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G = M6(2)⋊C2  order 128 = 27

6th semidirect product of M6(2) and C2 acting faithfully

p-group, metabelian, nilpotent (class 5), monomial

Aliases: D16.2C4, C16.17D4, C4.11D16, C8.5SD16, M6(2)⋊6C2, C22.3SD32, C16.5(C2×C4), (C2×C8).85D4, (C2×C4).13D8, (C2×D16).5C2, C8.4Q8⋊2C2, C8.17(C22⋊C4), (C2×C16).14C22, C4.17(D4⋊C4), C2.10(C2.D16), SmallGroup(128,151)

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

Derived series
C1 — C16 — M6(2)⋊C2
Chief series
C1 — C2 — C4 — C8 — C2×C8 — C2×C16 — C2×D16 — M6(2)⋊C2
Lower central
C1 — C2 — C4 — C8 — C16 — M6(2)⋊C2
Upper central
C1 — C2 — C2×C4 — C2×C8 — C2×C16 — M6(2)⋊C2
Jennings
C1 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C2 — C4 — C4 — C4 — C4 — C8 — C8 — C2×C16 — M6(2)⋊C2

Generators and relations for M6(2)⋊C2
 G = < a,b,c | a32=b2=c2=1, bab=a17, cac=a7b, bc=cb >

C1
C2
2C2
16C2
16C2
C22
C4
C4
8C22
8C22
16C22
16C22
C2×C4
C8
C8
4D4
4D4
8D4
8C23
8C8
C2×C8
C16
C16
2D8
2D8
4C2×D4
4D8
4M4(2)
D16
C2×C16
D16
2C32
2C8.C4
2D16
2C2×D8
C2×D16
C8.4Q8
M6(2)
M6(2)⋊C2

Character table of M6(2)⋊C2

 class 12A2B2C2D4A4B8A8B8C8D8E16A16B16C16D16E16F32A32B32C32D32E32F32G32H
 size 112161622224161622224444444444
ρ111111111111111111111111111    trivial
ρ21111111111-1-1111111-1-1-1-1-1-1-1-1    linear of order 2
ρ3111-1-111111-1-111111111111111    linear of order 2
ρ4111-1-11111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ511-1-11-1111-1i-i-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ611-1-11-1111-1-ii-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ711-11-1-1111-1i-i-1-1-1-111-ii-ii-iii-i    linear of order 4
ρ811-11-1-1111-1-ii-1-1-1-111i-ii-ii-i-ii    linear of order 4
ρ922-200-2222-2002222-2-200000000    orthogonal lifted from D4
ρ10222002222200-2-2-2-2-2-200000000    orthogonal lifted from D4
ρ1122-2002-200000√2-√2√2-√2-√2√2-ζ165+ζ163-ζ165+ζ163ζ1615-ζ169ζ1615-ζ169ζ165-ζ163-ζ1615+ζ169ζ165-ζ163-ζ1615+ζ169    orthogonal lifted from D16
ρ122220022-2-2-200000000√2√2-√2-√2√2-√2√2-√2    orthogonal lifted from D8
ρ132220022-2-2-200000000-√2-√2√2√2-√2√2-√2√2    orthogonal lifted from D8
ρ1422-2002-200000-√2√2-√2√2√2-√2-ζ1615+ζ169-ζ1615+ζ169-ζ165+ζ163-ζ165+ζ163ζ1615-ζ169ζ165-ζ163ζ1615-ζ169ζ165-ζ163    orthogonal lifted from D16
ρ1522-2002-200000√2-√2√2-√2-√2√2ζ165-ζ163ζ165-ζ163-ζ1615+ζ169-ζ1615+ζ169-ζ165+ζ163ζ1615-ζ169-ζ165+ζ163ζ1615-ζ169    orthogonal lifted from D16
ρ1622-2002-200000-√2√2-√2√2√2-√2ζ1615-ζ169ζ1615-ζ169ζ165-ζ163ζ165-ζ163-ζ1615+ζ169-ζ165+ζ163-ζ1615+ζ169-ζ165+ζ163    orthogonal lifted from D16
ρ1722-200-22-2-2200000000√-2-√-2-√-2√-2√-2√-2-√-2-√-2    complex lifted from SD16
ρ1822200-2-200000√2-√2√2-√2√2-√2ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16    complex lifted from SD32
ρ1922-200-22-2-2200000000-√-2√-2√-2-√-2-√-2-√-2√-2√-2    complex lifted from SD16
ρ2022200-2-200000√2-√2√2-√2√2-√2ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169    complex lifted from SD32
ρ2122200-2-200000-√2√2-√2√2-√2√2ζ167+ζ16ζ1615+ζ169ζ165+ζ163ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163ζ167+ζ16ζ1613+ζ1611    complex lifted from SD32
ρ2222200-2-200000-√2√2-√2√2-√2√2ζ1615+ζ169ζ167+ζ16ζ1613+ζ1611ζ165+ζ163ζ167+ζ16ζ1613+ζ1611ζ1615+ζ169ζ165+ζ163    complex lifted from SD32
ρ234-400000-2√22√2000-2ζ165+2ζ163-2ζ1615+2ζ1692ζ165-2ζ1632ζ1615-2ζ1690000000000    orthogonal faithful
ρ244-4000002√2-2√20002ζ1615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1692ζ165-2ζ1630000000000    orthogonal faithful
ρ254-400000-2√22√20002ζ165-2ζ1632ζ1615-2ζ169-2ζ165+2ζ163-2ζ1615+2ζ1690000000000    orthogonal faithful
ρ264-4000002√2-2√2000-2ζ1615+2ζ1692ζ165-2ζ1632ζ1615-2ζ169-2ζ165+2ζ1630000000000    orthogonal faithful

Smallest permutation representation of M6(2)⋊C2
►On 32 points
Generators in S32
(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 17)(3 19)(5 21)(7 23)(9 25)(11 27)(13 29)(15 31)

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

Matrix representation of M6(2)⋊C2 ►in GL4(𝔽97) generated by

893202
421020
54678755
8566658
,
96000
09600
421010
893201
,
0100
1000
4844262
3383271
G:=sub<GL(4,GF(97))| [89,42,54,85,32,10,67,66,0,2,87,65,2,0,55,8],[96,0,42,89,0,96,10,32,0,0,1,0,0,0,0,1],[0,1,48,33,1,0,44,83,0,0,26,2,0,0,2,71] >;
 

M6(2)⋊C2 in GAP, Magma, Sage, TeX 

M_6(2)\rtimes C_2
 
% in TeX
 
G:=Group("M6(2):C2");
 
// GroupNames label
 
G:=SmallGroup(128,151);
 
// by ID
 
G=gap.SmallGroup(128,151);
 
# by ID
 
G:=PCGroup([7,-2,2,-2,2,-2,-2,-2,56,85,422,891,604,1018,248,1684,851,242,4037,2028,124]);
 
// Polycyclic
 
G:=Group<a,b,c|a^32=b^2=c^2=1,b*a*b=a^17,c*a*c=a^7*b,b*c=c*b>;
 
// generators/relations
 

Export

Subgroup lattice of M6(2)⋊C2 in TeX
Character table of M6(2)⋊C2 in TeX

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