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G = C24.5D4  order 128 = 27

5th non-split extension by C24 of D4 acting faithfully

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

Aliases: C24.5D4, C23.1C42, C2.3C2≀C4, C23⋊C4⋊1C4, (C23×C4)⋊1C4, (C2×D4).1Q8, (C2×D4).39D4, C4.D4⋊5C4, C23.7(C4⋊C4), C24.18(C2×C4), (C22×C4).35D4, C23.1(C22⋊C4), (C22×D4).1C22, C22.42(C23⋊C4), C23.23D4.1C2, C2.3(C23.D4), C2.12(C23.9D4), C22.1(C2.C42), (C2×C4).1(C4⋊C4), (C2×C22⋊C4)⋊2C4, (C2×D4).43(C2×C4), (C2×C23⋊C4).1C2, (C2×C4).1(C22⋊C4), (C2×C4.D4).5C2, SmallGroup(128,122)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C24.5D4
Chief series
C1 — C2 — C22 — C23 — C24 — C22×D4 — C23.23D4 — C24.5D4
Lower central
C1 — C2 — C22 — C23 — C24.5D4
Upper central
C1 — C22 — C23 — C22×D4 — C24.5D4
Jennings
C1 — C2 — C22 — C22×D4 — C24.5D4

Generators and relations for C24.5D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=a, ab=ba, ac=ca, ad=da, eae-1=acd, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf-1=cd=dc, de=ed, df=fd, fef-1=abde3 >

Subgroups: 336 in 117 conjugacy classes, 32 normal (28 characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C2×C4, D4, C23, C23, C22⋊C4, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C23⋊C4, C23⋊C4, C4.D4, C4.D4, C2×C22⋊C4, C2×C22⋊C4, C2×M4(2), C23×C4, C22×D4, C23.23D4, C2×C23⋊C4, C2×C4.D4, C24.5D4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C42, C22⋊C4, C4⋊C4, C2.C42, C23⋊C4, C23.9D4, C2≀C4, C23.D4, C24.5D4

Character table of C24.5D4

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ21111111111-111-1-1-111-1-111-1-1-1-1    linear of order 2
ρ31111111111111111-1-111-1-1-1-1-1-1    linear of order 2
ρ41111111111-111-1-1-1-1-1-1-1-1-11111    linear of order 2
ρ51-11-1-111-11-1i1-1-ii-i-11-ii-11-iii-i    linear of order 4
ρ61-11-1-111-1-11-i-11i-iiii-ii-i-i1-11-1    linear of order 4
ρ71-11-1-111-1-11-i-11i-ii-i-i-iiii-11-11    linear of order 4
ρ81-11-1-111-11-1-i1-1i-ii-11i-i-11i-i-ii    linear of order 4
ρ91-11-1-111-1-11i-11-ii-i-i-ii-iii1-11-1    linear of order 4
ρ101-11-1-111-1-11i-11-ii-iiii-i-i-i-11-11    linear of order 4
ρ111-11-1-111-11-1i1-1-ii-i1-1-ii1-1i-i-ii    linear of order 4
ρ1211111111-1-1-1-1-1-1-1-1-ii11i-iii-i-i    linear of order 4
ρ1311111111-1-1-1-1-1-1-1-1i-i11-ii-i-iii    linear of order 4
ρ141-11-1-111-11-1-i1-1i-ii1-1i-i1-1-iii-i    linear of order 4
ρ1511111111-1-11-1-1111i-i-1-1-iiii-i-i    linear of order 4
ρ1611111111-1-11-1-1111-ii-1-1i-i-i-iii    linear of order 4
ρ17222222-2-2-2-20220000000000000    orthogonal lifted from D4
ρ18222222-2-2220-2-20000000000000    orthogonal lifted from D4
ρ192-22-2-22-222-20-220000000000000    orthogonal lifted from D4
ρ202-22-2-22-22-2202-20000000000000    symplectic lifted from Q8, Schur index 2
ρ214-44-44-400000000000000000000    orthogonal lifted from C23⋊C4
ρ224-4-44000000-200-2220000000000    orthogonal lifted from C2≀C4
ρ234-4-440000002002-2-20000000000    orthogonal lifted from C2≀C4
ρ244444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2544-4-40000002i00-2i-2i2i0000000000    complex lifted from C23.D4
ρ2644-4-4000000-2i002i2i-2i0000000000    complex lifted from C23.D4

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

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

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

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

Matrix representation of C24.5D4 ►in GL6(𝔽17)

1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
001000
000100
000010
000001
,
100000
010000
000100
001000
000001
000010
,
100000
010000
0016000
0001600
0000160
0000016
,
1150000
1160000
008999
008988
008889
009989
,
400000
4130000
001000
0001600
0000016
000010

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[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],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,8,8,8,9,0,0,9,9,8,9,0,0,9,8,8,8,0,0,9,8,9,9],[4,4,0,0,0,0,0,13,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;
 

C24.5D4 in GAP, Magma, Sage, TeX 

C_2^4._5D_4
 
% in TeX
 
G:=Group("C2^4.5D4");
 
// GroupNames label
 
G:=SmallGroup(128,122);
 
// by ID
 
G=gap.SmallGroup(128,122);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,2,-2,-2,56,85,120,758,570,521,4037]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*c*d,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*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*d*e^3>;
 
// generators/relations
 

Export

Character table of C24.5D4 in TeX

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