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G = C42.406C23  order 128 = 27

267th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.406C23, C4.1122+ (1+4), C4⋊D8⋊27C2, C8⋊4D4⋊10C2, C8⋊3D4⋊13C2, C4⋊C8⋊20C22, C4⋊C4.129D4, (C4×C8)⋊10C22, (C2×D8)⋊6C22, D4⋊D4⋊28C2, C22⋊D8⋊23C2, C2.30(D4○D8), (C4×D4)⋊15C22, C4⋊1D4⋊8C22, C22⋊C4.21D4, C8⋊C4⋊11C22, D4.2D4⋊25C2, C4⋊C4.159C23, (C2×C4).418C24, (C2×C8).161C23, Q8⋊C4⋊6C22, C23.290(C2×D4), C42.C2⋊5C22, D4⋊C4⋊32C22, (C2×SD16)⋊24C22, (C2×D4).167C23, C4⋊D4.43C22, C22⋊C8.53C22, (C2×Q8).155C23, C22.29C24⋊16C2, (C22×C4).306C23, C4.4D4.38C22, C22.678(C22×D4), C22.34C24⋊5C2, C42.7C22⋊10C2, C42.29C22⋊3C2, C42.78C22⋊1C2, (C22×D4).392C22, C42⋊C2.157C22, C2.89(C22.29C24), (C2×C4).547(C2×D4), (C2×C4○D4).177C22, SmallGroup(128,1952)

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

Derived series
C1 — C2×C4 — C42.406C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — C22.29C24 — C42.406C23
Lower central
C1 — C2 — C2×C4 — C42.406C23
Upper central
C1 — C22 — C42⋊C2 — C42.406C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.406C23

Subgroups: 540 in 210 conjugacy classes, 84 normal (32 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×2], C4 [×8], C22, C22 [×22], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×9], D4 [×22], Q8 [×2], C23, C23 [×9], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×2], C2×C8 [×4], D8 [×6], SD16 [×2], C22×C4, C22×C4 [×3], C2×D4 [×3], C2×D4 [×2], C2×D4 [×12], C2×Q8, C4○D4 [×2], C24, C4×C8, C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4⋊C8 [×2], C42⋊C2, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×2], C4⋊D4 [×4], C22.D4 [×2], C4.4D4 [×2], C42.C2, C4⋊1D4 [×3], C2×D8 [×6], C2×SD16 [×2], C22×D4, C2×C4○D4, C42.7C22, C22⋊D8 [×2], D4⋊D4 [×2], C4⋊D8 [×2], D4.2D4 [×2], C42.78C22, C42.29C22, C8⋊4D4, C8⋊3D4, C22.29C24, C22.34C24, C42.406C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ (1+4) [×2], C22.29C24, D4○D8 [×2], C42.406C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=c2=d2=e2=1, ab=ba, cac=dad=a-1, eae=ab2, cbc=dbd=b-1, be=eb, dcd=bc, ece=a2c, de=ed >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

00100000
00010000
160000000
016000000
00000010
00000001
00001000
00000100
,
115000000
116000000
001150000
001160000
00000100
000016000
00000001
000000160
,
000110000
001400000
011000000
140000000
0000141400
000014300
0000001414
000000143
,
115000000
016000000
001620000
00010000
00000100
00001000
00000001
00000010
,
10000000
01000000
001600000
000160000
00001000
00000100
000000160
000000016

G:=sub<GL(8,GF(17))| [0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,0,14,0,0,0,0,0,0,11,0,0,0,0,0,0,14,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3,0,0,0,0,0,0,0,0,14,14,0,0,0,0,0,0,14,3],[1,0,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;
 

Character table of C42.406C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J8A8B8C8D8E8F
 size 11114888882244444888444488
ρ111111111111111111111111111    trivial
ρ21111-1-11-1-1111-1-111-1-111-11-111-1    linear of order 2
ρ31111111-1-1111-1-1-1-111-1-1-11-11-11    linear of order 2
ρ41111-1-111111111-1-1-1-1-1-11111-1-1    linear of order 2
ρ51111-1-1-1-1111111-1-1-11-11-1-1-1-111    linear of order 2
ρ6111111-11-1111-1-1-1-11-1-111-11-11-1    linear of order 2
ρ71111-1-1-11-1111-1-111-111-11-11-1-11    linear of order 2
ρ8111111-1-1111111111-11-1-1-1-1-1-1-1    linear of order 2
ρ91111-111-11-111-1-111-1-1-111-11-1-11    linear of order 2
ρ1011111-111-1-111111111-11-1-1-1-1-1-1    linear of order 2
ρ111111-1111-1-11111-1-1-1-11-1-1-1-1-111    linear of order 2
ρ1211111-11-11-111-1-1-1-1111-11-11-11-1    linear of order 2
ρ1311111-1-111-111-1-1-1-11-111-11-11-11    linear of order 2
ρ141111-11-1-1-1-11111-1-1-11111111-1-1    linear of order 2
ρ1511111-1-1-1-1-11111111-1-1-1111111    linear of order 2
ρ161111-11-111-111-1-111-11-1-1-11-111-1    linear of order 2
ρ172222200000-2-22-2-22-2000000000    orthogonal lifted from D4
ρ182222200000-2-2-222-2-2000000000    orthogonal lifted from D4
ρ192222-200000-2-22-22-22000000000    orthogonal lifted from D4
ρ202222-200000-2-2-22-222000000000    orthogonal lifted from D4
ρ214-44-40000004-400000000000000    orthogonal lifted from 2+ (1+4)
ρ224-44-4000000-4400000000000000    orthogonal lifted from 2+ (1+4)
ρ2344-4-400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ2444-4-400000000000000002√202√2000    orthogonal lifted from D4○D8
ρ254-4-44000000000000000002√202√200    orthogonal lifted from D4○D8
ρ264-4-44000000000000000002√202√200    orthogonal lifted from D4○D8

In GAP, Magma, Sage, TeX 

C_4^2._{406}C_2^3
 
% in TeX
 
G:=Group("C4^2.406C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,1952);
 
// by ID
 
G=gap.SmallGroup(128,1952);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,675,1018,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=c^2=d^2=e^2=1,a*b=b*a,c*a*c=d*a*d=a^-1,e*a*e=a*b^2,c*b*c=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*c,d*e=e*d>;
 
// generators/relations
 

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