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G = C42.12C23  order 128 = 27

12nd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.12C23, M4(2).5C23, 2- (1+4)⋊6C22, 2+ (1+4)⋊6C22, C4○D4⋊16D4, (C2×D4)⋊23D4, (C2×Q8)⋊19D4, C4≀C2⋊3C22, D4⋊4D4⋊4C2, D4.55(C2×D4), Q8.55(C2×D4), D4.8D4⋊4C2, C4⋊1D4⋊4C22, C8⋊C22⋊9C22, (C2×C4).13C24, C4○D4.8C23, C23.23(C2×D4), C4.58(C22×D4), C4.108C22≀C2, C2.C25⋊2C2, (C2×D4).37C23, C4.4D4⋊3C22, (C22×C4).112D4, (C2×Q8).29C23, C22.29C24⋊6C2, C42⋊C22⋊6C2, C22.23C22≀C2, C4.D4⋊10C22, (C22×D4)⋊20C22, C22.37(C22×D4), C42⋊C2⋊10C22, C4.10D4⋊10C22, (C2×M4(2))⋊11C22, (C22×C4).283C23, M4(2).8C22⋊2C2, (C2×C8⋊C22)⋊13C2, (C2×C4).461(C2×D4), C2.58(C2×C22≀C2), (C2×C4○D4).109C22, SmallGroup(128,1753)

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

Derived series
C1 — C2×C4 — C42.12C23
Chief series
C1 — C2 — C22 — C2×C4 — C22×C4 — C2×C4○D4 — C2.C25 — C42.12C23
Lower central
C1 — C2 — C2×C4 — C42.12C23
Upper central
C1 — C2 — C22×C4 — C42.12C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.12C23

Subgroups: 780 in 368 conjugacy classes, 106 normal (18 characteristic)
C1, C2, C2 [×11], C4 [×2], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×25], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×28], D4 [×4], D4 [×38], Q8 [×4], Q8 [×8], C23, C23 [×2], C23 [×12], C42 [×2], C22⋊C4 [×5], C4⋊C4, C2×C8 [×2], M4(2) [×4], M4(2) [×2], D8 [×8], SD16 [×8], C22×C4, C22×C4 [×2], C22×C4 [×6], C2×D4, C2×D4 [×6], C2×D4 [×29], C2×Q8, C2×Q8 [×2], C2×Q8 [×6], C4○D4 [×8], C4○D4 [×36], C24, C4.D4 [×2], C4.10D4 [×2], C4≀C2 [×8], C42⋊C2, C22≀C2 [×2], C4⋊D4 [×2], C4.4D4 [×2], C4⋊1D4 [×2], C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C8⋊C22 [×8], C8⋊C22 [×4], C22×D4, C2×C4○D4, C2×C4○D4 [×2], C2×C4○D4 [×6], 2+ (1+4) [×2], 2+ (1+4) [×4], 2- (1+4) [×2], 2- (1+4) [×2], M4(2).8C22, C42⋊C22 [×2], D4⋊4D4 [×4], D4.8D4 [×4], C22.29C24, C2×C8⋊C22 [×2], C2.C25, C42.12C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×12], C23 [×15], C2×D4 [×18], C24, C22≀C2 [×4], C22×D4 [×3], C2×C22≀C2, C42.12C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=c2=d2=1, e2=b2, cac=ab=ba, dad=a-1, eae-1=ab2, cbc=dbd=b-1, be=eb, dcd=b-1c, ce=ec, ede-1=b2d >

Permutation representations
►On 16 points - transitive group 16T269
Generators in S16
(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

(3 4)(5 6)(9 12)(10 11)(13 16)(14 15)

(1 6 2 5)(3 8 4 7)(9 15 11 13)(10 14 12 16)
 
G:=sub<Sym(16)| (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,9)(2,11)(3,12)(4,10)(5,13)(6,15)(7,14)(8,16), (3,4)(5,6)(9,12)(10,11)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,15,11,13)(10,14,12,16)>;
 
G:=Group( (5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,4,2,3)(5,8,6,7)(9,12,11,10)(13,14,15,16), (1,9)(2,11)(3,12)(4,10)(5,13)(6,15)(7,14)(8,16), (3,4)(5,6)(9,12)(10,11)(13,16)(14,15), (1,6,2,5)(3,8,4,7)(9,15,11,13)(10,14,12,16) );
 
G=PermutationGroup([(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,4,2,3),(5,8,6,7),(9,12,11,10),(13,14,15,16)], [(1,9),(2,11),(3,12),(4,10),(5,13),(6,15),(7,14),(8,16)], [(3,4),(5,6),(9,12),(10,11),(13,16),(14,15)], [(1,6,2,5),(3,8,4,7),(9,15,11,13),(10,14,12,16)])
 
G:=TransitiveGroup(16,269);
 

►On 16 points - transitive group 16T282
Generators in S16
(1 2)(3 4)(5 6)(7 8)(9 10 11 12)(13 14 15 16)

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

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

(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)

(1 6 7 3)(2 4 8 5)(9 14 11 16)(10 13 12 15)
 
