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G = C42.471C23  order 128 = 27

332nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.471C23, C4.672+ (1+4), (C4×D8)⋊42C2, C8⋊2D4⋊27C2, C8⋊D4⋊42C2, C8⋊6D4⋊10C2, C4⋊C8⋊38C22, (C4×C8)⋊37C22, C4⋊C4.161D4, D4.Q8⋊38C2, D4⋊D4⋊47C2, C22⋊D8⋊33C2, D4⋊5D4⋊11C2, (C2×D4).175D4, C2.51(D4○D8), (C4×D4)⋊28C22, C4.Q8⋊28C22, C2.D8⋊39C22, D4.28(C4○D4), D4.2D4⋊44C2, C4⋊C4.414C23, C4⋊D4⋊19C22, C22⋊C8⋊34C22, (C2×C4).514C24, (C2×C8).102C23, C22⋊C4.171D4, C23.331(C2×D4), D4⋊C4⋊96C22, Q8⋊C4⋊10C22, (C2×SD16)⋊34C22, (C2×D4).240C23, (C2×D8).140C22, C22.D8⋊29C2, (C2×Q8).225C23, C2.150(D4⋊5D4), C42.C2⋊11C22, C22⋊Q8.88C22, C23.37D4⋊16C2, C23.20D4⋊39C2, C23.36D4⋊21C2, (C2×M4(2))⋊30C22, (C22×C4).327C23, C4.4D4.70C22, C22.774(C22×D4), C2.90(D8⋊C22), C22.47C24⋊6C2, (C22×D4).415C22, C42.78C22⋊11C2, C42⋊C2.194C22, (C2×C4⋊C4)⋊61C22, C4.239(C2×C4○D4), (C2×C4).927(C2×D4), (C2×C4○D4).216C22, SmallGroup(128,2054)

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

Derived series
C1 — C2×C4 — C42.471C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C22×D4 — D4⋊5D4 — C42.471C23
Lower central
C1 — C2 — C2×C4 — C42.471C23
Upper central
C1 — C22 — C4×D4 — C42.471C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.471C23

Subgroups: 472 in 207 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×6], C4 [×2], C4 [×9], C22, C22 [×20], C8 [×4], C2×C4 [×5], C2×C4 [×14], D4 [×2], D4 [×14], Q8 [×2], C23 [×2], C23 [×8], C42, C42, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×4], C2×C8 [×4], M4(2) [×2], D8 [×3], SD16, C22×C4 [×2], C22×C4 [×4], C2×D4 [×4], C2×D4 [×8], C2×Q8, C4○D4 [×3], C24, C4×C8, C22⋊C8 [×2], D4⋊C4 [×7], Q8⋊C4 [×3], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4×D4 [×3], C4×D4, C22≀C2, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C42.C2, C42⋊2C2, C2×M4(2) [×2], C2×D8 [×2], C2×SD16, C22×D4, C2×C4○D4, C23.36D4, C23.37D4, C8⋊6D4, C4×D8, C22⋊D8, D4⋊D4, D4.2D4, C8⋊D4, C8⋊2D4, D4.Q8, C22.D8, C23.20D4, C42.78C22, D4⋊5D4, C22.47C24, C42.471C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C22×D4, C2×C4○D4, 2+ (1+4), D4⋊5D4, D8⋊C22, D4○D8, C42.471C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=a2, ab=ba, cac-1=a-1b2, dad=ab2, eae=a-1, cbc-1=dbd=b-1, be=eb, dcd=bc, ece=a2c, ede=b2d >

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

16150000
110000
0001300
004000
000004
0000130
,
100000
010000
000100
0016000
0000016
000010
,
400000
13130000
000010
000001
001000
000100
,
100000
010000
001000
0001600
0000016
0000160
,
16150000
010000
0001300
004000
0000013
000040

G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,4,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[4,13,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,15,1,0,0,0,0,0,0,0,4,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,13,0] >;
 

Character table of C42.471C23

 class 12A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M8A8B8C8D8E8F
 size 11114444882222444448888444488
ρ111111111111111111111111111111    trivial
ρ2111111-1-11-111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-1111-111111-11-11-1-1-111111-1-1    linear of order 2
ρ41111-1-1-1-11111111-11-11-1-11-1-1-1-1-111    linear of order 2
ρ511111-111-1-1-111-1-1-11111-11-1-111-1-11    linear of order 2
ρ611111-1-1-1-11-111-1-1-11111-1-111-1-111-1    linear of order 2
ρ71111-1111-11-111-1-111-11-11-1-1-111-11-1    linear of order 2
ρ81111-11-1-1-1-1-111-1-111-11-11111-1-11-11    linear of order 2
ρ91111-1-1-1-1-1111111-1-1-1-1111-11111-1-1    linear of order 2
ρ101111-1-111-1-111111-1-1-1-111-11-1-1-1-111    linear of order 2
ρ11111111-1-1-1-1111111-11-1-1-1-1-1111111    linear of order 2
ρ1211111111-11111111-11-1-1-111-1-1-1-1-1-1    linear of order 2
ρ131111-11-1-11-1-111-1-11-1-1-11-111-111-11-1    linear of order 2
ρ141111-111111-111-1-11-1-1-11-1-1-11-1-11-11    linear of order 2
ρ1511111-1-1-111-111-1-1-1-11-1-11-11-111-1-11    linear of order 2
ρ1611111-1111-1-111-1-1-1-11-1-111-11-1-111-1    linear of order 2
ρ1722222-200002-2-22-220-200000000000    orthogonal lifted from D4
ρ182222-2200002-2-22-2-20200000000000    orthogonal lifted from D4
ρ192222-2-20000-2-2-2-2220200000000000    orthogonal lifted from D4
ρ202222220000-2-2-2-22-20-200000000000    orthogonal lifted from D4
ρ212-22-2002-2000-220002i02i000002i2i000    complex lifted from C4○D4
ρ222-22-200-22000-220002i02i000002i2i000    complex lifted from C4○D4
ρ232-22-200-22000-220002i02i000002i2i000    complex lifted from C4○D4
ρ242-22-2002-2000-220002i02i000002i2i000    complex lifted from C4○D4
ρ254-44-400000004-40000000000000000    orthogonal lifted from 2+ (1+4)
ρ2644-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ2744-4-400000000000000000002√2002√200    orthogonal lifted from D4○D8
ρ284-4-440000004i004i000000000000000    complex lifted from D8⋊C22
ρ294-4-440000004i004i000000000000000    complex lifted from D8⋊C22

In GAP, Magma, Sage, TeX 

C_4^2._{471}C_2^3
 
% in TeX
 
G:=Group("C4^2.471C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2054);
 
// by ID
 
G=gap.SmallGroup(128,2054);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,2019,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=d^2=e^2=1,c^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a*b^2,e*a*e=a^-1,c*b*c^-1=d*b*d=b^-1,b*e=e*b,d*c*d=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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