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G = C23.1M4(2)  order 128 = 27

1st non-split extension by C23 of M4(2) acting via M4(2)/C4=C4

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

Aliases: C23.1M4(2), (C2xD4).2C8, (C2xC8).16D4, (C2xQ8).2C8, C23.C8:5C2, (C22xC8).6C4, C4.38(C23:C4), C2.11(C23:C8), C4.14(C4.D4), C22.16(C22:C8), (C2xM4(2)).144C22, (C2xC4).3(C2xC8), (C2xC4oD4).2C4, (C22xC4).59(C2xC4), (C22xC8):C2.9C2, (C2xC4).343(C22:C4), SmallGroup(128,53)

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

Derived series
C1 — C2xC4 — C23.1M4(2)
Chief series
C1 — C2 — C4 — C2xC4 — C2xC8 — C2xM4(2) — (C22xC8):C2 — C23.1M4(2)
Lower central
C1 — C2 — C22 — C2xC4 — C23.1M4(2)
Upper central
C1 — C4 — C2xC4 — C2xM4(2) — C23.1M4(2)
Jennings
C1 — C2 — C2 — C2 — C2 — C4 — C2xC4 — C2xM4(2) — C23.1M4(2)

Generators and relations for C23.1M4(2)
 G = < a,b,c,d,e | a2=b2=c2=e2=1, d8=c, dad-1=ab=ba, eae=ac=ca, dbd-1=bc=cb, be=eb, cd=dc, ce=ec, ede=abcd5 >

Subgroups: 136 in 56 conjugacy classes, 20 normal (16 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2xC4, C2xC4, D4, Q8, C23, C23, C16, C2xC8, C2xC8, M4(2), C22xC4, C22xC4, C2xD4, C2xD4, C2xQ8, C4oD4, C22:C8, M5(2), C22xC8, C2xM4(2), C2xC4oD4, C23.C8, (C22xC8):C2, C23.1M4(2)
Quotients: C1, C2, C4, C22, C8, C2xC4, D4, C22:C4, C2xC8, M4(2), C22:C8, C23:C4, C4.D4, C23:C8, C23.1M4(2)

Character table of C23.1M4(2)

 class 12A2B2C2D4A4B4C4D4E8A8B8C8D8E8F8G8H16A16B16C16D16E16F16G16H
 size 11248112484444444488888888
ρ111111111111111111111111111    trivial
ρ21111-11111-1-1111-11-1-1-111-111-1-1    linear of order 2
ρ31111-11111-1-1111-11-1-11-1-11-1-111    linear of order 2
ρ4111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ51111111111-1-1-1-1-1-1-1-1ii-i-i-ii-ii    linear of order 4
ρ61111-11111-11-1-1-11-111-ii-ii-iii-i    linear of order 4
ρ71111111111-1-1-1-1-1-1-1-1-i-iiii-ii-i    linear of order 4
ρ81111-11111-11-1-1-11-111i-ii-ii-i-ii    linear of order 4
ρ9111-11-1-1-11-1i-i-ii-iii-iζ87ζ83ζ85ζ85ζ8ζ87ζ8ζ83    linear of order 8
ρ10111-11-1-1-11-1-iii-ii-i-iiζ85ζ8ζ87ζ87ζ83ζ85ζ83ζ8    linear of order 8
ρ11111-11-1-1-11-1i-i-ii-iii-iζ83ζ87ζ8ζ8ζ85ζ83ζ85ζ87    linear of order 8
ρ12111-1-1-1-1-111iii-i-i-ii-iζ8ζ8ζ87ζ83ζ83ζ85ζ87ζ85    linear of order 8
ρ13111-1-1-1-1-111-i-i-iiii-iiζ87ζ87ζ8ζ85ζ85ζ83ζ8ζ83    linear of order 8
ρ14111-11-1-1-11-1-iii-ii-i-iiζ8ζ85ζ83ζ83ζ87ζ8ζ87ζ85    linear of order 8
ρ15111-1-1-1-1-111-i-i-iiii-iiζ83ζ83ζ85ζ8ζ8ζ87ζ85ζ87    linear of order 8
ρ16111-1-1-1-1-111iii-i-i-ii-iζ85ζ85ζ83ζ87ζ87ζ8ζ83ζ8    linear of order 8
ρ17222-20222-2002-220-20000000000    orthogonal lifted from D4
ρ18222-20222-200-22-2020000000000    orthogonal lifted from D4
ρ1922220-2-2-2-200-2i2i2i0-2i0000000000    complex lifted from M4(2)
ρ2022220-2-2-2-2002i-2i-2i02i0000000000    complex lifted from M4(2)
ρ2144-40044-4000000000000000000    orthogonal lifted from C23:C4
ρ2244-400-4-44000000000000000000    orthogonal lifted from C4.D4
ρ234-4000-4i4i0002ζ850002ζ8302ζ82ζ8700000000    complex faithful
ρ244-4000-4i4i0002ζ80002ζ8702ζ852ζ8300000000    complex faithful
ρ254-40004i-4i0002ζ830002ζ8502ζ872ζ800000000    complex faithful
ρ264-40004i-4i0002ζ870002ζ802ζ832ζ8500000000    complex faithful

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

(2 10)(4 12)(6 14)(8 16)(17 25)(19 27)(21 29)(23 31)

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

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

Matrix representation of C23.1M4(2) ►in GL4(F17) generated by

0200
9000
0002
0090
,
1000
0100
00160
00016
,
16000
01600
00160
00016
,
0010
00016
15000
01500
,
1000
01600
0008
00150
G:=sub<GL(4,GF(17))| [0,9,0,0,2,0,0,0,0,0,0,9,0,0,2,0],[1,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[0,0,15,0,0,0,0,15,1,0,0,0,0,16,0,0],[1,0,0,0,0,16,0,0,0,0,0,15,0,0,8,0] >;
 

C23.1M4(2) in GAP, Magma, Sage, TeX 

C_2^3._1M_4(2)
 
% in TeX
 
G:=Group("C2^3.1M4(2)");
 
// GroupNames label
 
G:=SmallGroup(128,53);
 
// by ID
 
G=gap.SmallGroup(128,53);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,2,-2,56,85,422,723,352,1242,521,136,2804,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^2=e^2=1,d^8=c,d*a*d^-1=a*b=b*a,e*a*e=a*c=c*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=a*b*c*d^5>;
 
// generators/relations
 

Export

Character table of C23.1M4(2) in TeX

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