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G = C42.516C23  order 128 = 27

377th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.516C23, C4.372- (1+4), (Q82)⋊5C2, C4⋊C4.181D4, (C4×Q16)⋊45C2, C8⋊4Q8⋊11C2, D4⋊Q8⋊45C2, C4.Q16⋊44C2, C2.65(D4○D8), C4⋊SD16.2C2, (C2×Q8).247D4, C4⋊C8.140C22, C4⋊C4.441C23, (C2×C8).121C23, (C2×C4).567C24, (C4×C8).234C22, Q8.39(C4○D4), Q8.D4⋊47C2, C4.4D8.10C2, C4⋊Q8.196C22, SD16⋊C4⋊47C2, C8⋊C4.66C22, C2.75(Q8⋊5D4), (C4×D4).205C22, (C2×D4).276C23, C4.81(C8.C22), (C2×Q8).406C23, (C4×Q8).198C22, C2.D8.208C22, D4⋊C4.90C22, C4⋊1D4.103C22, (C2×Q16).144C22, (C2×SD16).72C22, C4.4D4.83C22, C22.827(C22×D4), Q8⋊C4.211C22, C42.28C22⋊24C2, C22.53C24.5C2, C4.268(C2×C4○D4), (C2×C4).643(C2×D4), C2.89(C2×C8.C22), SmallGroup(128,2107)

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

Derived series
C1 — C2×C4 — C42.516C23
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — Q82 — C42.516C23
Lower central
C1 — C2 — C2×C4 — C42.516C23
Upper central
C1 — C22 — C4×Q8 — C42.516C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.516C23

Subgroups: 336 in 175 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×2], C4 [×12], C22, C22 [×6], C8 [×4], C2×C4 [×3], C2×C4 [×4], C2×C4 [×8], D4 [×7], Q8 [×2], Q8 [×8], C23 [×2], C42, C42 [×2], C42 [×4], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×7], C2×C8 [×2], C2×C8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×2], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×3], C4×C8, C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C4⋊C8 [×2], C2.D8, C2.D8 [×2], C4×D4 [×2], C4×D4, C4×Q8 [×3], C4×Q8 [×2], C4×Q8, C22.D4 [×2], C4.4D4 [×2], C4.4D4, C4⋊1D4, C4⋊Q8, C4⋊Q8 [×2], C4⋊Q8 [×3], C2×SD16 [×2], C2×Q16, C4×Q16, SD16⋊C4 [×2], C8⋊4Q8, C4⋊SD16, Q8.D4 [×2], D4⋊Q8 [×2], C4.Q16, C4.4D8, C42.28C22 [×2], Q82, C22.53C24, C42.516C23

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C8.C22 [×2], C22×D4, C2×C4○D4, 2- (1+4), Q8⋊5D4, C2×C8.C22, D4○D8, C42.516C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2c, ece-1=bc, ede-1=b2d >

Smallest permutation representation
►On 64 points
Generators in S64
(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)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 23 25 17)(2 24 26 18)(3 21 27 19)(4 22 28 20)(5 15 9 64)(6 16 10 61)(7 13 11 62)(8 14 12 63)(29 37 41 35)(30 38 42 36)(31 39 43 33)(32 40 44 34)(45 53 51 57)(46 54 52 58)(47 55 49 59)(48 56 50 60)

(1 55 25 59)(2 56 26 60)(3 53 27 57)(4 54 28 58)(5 36 9 38)(6 33 10 39)(7 34 11 40)(8 35 12 37)(13 44 62 32)(14 41 63 29)(15 42 64 30)(16 43 61 31)(17 49 23 47)(18 50 24 48)(19 51 21 45)(20 52 22 46)

(1 57 27 55)(2 60 28 54)(3 59 25 53)(4 58 26 56)(5 44 11 30)(6 43 12 29)(7 42 9 32)(8 41 10 31)(13 36 64 40)(14 35 61 39)(15 34 62 38)(16 33 63 37)(17 51 21 47)(18 50 22 46)(19 49 23 45)(20 52 24 48)

(1 41 25 29)(2 42 26 30)(3 43 27 31)(4 44 28 32)(5 56 9 60)(6 53 10 57)(7 54 11 58)(8 55 12 59)(13 46 62 52)(14 47 63 49)(15 48 64 50)(16 45 61 51)(17 35 23 37)(18 36 24 38)(19 33 21 39)(20 34 22 40)
 
G:=sub<Sym(64)| (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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,25,59)(2,56,26,60)(3,53,27,57)(4,54,28,58)(5,36,9,38)(6,33,10,39)(7,34,11,40)(8,35,12,37)(13,44,62,32)(14,41,63,29)(15,42,64,30)(16,43,61,31)(17,49,23,47)(18,50,24,48)(19,51,21,45)(20,52,22,46), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40)>;
 
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)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,23,25,17)(2,24,26,18)(3,21,27,19)(4,22,28,20)(5,15,9,64)(6,16,10,61)(7,13,11,62)(8,14,12,63)(29,37,41,35)(30,38,42,36)(31,39,43,33)(32,40,44,34)(45,53,51,57)(46,54,52,58)(47,55,49,59)(48,56,50,60), (1,55,25,59)(2,56,26,60)(3,53,27,57)(4,54,28,58)(5,36,9,38)(6,33,10,39)(7,34,11,40)(8,35,12,37)(13,44,62,32)(14,41,63,29)(15,42,64,30)(16,43,61,31)(17,49,23,47)(18,50,24,48)(19,51,21,45)(20,52,22,46), (1,57,27,55)(2,60,28,54)(3,59,25,53)(4,58,26,56)(5,44,11,30)(6,43,12,29)(7,42,9,32)(8,41,10,31)(13,36,64,40)(14,35,61,39)(15,34,62,38)(16,33,63,37)(17,51,21,47)(18,50,22,46)(19,49,23,45)(20,52,24,48), (1,41,25,29)(2,42,26,30)(3,43,27,31)(4,44,28,32)(5,56,9,60)(6,53,10,57)(7,54,11,58)(8,55,12,59)(13,46,62,52)(14,47,63,49)(15,48,64,50)(16,45,61,51)(17,35,23,37)(18,36,24,38)(19,33,21,39)(20,34,22,40) );
 
