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G = C23.4D8  order 128 = 27

4th non-split extension by C23 of D8 acting via D8/C2=D4

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

Aliases: C23.4D8, C24.2D4, C23.4SD16, C23⋊C8.3C2, (C22×C4).9D4, C22.14C4≀C2, C2.C42⋊4C4, C2.4(C42⋊3C4), C23.9D4.3C2, C22.55(C23⋊C4), C23.11D4.1C2, C2.4(C23.D4), C2.9(C22.SD16), C22.18(D4⋊C4), C23.155(C22⋊C4), (C2×C4⋊C4)⋊2C4, (C22×C4).2(C2×C4), (C2×C22⋊C4).83C22, SmallGroup(128,76)

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

Derived series
C1 — C22×C4 — C23.4D8
Chief series
C1 — C2 — C22 — C23 — C24 — C2×C22⋊C4 — C23.11D4 — C23.4D8
Lower central
C1 — C2 — C23 — C22×C4 — C23.4D8
Upper central
C1 — C22 — C23 — C2×C22⋊C4 — C23.4D8
Jennings
C1 — C22 — C23 — C2×C22⋊C4 — C23.4D8

Generators and relations for C23.4D8
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=dc=cd, f2=a, ab=ba, ac=ca, ad=da, eae-1=abcd, af=fa, bc=cb, ebe-1=fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=ace3 >

Subgroups: 228 in 73 conjugacy classes, 20 normal (all characteristic)
C1, C2, C2, C4, C22, C22, C8, C2×C4, C23, C23, C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C24, C2.C42, C2.C42, C22⋊C8, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C23⋊C8, C23.9D4, C23.11D4, C23.4D8
Quotients: C1, C2, C4, C22, C2×C4, D4, C22⋊C4, D8, SD16, C23⋊C4, D4⋊C4, C4≀C2, C22.SD16, C23.D4, C42⋊3C4, C23.4D8

Character table of C23.4D8

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11112244448888888888888
ρ111111111111111111111111    trivial
ρ21111111111-1-1-1-11-1-1-1-11111    linear of order 2
ρ31111111111-11-111-1-111-1-1-1-1    linear of order 2
ρ411111111111-11-1111-1-1-1-1-1-1    linear of order 2
ρ5111111-1-1-1-1-i-1i11i-i1-1-iii-i    linear of order 4
ρ6111111-1-1-1-1i1-i-11-ii-11-iii-i    linear of order 4
ρ7111111-1-1-1-1-i1i-11i-i-11i-i-ii    linear of order 4
ρ8111111-1-1-1-1i-1-i11-ii1-1i-i-ii    linear of order 4
ρ922222222-2-20000-200000000    orthogonal lifted from D4
ρ10222222-2-2220000-200000000    orthogonal lifted from D4
ρ112-22-22-2-2200000000000√2-√2√2-√2    orthogonal lifted from D8
ρ122-22-22-2-2200000000000-√2√2-√2√2    orthogonal lifted from D8
ρ132-22-2-22002i-2i1-i0-1-i001+i-1+i000000    complex lifted from C4≀C2
ρ142-22-2-2200-2i2i-1-i01-i00-1+i1+i000000    complex lifted from C4≀C2
ρ152-22-2-22002i-2i-1+i01+i00-1-i1-i000000    complex lifted from C4≀C2
ρ162-22-22-22-200000000000√-2√-2-√-2-√-2    complex lifted from SD16
ρ172-22-2-2200-2i2i1+i0-1+i001-i-1-i000000    complex lifted from C4≀C2
ρ182-22-22-22-200000000000-√-2-√-2√-2√-2    complex lifted from SD16
ρ194444-4-400000000000000000    orthogonal lifted from C23⋊C4
ρ2044-4-40000000-2i0000002i0000    complex lifted from C42⋊3C4
ρ214-4-440000000002i000-2i00000    complex lifted from C23.D4
ρ2244-4-400000002i000000-2i0000    complex lifted from C42⋊3C4
ρ234-4-44000000000-2i0002i00000    complex lifted from C23.D4

Smallest permutation representation of C23.4D8
►On 32 points
Generators in S32
(2 29)(3 19)(4 16)(6 25)(7 23)(8 12)(10 22)(11 26)(14 18)(15 30)(20 31)(24 27)

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

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

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

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

Matrix representation of C23.4D8 ►in GL6(𝔽17)

100000
9160000
001000
000100
0000160
0000016
,
100000
010000
000100
001000
000001
000010
,
1600000
0160000
0016000
0001600
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
290000
0150000
0022215
001515215
0021522
002151515
,
1600000
5130000
0016000
000100
000001
0000160

G:=sub<GL(6,GF(17))| [1,9,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[2,0,0,0,0,0,9,15,0,0,0,0,0,0,2,15,2,2,0,0,2,15,15,15,0,0,2,2,2,15,0,0,15,15,2,15],[16,5,0,0,0,0,0,13,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0] >;
 

C23.4D8 in GAP, Magma, Sage, TeX 

C_2^3._4D_8
 
% in TeX
 
G:=Group("C2^3.4D8");
 
// GroupNames label
 
G:=SmallGroup(128,76);
 
// by ID
 
G=gap.SmallGroup(128,76);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,387,184,794,521,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d*c=c*d,f^2=a,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*c*d,a*f=f*a,b*c=c*b,e*b*e^-1=f*b*f^-1=b*d=d*b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=a*c*e^3>;
 
// generators/relations
 

Export

Character table of C23.4D8 in TeX

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