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G = M4(2)⋊C23  order 128 = 27

3rd semidirect product of M4(2) and C23 acting via C23/C2=C22

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

Aliases: C42⋊1C23, C24.44D4, M4(2)⋊3C23, 2+ (1+4).10C22, C4○D4⋊6D4, (C2×D4)⋊52D4, (C2×Q8)⋊39D4, C4≀C2⋊1C22, D4⋊4D4⋊3C2, D4.53(C2×D4), (C2×D4)⋊4C23, Q8.53(C2×D4), (C2×Q8)⋊4C23, C4.86C22≀C2, D4.9D4⋊3C2, C4⋊1D4⋊3C22, C8⋊C22⋊7C22, (C2×C4).11C24, C4○D4.6C23, C4.56(C22×D4), C23.22(C2×D4), D8⋊C22⋊6C2, C4.D4⋊9C22, C4.4D4⋊1C22, C8.C22⋊8C22, C42⋊C2⋊9C22, C22.29C24⋊5C2, C42⋊C22⋊4C2, C22.60C22≀C2, (C2×2+ (1+4))⋊4C2, C22.35(C22×D4), (C2×M4(2))⋊10C22, (C22×C4).281C23, (C22×D4).331C22, (C2×C4.D4)⋊9C2, (C2×C4).460(C2×D4), (C2×C4○D4)⋊7C22, C2.56(C2×C22≀C2), SmallGroup(128,1751)

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×D4 — 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: 844 in 381 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×12], C4 [×4], C4 [×7], C22 [×3], C22 [×34], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×23], D4 [×4], D4 [×42], Q8 [×4], Q8 [×4], C23, C23 [×4], C23 [×21], C42 [×2], C22⋊C4 [×5], C4⋊C4, C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×4], SD16 [×8], Q16 [×4], C22×C4, C22×C4 [×5], C2×D4, C2×D4 [×6], C2×D4 [×47], C2×Q8, C2×Q8 [×2], C4○D4 [×8], C4○D4 [×24], C24 [×2], C24 [×2], C4.D4 [×4], C4≀C2 [×8], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C2×M4(2) [×2], C4○D8 [×4], C8⋊C22 [×4], C8⋊C22 [×2], C8.C22 [×4], C8.C22 [×2], C22×D4, C22×D4 [×4], C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×2], 2+ (1+4) [×4], 2+ (1+4) [×6], C2×C4.D4, C42⋊C22 [×2], D4⋊4D4 [×4], D4.9D4 [×4], C22.29C24, 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=d2=e2=1, bab=eae=a5, cac=a3, dad=a5b, cbc=a4b, bd=db, be=eb, cd=dc, ce=ec, de=ed >

Permutation representations
►On 16 points - transitive group 16T214
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(1 10)(2 15)(3 12)(4 9)(5 14)(6 11)(7 16)(8 13)

(1 3)(2 6)(5 7)(9 13)(10 16)(12 14)

(1 3)(2 13)(4 11)(5 7)(6 9)(8 15)(10 12)(14 16)

(1 5)(3 7)(10 14)(12 16)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (1,3)(2,6)(5,7)(9,13)(10,16)(12,14), (1,3)(2,13)(4,11)(5,7)(6,9)(8,15)(10,12)(14,16), (1,5)(3,7)(10,14)(12,16)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (1,10)(2,15)(3,12)(4,9)(5,14)(6,11)(7,16)(8,13), (1,3)(2,6)(5,7)(9,13)(10,16)(12,14), (1,3)(2,13)(4,11)(5,7)(6,9)(8,15)(10,12)(14,16), (1,5)(3,7)(10,14)(12,16) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(1,10),(2,15),(3,12),(4,9),(5,14),(6,11),(7,16),(8,13)], [(1,3),(2,6),(5,7),(9,13),(10,16),(12,14)], [(1,3),(2,13),(4,11),(5,7),(6,9),(8,15),(10,12),(14,16)], [(1,5),(3,7),(10,14),(12,16)])
 
G:=TransitiveGroup(16,214);
 

►On 16 points - transitive group 16T262
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(10 14)(12 16)

(1 14)(2 9)(3 12)(4 15)(5 10)(6 13)(7 16)(8 11)

(3 7)(4 8)(11 15)(12 16)

(1 13)(2 10)(3 15)(4 12)(5 9)(6 14)(7 11)(8 16)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11), (3,7)(4,8)(11,15)(12,16), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,14)(2,9)(3,12)(4,15)(5,10)(6,13)(7,16)(8,11), (3,7)(4,8)(11,15)(12,16), (1,13)(2,10)(3,15)(4,12)(5,9)(6,14)(7,11)(8,16) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(1,14),(2,9),(3,12),(4,15),(5,10),(6,13),(7,16),(8,11)], [(3,7),(4,8),(11,15),(12,16)], [(1,13),(2,10),(3,15),(4,12),(5,9),(6,14),(7,11),(8,16)])
 
G:=TransitiveGroup(16,262);
 

►On 16 points - transitive group 16T288
Generators in S16
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)