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),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,23,25,17),(2,24,26,18),(3,21,27,19),(4,22,28,20),(5,15,9,64),(6,16,10,61),(7,13,11,62),(8,14,12,63),(29,37,41,35),(30,38,42,36),(31,39,43,33),(32,40,44,34),(45,53,51,57),(46,54,52,58),(47,55,49,59),(48,56,50,60)], [(1,55,25,59),(2,56,26,60),(3,53,27,57),(4,54,28,58),(5,36,9,38),(6,33,10,39),(7,34,11,40),(8,35,12,37),(13,44,62,32),(14,41,63,29),(15,42,64,30),(16,43,61,31),(17,49,23,47),(18,50,24,48),(19,51,21,45),(20,52,22,46)], [(1,57,27,55),(2,60,28,54),(3,59,25,53),(4,58,26,56),(5,44,11,30),(6,43,12,29),(7,42,9,32),(8,41,10,31),(13,36,64,40),(14,35,61,39),(15,34,62,38),(16,33,63,37),(17,51,21,47),(18,50,22,46),(19,49,23,45),(20,52,24,48)], [(1,41,25,29),(2,42,26,30),(3,43,27,31),(4,44,28,32),(5,56,9,60),(6,53,10,57),(7,54,11,58),(8,55,12,59),(13,46,62,52),(14,47,63,49),(15,48,64,50),(16,45,61,51),(17,35,23,37),(18,36,24,38),(19,33,21,39),(20,34,22,40)])
 

Matrix representation ►G ⊆ GL6(𝔽17)

16150000
110000
001000
000100
000010
000001
,
100000
010000
000100
0016000
0000016
000010
,
480000
13130000
0050152
000121515
00151505
0021550
,
480000
0130000
00120215
000121515
0021550
00151505
,
1600000
0160000
000010
000001
0016000
0001600

G:=sub<GL(6,GF(17))| [16,1,0,0,0,0,15,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[4,13,0,0,0,0,8,13,0,0,0,0,0,0,5,0,15,2,0,0,0,12,15,15,0,0,15,15,0,5,0,0,2,15,5,0],[4,0,0,0,0,0,8,13,0,0,0,0,0,0,12,0,2,15,0,0,0,12,15,15,0,0,2,15,5,0,0,0,15,15,0,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;
 

Character table of C42.516C23

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D8E8F
 size 11118822224444444448888444488
ρ111111111111111111111111111111    trivial
ρ211111-1-11-11-11-11-11-11-111-1-11-11-1-11    linear of order 2
ρ31111111111-1-111-1-1-11-1-1-1-11-1-1-1-111    linear of order 2
ρ411111-1-11-111-1-111-1111-1-11-1-11-11-11    linear of order 2
ρ51111-1-11111-1-11-1-1-11-11111-1-1-1-1-111    linear of order 2
ρ61111-11-11-111-1-1-11-1-1-1-111-11-11-11-11    linear of order 2
ρ71111-1-11111111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-11-11-11-11-1-1-111-11-1-1111-11-1-11    linear of order 2
ρ91111-11-11-111-1-111-1-11-1-111-11-11-11-1    linear of order 2
ρ101111-1-11111-1-111-1-1111-11-111111-1-1    linear of order 2
ρ111111-11-11-11-11-11-111111-1-1-1-11-111-1    linear of order 2
ρ121111-1-11111111111-11-11-111-1-1-1-1-1-1    linear of order 2
ρ1311111-1-11-11-11-1-1-11-1-1-1-1111-11-111-1    linear of order 2
ρ141111111111111-1111-11-11-1-1-1-1-1-1-1-1    linear of order 2
ρ1511111-1-11-111-1-1-11-11-111-1-111-11-11-1    linear of order 2
ρ161111111111-1-11-1-1-1-1-1-11-11-11111-1-1    linear of order 2
ρ17222200-2-2-2-2-22202-20000000000000    orthogonal lifted from D4
ρ182222002-22-2-2-2-20220000000000000    orthogonal lifted from D4
ρ19222200-2-2-2-22-220-220000000000000    orthogonal lifted from D4
ρ202222002-22-222-20-2-20000000000000    orthogonal lifted from D4
ρ212-22-200020-2000-2002i22i000002i02i00    complex lifted from C4○D4
ρ222-22-200020-20002002i-22i000002i02i00    complex lifted from C4○D4
ρ232-22-200020-2000-2002i22i000002i02i00    complex lifted from C4○D4
ρ242-22-200020-20002002i-22i000002i02i00    complex lifted from C4○D4
ρ2544-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8
ρ2644-4-400000000000000000002√202√2000    orthogonal lifted from D4○D8
ρ274-44-4000-4040000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ284-4-440040-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ294-4-4400-40400000000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{516}C_2^3
 
% in TeX
 
G:=Group("C4^2.516C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2107);
 
// by ID
 
G=gap.SmallGroup(128,2107);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,568,758,723,352,346,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=e^2=b^2,d^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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