Copied to
clipboard

?

G = M4(2).C23  order 128 = 27

4th non-split extension by M4(2) of C23 acting via C23/C2=C22

p-group, metabelian, nilpotent (class 3), monomial, rational

Aliases: C42.11C23, M4(2).4C23, 2- (1+4).8C22, C4⋊Q8⋊4C22, C4≀C2⋊2C22, C4○D4.18D4, D4.54(C2×D4), Q8.54(C2×D4), C4.87C22≀C2, (C2×D4).300D4, D4.8D4⋊3C2, C8⋊C22⋊8C22, (C2×C4).12C24, (C2×Q8).235D4, C4○D4.7C23, C4.57(C22×D4), D4.10D4⋊3C2, D8⋊C22⋊7C2, (C2×D4).36C23, C4.4D4⋊2C22, C23.237(C2×D4), (C22×C4).111D4, C8.C22⋊9C22, (C2×Q8).28C23, C42⋊C22⋊5C2, C22.61C22≀C2, C4.10D4⋊9C22, (C2×2- (1+4))⋊4C2, C22.36(C22×D4), (C22×C4).282C23, C42⋊C2.99C22, C23.38C23⋊5C2, (C2×M4(2)).46C22, (C22×Q8).266C22, (C2×C4).25(C2×D4), C2.57(C2×C22≀C2), (C2×C4.10D4)⋊9C2, (C2×C4○D4).108C22, SmallGroup(128,1752)

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

Derived series
C1 — C2×C4 — M4(2).C23
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C22×Q8 — C2×2- (1+4) — M4(2).C23
Lower central
C1 — C2 — C2×C4 — M4(2).C23
Upper central
C1 — C2 — C22×C4 — M4(2).C23
Jennings
C1 — C2 — C2 — C2×C4 — M4(2).C23

Subgroups: 636 in 349 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×8], C4 [×4], C4 [×11], C22 [×3], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×35], D4 [×4], D4 [×22], Q8 [×4], Q8 [×20], C23, C23 [×3], C42 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×2], C22×C4 [×7], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8 [×6], C2×Q8 [×22], C4○D4 [×8], C4○D4 [×40], C4.10D4 [×4], C4≀C2 [×8], C42⋊C2, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C4○D8 [×4], C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C22×Q8, C22×Q8 [×2], C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×4], 2- (1+4) [×4], 2- (1+4) [×6], C2×C4.10D4, C42⋊C22 [×2], D4.8D4 [×4], D4.10D4 [×4], C23.38C23, D8⋊C22 [×2], C2×2- (1+4), M4(2).C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, M4(2).C23

Generators and relations
 G = < a,b,c,d,e | a8=b2=c2=1, d2=a4, e2=a6, bab=a5, cac=a3, dad-1=a5b, ae=ea, cbc=ebe-1=a4b, bd=db, dcd-1=a4c, ece-1=a6c, ede-1=a4bd >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

000000130
000000134
000013000
000013400
40000000
413000000
001300000
001340000
,
115000000
016000000
001150000
000160000
000011500
000001600
000000115
000000016
,
49000000
413000000
001380000
001340000
00000049
000000413
00004900
000041300
,
138000000
04000000
00490000
000130000
000013000
000001300
00000040
00000004
,
00004900
000041300
000000138
000000134
00490000
004130000
49000000
413000000

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

Character table of M4(2).C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11222444482222444444448888888
ρ111111111111111111111111111111    trivial
ρ211111-1-1-1-1-11111111-1-1-1-11-1-1-11111    linear of order 2
ρ311-11-11-1-111-1-111-1-111-1-1111-1-111-1-1    linear of order 2
ρ411-11-1-111-1-1-1-111-1-11-111-11-11111-1-1    linear of order 2
ρ511-11-111-1-11-1-11111-1-11-11-1-1-111-11-1    linear of order 2
ρ611-11-1-1-111-1-1-11111-11-11-1-111-11-11-1    linear of order 2
ρ7111111-11-111111-1-1-1-1-111-1-11-11-1-11    linear of order 2
ρ811111-11-11-11111-1-1-111-1-1-11-111-1-11    linear of order 2
ρ9111111111-1111111111111-1-1-1-1-1-1-1    linear of order 2
ρ1011111-1-1-1-111111111-1-1-1-11111-1-1-1-1    linear of order 2
ρ1111-11-11-1-11-1-1-111-1-111-1-111-111-1-111    linear of order 2
ρ1211-11-1-111-11-1-111-1-11-111-111-1-1-1-111    linear of order 2
ρ1311-11-111-1-1-1-1-11111-1-11-11-111-1-11-11    linear of order 2
ρ1411-11-1-1-1111-1-11111-11-11-1-1-1-11-11-11    linear of order 2
ρ15111111-11-1-11111-1-1-1-1-111-11-11-111-1    linear of order 2
ρ1611111-11-1111111-1-1-111-1-1-1-11-1-111-1    linear of order 2
ρ17222-2-220200-22-2200000-2-200000000    orthogonal lifted from D4
ρ1822-22-20000022-2-22-220000-20000000    orthogonal lifted from D4
ρ19222-2-2020202-22-2000-2-20000000000    orthogonal lifted from D4
ρ2022-2-22-202002-2-2200000-2200000000    orthogonal lifted from D4
ρ212222200000-2-2-2-22-2-2000020000000    orthogonal lifted from D4
ρ2222-2-22020-20-222-20002-20000000000    orthogonal lifted from D4
ρ23222-2-2-20-200-22-22000002200000000    orthogonal lifted from D4
ρ24222-2-20-20-202-22-2000220000000000    orthogonal lifted from D4
ρ252222200000-2-2-2-2-2220000-20000000    orthogonal lifted from D4
ρ2622-22-20000022-2-2-22-2000020000000    orthogonal lifted from D4
ρ2722-2-2220-2002-2-22000002-200000000    orthogonal lifted from D4
ρ2822-2-220-2020-222-2000-220000000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

In GAP, Magma, Sage, TeX 

M_{4(2)}.C_2^3
 
% in TeX
 
G:=Group("M4(2).C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1752);
 
// by ID
 
G=gap.SmallGroup(128,1752);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2019,2804,1411,718,172,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^8=b^2=c^2=1,d^2=a^4,e^2=a^6,b*a*b=a^5,c*a*c=a^3,d*a*d^-1=a^5*b,a*e=e*a,c*b*c=e*b*e^-1=a^4*b,b*d=d*b,d*c*d^-1=a^4*c,e*c*e^-1=a^6*c,e*d*e^-1=a^4*b*d>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