(2 6)(4 8)(10 14)(12 16)

(1 16)(2 11)(3 14)(4 9)(5 12)(6 15)(7 10)(8 13)

(3 7)(4 8)(9 13)(10 14)

(2 6)(4 8)(9 13)(11 15)
 
G:=sub<Sym(16)| (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13), (3,7)(4,8)(9,13)(10,14), (2,6)(4,8)(9,13)(11,15)>;
 
G:=Group( (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16), (2,6)(4,8)(10,14)(12,16), (1,16)(2,11)(3,14)(4,9)(5,12)(6,15)(7,10)(8,13), (3,7)(4,8)(9,13)(10,14), (2,6)(4,8)(9,13)(11,15) );
 
G=PermutationGroup([(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16)], [(2,6),(4,8),(10,14),(12,16)], [(1,16),(2,11),(3,14),(4,9),(5,12),(6,15),(7,10),(8,13)], [(3,7),(4,8),(9,13),(10,14)], [(2,6),(4,8),(9,13),(11,15)])
 
G:=TransitiveGroup(16,288);
 

Matrix representation ►G ⊆ GL8(ℤ)

0000000-1
00000010
00000-100
00001000
01000000
-10000000
000-10000
00100000
,
01000000
10000000
00010000
00100000
00000100
00001000
00000001
00000010
,
00100000
000-10000
10000000
0-1000000
00001000
00000-100
000000-10
00000001
,
00100000
00010000
10000000
01000000
00000001
00000010
00000100
00001000
,
10000000
01000000
00100000
00010000
0000-1000
00000-100
000000-10
0000000-1

G:=sub<GL(8,Integers())| [0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,-1] >;
 

Character table of M4(2)⋊C23

 class 12A2B2C2D2E2F2G2H2I2J2K2L2M4A4B4C4D4E4F4G4H4I4J4K8A8B8C8D
 size 11222444444448222244448888888
ρ111111111111111111111111111111    trivial
ρ211111-111-1-1-1-1-1-111111-1-11-1111-1-11    linear of order 2
ρ311111-1-1-1-11-1-1111111-111-11-1-11-1-11    linear of order 2
ρ4111111-1-11-111-1-11111-1-1-1-1-1-1-11111    linear of order 2
ρ511-11-1-11-11-1-1111-111-1-1-111-1-1111-1-1    linear of order 2
ρ611-11-111-1-111-1-1-1-111-1-11-111-111-11-1    linear of order 2
ρ711-11-11-11-1-11-111-111-11-11-1-11-11-11-1    linear of order 2
ρ811-11-1-1-1111-11-1-1-111-111-1-111-111-1-1    linear of order 2
ρ91111111111111-111111111-1-1-1-1-1-1-1    linear of order 2
ρ1011111-111-1-1-1-1-1111111-1-111-1-1-111-1    linear of order 2
ρ1111111-1-1-1-11-1-11-11111-111-1-111-111-1    linear of order 2
ρ12111111-1-11-111-111111-1-1-1-1111-1-1-1-1    linear of order 2
ρ1311-11-1-11-11-1-111-1-111-1-1-11111-1-1-111    linear of order 2
ρ1411-11-111-1-111-1-11-111-1-11-11-11-1-11-11    linear of order 2
ρ1511-11-11-11-1-11-11-1-111-11-11-11-11-11-11    linear of order 2
ρ1611-11-1-1-1111-11-11-111-111-1-1-1-11-1-111    linear of order 2
ρ1722-2-220-22000000-2-222-20020000000    orthogonal lifted from D4
ρ1822-2-220000200-2022-2-20-2200000000    orthogonal lifted from D4
ρ1922-2-220000-2002022-2-202-200000000    orthogonal lifted from D4
ρ2022-22-2200-20-22002-2-2200000000000    orthogonal lifted from D4
ρ21222-2-20-2-20000002-22-220020000000    orthogonal lifted from D4
ρ22222-2-20000-200-20-22-2202200000000    orthogonal lifted from D4
ρ23222-2-2000020020-22-220-2-200000000    orthogonal lifted from D4
ρ242222220020-2-200-2-2-2-200000000000    orthogonal lifted from D4
ρ2522-2-2202-2000000-2-222200-20000000    orthogonal lifted from D4
ρ2622222-200-202200-2-2-2-200000000000    orthogonal lifted from D4
ρ2722-22-2-200202-2002-2-2200000000000    orthogonal lifted from D4
ρ28222-2-20220000002-22-2-200-20000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    orthogonal faithful

In GAP, Magma, Sage, TeX 

M_{4(2)}\rtimes C_2^3
 
% in TeX
 
G:=Group("M4(2):C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1751);
 
// by ID
 
G=gap.SmallGroup(128,1751);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,2804,1411,718,172,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^8=b^2=c^2=d^2=e^2=1,b*a*b=e*a*e=a^5,c*a*c=a^3,d*a*d=a^5*b,c*b*c=a^4*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,d*e=e*d>;
 
// generators/relations
 

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