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G = C23.15M4(2)  order 128 = 27

2nd non-split extension by C23 of M4(2) acting via M4(2)/C22=C4

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

Aliases: C23.15M4(2), (C22×C4)⋊2C8, C23⋊C8.1C2, (C23×C4).3C4, (C22×C4).2D4, C2.7(C23⋊C8), C24.16(C2×C4), C23.14(C2×C8), C22.12(C22⋊C8), C22.36(C23⋊C4), C23.34D4.1C2, C2.1(C23.D4), C22.7(C4.D4), C23.148(C22⋊C4), (C2×C22⋊C4).4C4, (C2×C22⋊C4).2C22, SmallGroup(128,49)

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

Derived series
C1 — C23 — C23.15M4(2)
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C22⋊C4 — C23.34D4 — C23.15M4(2)
Lower central
C1 — C2 — C22 — C23 — C23.15M4(2)
Upper central
C1 — C22 — C23 — C2×C22⋊C4 — C23.15M4(2)
Jennings
C1 — C22 — C23 — C2×C22⋊C4 — C23.15M4(2)

Generators and relations for C23.15M4(2)
 G = < a,b,c,d,e | a2=b2=c2=d8=1, e2=c, ab=ba, ac=ca, dad-1=abc, ae=ea, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede-1=abd5 >

Subgroups: 208 in 75 conjugacy classes, 22 normal (14 characteristic)
C1, C2, C2, C2, C4, C22, C22, C8, C2×C4, C23, C23, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, C2.C42, C22⋊C8, C2×C22⋊C4, C23×C4, C23⋊C8, C23.34D4, C23.15M4(2)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, C22⋊C4, C2×C8, M4(2), C22⋊C8, C23⋊C4, C4.D4, C23⋊C8, C23.D4, C23.15M4(2)

Character table of C23.15M4(2)

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F8G8H
 size 11112244444444448888888888
ρ111111111111111111111111111    trivial
ρ2111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ311111111-1111-1-11-1-1-111-11-1-1-11    linear of order 2
ρ411111111-1111-1-11-1-1-1-1-11-1111-1    linear of order 2
ρ5111111111-1-1-111-11-1-1-ii-i-ii-iii    linear of order 4
ρ6111111111-1-1-111-11-1-1i-iii-ii-i-i    linear of order 4
ρ711111111-1-1-1-1-1-1-1-111-iii-i-ii-ii    linear of order 4
ρ811111111-1-1-1-1-1-1-1-111i-i-iii-ii-i    linear of order 4
ρ91-11-1-111-1-1i-ii11-i-1-iiζ85ζ87ζ8ζ8ζ87ζ85ζ83ζ83    linear of order 8
ρ101-11-1-111-1-1i-ii11-i-1-iiζ8ζ83ζ85ζ85ζ83ζ8ζ87ζ87    linear of order 8
ρ111-11-1-111-11-ii-i-1-1i1-iiζ83ζ8ζ83ζ87ζ85ζ87ζ8ζ85    linear of order 8
ρ121-11-1-111-11-ii-i-1-1i1-iiζ87ζ85ζ87ζ83ζ8ζ83ζ85ζ8    linear of order 8
ρ131-11-1-111-11i-ii-1-1-i1i-iζ85ζ87ζ85ζ8ζ83ζ8ζ87ζ83    linear of order 8
ρ141-11-1-111-1-1-ii-i11i-1i-iζ83ζ8ζ87ζ87ζ8ζ83ζ85ζ85    linear of order 8
ρ151-11-1-111-11i-ii-1-1-i1i-iζ8ζ83ζ8ζ85ζ87ζ85ζ83ζ87    linear of order 8
ρ161-11-1-111-1-1-ii-i11i-1i-iζ87ζ85ζ83ζ83ζ85ζ87ζ8ζ8    linear of order 8
ρ17222222-2-20-2-2200200000000000    orthogonal lifted from D4
ρ18222222-2-2022-200-200000000000    orthogonal lifted from D4
ρ192-22-2-22-2202i-2i-2i002i00000000000    complex lifted from M4(2)
ρ202-22-2-22-220-2i2i2i00-2i00000000000    complex lifted from M4(2)
ρ214-44-44-400000000000000000000    orthogonal lifted from C4.D4
ρ224444-4-400000000000000000000    orthogonal lifted from C23⋊C4
ρ2344-4-400002i0002i-2i0-2i0000000000    complex lifted from C23.D4
ρ244-4-4400002i000-2i2i0-2i0000000000    complex lifted from C23.D4
ρ2544-4-40000-2i000-2i2i02i0000000000    complex lifted from C23.D4
ρ264-4-440000-2i0002i-2i02i0000000000    complex lifted from C23.D4

Smallest permutation representation of C23.15M4(2)
►On 32 points
Generators in S32
(1 29)(2 9)(3 14)(4 28)(5 25)(6 13)(7 10)(8 32)(11 22)(12 19)(15 18)(16 23)(17 27)(20 26)(21 31)(24 30)

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

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

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

Matrix representation of C23.15M4(2) ►in GL6(𝔽17)

100000
010000
0001600
0016000
0000016
0000160
,
100000
010000
001000
000100
0000160
0000016
,
100000
010000
0016000
0001600
0000160
0000016
,
940000
080000
0000130
0000013
0001600
001000
,
1600000
1310000
0013000
0001300
0000013
0000130

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

C23.15M4(2) in GAP, Magma, Sage, TeX 

C_2^3._{15}M_4(2)
 
% in TeX
 
G:=Group("C2^3.15M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,49);
 
// by ID
 
G=gap.SmallGroup(128,49);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,456,422,346,521,136,2804]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=c,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*b*d^5>;
 
// generators/relations
 

Export

Character table of C23.15M4(2) in TeX

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