-1111-1-1111-1-11-1-1-111-1    linear of order 2
ρ16111111111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ172222-2-2-2-22-2-222-2000000000000    orthogonal lifted from D4
ρ18222222-2-2-2-2-2-222000000000000    orthogonal lifted from D4
ρ192222-2-222-2-2-22-22000000000000    orthogonal lifted from D4
ρ20222222222-2-2-2-2-2000000000000    orthogonal lifted from D4
ρ214-44-400000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-4000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-4-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-44004-4000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-4400-44000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-44-400000000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C24.128D4
On 32 points
Generators in S32
(2 6)(4 8)(9 21)(10 18)(11 23)(12 20)(13 17)(14 22)(15 19)(16 24)(25 29)(27 31)
(1 5)(2 27)(3 7)(4 29)(6 31)(8 25)(9 13)(10 22)(11 15)(12 24)(14 18)(16 20)(17 21)(19 23)(26 30)(28 32)
(1 30)(2 31)(3 32)(4 25)(5 26)(6 27)(7 28)(8 29)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 24)
(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 10 5 14)(2 9 6 13)(3 16 7 12)(4 15 8 11)(17 27 21 31)(18 26 22 30)(19 25 23 29)(20 32 24 28)

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

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

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

Matrix representation of C24.128D4 in GL8(𝔽17)

10000000
1616000000
001600000
00110000
00001000
00000100
0000160160
0000160016
,
10000000
01000000
001600000
000160000
000016000
000001600
00001010
00001001
,
160000000
016000000
001600000
000160000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000016000
000001600
000000160
000000016
,
0016150000
00010000
12000000
016000000
000013009
000000413
00004004
00000404
,
00100000
00010000
10000000
01000000
0000160150
000000161
00001010
000011610

G:=sub<GL(8,GF(17))| [1,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[0,0,1,0,0,0,0,0,0,0,2,16,0,0,0,0,16,0,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,13,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,0,9,13,4,4],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,1,1,0,0,0,0,0,0,0,16,0,0,0,0,15,16,1,1,0,0,0,0,0,1,0,0] >;

C24.128D4 in GAP, Magma, Sage, TeX

C_2^4._{128}D_4
% in TeX

G:=Group("C2^4.128D4");
// GroupNames label

G:=SmallGroup(128,1927);
// by ID

G=gap.SmallGroup(128,1927);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,891,352,675,4037,1027,124]);
// Polycyclic

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

Export

Character table of C24.128D4 in TeX

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