G:=sub<Sym(16)| (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,7,6)(2,4,8,5)(9,14,11,16)(10,15,12,13), (1,9)(2,15)(3,16)(4,10)(5,12)(6,14)(7,11)(8,13), (3,6)(4,5)(9,14)(10,13)(11,16)(12,15), (1,6,7,3)(2,4,8,5)(9,14,11,16)(10,13,12,15)>;
 
G:=Group( (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16), (1,3,7,6)(2,4,8,5)(9,14,11,16)(10,15,12,13), (1,9)(2,15)(3,16)(4,10)(5,12)(6,14)(7,11)(8,13), (3,6)(4,5)(9,14)(10,13)(11,16)(12,15), (1,6,7,3)(2,4,8,5)(9,14,11,16)(10,13,12,15) );
 
G=PermutationGroup([(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16)], [(1,3,7,6),(2,4,8,5),(9,14,11,16),(10,15,12,13)], [(1,9),(2,15),(3,16),(4,10),(5,12),(6,14),(7,11),(8,13)], [(3,6),(4,5),(9,14),(10,13),(11,16),(12,15)], [(1,6,7,3),(2,4,8,5),(9,14,11,16),(10,13,12,15)])
 
G:=TransitiveGroup(16,282);
 

Matrix representation ►G ⊆ GL8(ℤ)

00-100000
00010000
10000000
0-1000000
00000001
00000010
00000100
00001000
,
01000000
-10000000
000-10000
00100000
00000-100
00001000
00000001
000000-10
,
00001000
00000100
00000010
00000001
10000000
01000000
00100000
00010000
,
10000000
0-1000000
00-100000
00010000
00000100
00001000
00000001
00000010
,
0-1000000
10000000
000-10000
00100000
00000-100
00001000
0000000-1
00000010

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

Character table of C42.12C23

 class 12A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L8A8B8C8D
 size 11222444444882222444444888888
ρ111111111111111111111111111111    trivial
ρ211-11-1-11-1-11-11-1-111-1111-11-1-111-1-11    linear of order 2
ρ311111111111-1-11111111111-1-1-1-1-1-1    linear of order 2
ρ411-11-1-11-1-11-1-11-111-1111-11-11-1-111-1    linear of order 2
ρ511111-11-1-1-11-1-11111-1-1-1-1111111-1-1    linear of order 2
ρ611-11-11111-1-1-11-111-1-1-1-111-1-111-11-1    linear of order 2
ρ711111-11-1-1-11111111-1-1-1-111-1-1-1-111    linear of order 2
ρ811-11-11111-1-11-1-111-1-1-1-111-11-1-11-11    linear of order 2
ρ911-11-1-1-11-1-11-11-111-1-1111-111-11-1-11    linear of order 2
ρ10111111-1-11-1-1-1-11111-111-1-1-1-1-11111    linear of order 2
ρ1111-11-1-1-11-1-111-1-111-1-1111-11-11-111-1    linear of order 2
ρ12111111-1-11-1-1111111-111-1-1-111-1-1-1-1    linear of order 2
ρ1311-11-11-1-11111-1-111-11-1-1-1-111-11-11-1    linear of order 2
ρ1411111-1-11-11-11111111-1-11-1-1-1-111-1-1    linear of order 2
ρ1511-11-11-1-1111-11-111-11-1-1-1-11-11-11-11    linear of order 2
ρ1611111-1-11-11-1-1-111111-1-11-1-111-1-111    linear of order 2
ρ1722-2-2202000200-2-2220000-2-2000000    orthogonal lifted from D4
ρ1822-2-220-2000-200-2-222000022000000    orthogonal lifted from D4
ρ19222-2-200-20-2000-22-22200200000000    orthogonal lifted from D4
ρ2022-2-2200-20200022-2-2-200200000000    orthogonal lifted from D4
ρ2122-22-2-200200002-2-220-22000000000    orthogonal lifted from D4
ρ2222222-20020000-2-2-2-202-2000000000    orthogonal lifted from D4
ρ23222-2-202000-2002-22-20000-22000000    orthogonal lifted from D4
ρ2422222200-20000-2-2-2-20-22000000000    orthogonal lifted from D4
ρ2522-22-2200-200002-2-2202-2000000000    orthogonal lifted from D4
ρ26222-2-20-20002002-22-200002-2000000    orthogonal lifted from D4
ρ2722-2-220020-200022-2-2200-200000000    orthogonal lifted from D4
ρ28222-2-200202000-22-22-200-200000000    orthogonal lifted from D4
ρ298-8000000000000000000000000000    orthogonal faithful

In GAP, Magma, Sage, TeX 

C_4^2._{12}C_2^3
 
% in TeX
 
G:=Group("C4^2.12C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1753);
 
// by ID
 
G=gap.SmallGroup(128,1753);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,2019,248,2804,1411,718,172,2028]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=1,e^2=b^2,c*a*c=a*b=b*a,d*a*d=a^-1,e*a*e^-1=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b^-1*c,c*e=e*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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