Copied to
clipboard

G = C42.518C23order 128 = 27

379th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.518C23, C4.392- 1+4, C4⋊C4.183D4, Q83Q88C2, (C4×Q16)⋊46C2, C84Q813C2, Q8.Q850C2, D4.Q8.2C2, C4.Q1645C2, C42Q1643C2, (C2×Q8).139D4, C2.65(Q8○D8), Q16⋊C429C2, C4⋊C4.266C23, C4⋊C8.142C22, (C2×C8).209C23, (C4×C8).235C22, (C2×C4).569C24, Q8.40(C4○D4), C4⋊Q8.198C22, Q8.D4.3C2, C8⋊C4.68C22, C4.Q8.76C22, C2.77(Q85D4), SD16⋊C4.2C2, (C4×D4).207C22, (C2×D4).277C23, (C4×Q8).200C22, (C2×Q8).262C23, (C2×Q16).93C22, C2.D8.209C22, D4⋊C4.92C22, (C2×SD16).74C22, C4.4D4.85C22, C22.829(C22×D4), C42.C2.70C22, Q8⋊C4.213C22, C2.104(D8⋊C22), C42.30C2214C2, C42.28C22.2C2, C42.78C22.3C2, C22.50C24.10C2, C4.270(C2×C4○D4), (C2×C4).645(C2×D4), SmallGroup(128,2109)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.518C23
C1C2C4C2×C4C42C4×Q8Q83Q8 — C42.518C23
C1C2C2×C4 — C42.518C23
C1C22C4×Q8 — C42.518C23
C1C2C2C2×C4 — C42.518C23

Generators and relations for C42.518C23
 G = < a,b,c,d,e | a4=b4=1, c2=e2=b2, d2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc-1=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=b2d >

Subgroups: 280 in 166 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×14], C22, C22 [×3], C8 [×4], C2×C4 [×7], C2×C4 [×9], D4 [×2], Q8 [×2], Q8 [×8], C23, C42 [×3], C42 [×5], C22⋊C4 [×5], C4⋊C4 [×7], C4⋊C4 [×12], C2×C8 [×4], SD16 [×2], Q16 [×4], C22×C4, C2×D4, C2×Q8 [×4], C2×Q8, C4×C8, C8⋊C4 [×2], D4⋊C4 [×3], Q8⋊C4 [×7], C4⋊C8 [×3], C4.Q8, C2.D8 [×2], C42⋊C2, C4×D4, C4×Q8 [×6], C4×Q8, C22⋊Q8, C4.4D4 [×2], C42.C2 [×2], C42.C2 [×2], C422C2 [×2], C4⋊Q8 [×2], C4⋊Q8, C2×SD16, C2×Q16 [×2], C4×Q16, SD16⋊C4, Q16⋊C4, C84Q8, C42Q16, Q8.D4 [×2], C4.Q16, D4.Q8, Q8.Q8, C42.78C22, C42.28C22, C42.30C22, C22.50C24, Q83Q8, C42.518C23
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, Q85D4, D8⋊C22, Q8○D8, C42.518C23

Character table of C42.518C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R8A8B8C8D8E8F
 size 11118222244444444488888444488
ρ111111111111111111111111111111    trivial
ρ2111111111111-1111-11-11-11-1-1-1-1-1-1-1    linear of order 2
ρ311111-11-11-1-1-11-11-111-1-111-11-11-11-1    linear of order 2
ρ411111-11-11-1-1-1-1-11-1-111-1-111-11-11-11    linear of order 2
ρ511111-11-1111-1-11-11-1-11-11-1-11-11-1-11    linear of order 2
ρ611111-11-1111-111-111-1-1-1-1-11-11-111-1    linear of order 2
ρ7111111111-1-11-1-1-1-1-1-1-111-111111-1-1    linear of order 2
ρ8111111111-1-111-1-1-11-111-1-1-1-1-1-1-111    linear of order 2
ρ91111-1-11-11-11-1-1-111-1111-1-111-11-11-1    linear of order 2
ρ101111-1-11-11-11-11-11111-111-1-1-11-11-11    linear of order 2
ρ111111-111111-11-111-1-11-1-1-1-1-1111111    linear of order 2
ρ121111-111111-11111-1111-11-11-1-1-1-1-1-1    linear of order 2
ρ131111-11111-1111-1-111-11-1-11-11111-1-1    linear of order 2
ρ141111-11111-111-1-1-11-1-1-1-1111-1-1-1-111    linear of order 2
ρ151111-1-11-111-1-111-1-11-1-11-1111-11-1-11    linear of order 2
ρ161111-1-11-111-1-1-11-1-1-1-11111-1-11-111-1    linear of order 2
ρ17222202-22-2-20-202200-200000000000    orthogonal lifted from D4
ρ1822220-2-2-2-2-20202-200200000000000    orthogonal lifted from D4
ρ19222202-22-220-20-2-200200000000000    orthogonal lifted from D4
ρ2022220-2-2-2-22020-2200-200000000000    orthogonal lifted from D4
ρ212-22-20020-202i0200-2i-200000002i0-2i00    complex lifted from C4○D4
ρ222-22-20020-20-2i02002i-20000000-2i02i00    complex lifted from C4○D4
ρ232-22-20020-20-2i0-2002i200000002i0-2i00    complex lifted from C4○D4
ρ242-22-20020-202i0-200-2i20000000-2i02i00    complex lifted from C4○D4
ρ254-44-400-40400000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-40000000000000000000220-22000    symplectic lifted from Q8○D8, Schur index 2
ρ2744-4-40000000000000000000-22022000    symplectic lifted from Q8○D8, Schur index 2
ρ284-4-440-4i04i000000000000000000000    complex lifted from D8⋊C22
ρ294-4-4404i0-4i000000000000000000000    complex lifted from D8⋊C22

Smallest permutation representation of C42.518C23
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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 63)(6 9 16 64)(7 10 13 61)(8 11 14 62)(29 36 37 41)(30 33 38 42)(31 34 39 43)(32 35 40 44)(45 51 56 60)(46 52 53 57)(47 49 54 58)(48 50 55 59)
(1 55 20 48)(2 56 17 45)(3 53 18 46)(4 54 19 47)(5 39 15 31)(6 40 16 32)(7 37 13 29)(8 38 14 30)(9 35 64 44)(10 36 61 41)(11 33 62 42)(12 34 63 43)(21 57 26 52)(22 58 27 49)(23 59 28 50)(24 60 25 51)
(1 56 18 47)(2 48 19 53)(3 54 20 45)(4 46 17 55)(5 35 13 42)(6 43 14 36)(7 33 15 44)(8 41 16 34)(9 31 62 37)(10 38 63 32)(11 29 64 39)(12 40 61 30)(21 49 28 60)(22 57 25 50)(23 51 26 58)(24 59 27 52)
(1 40 20 32)(2 37 17 29)(3 38 18 30)(4 39 19 31)(5 49 15 58)(6 50 16 59)(7 51 13 60)(8 52 14 57)(9 48 64 55)(10 45 61 56)(11 46 62 53)(12 47 63 54)(21 42 26 33)(22 43 27 34)(23 44 28 35)(24 41 25 36)

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

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

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

Matrix representation of C42.518C23 in GL8(𝔽17)

1615000000
11000000
0016150000
00110000
00000010
00000001
00001000
00000100
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
6141500000
1090150000
1131130000
78780000
00000550
000050012
00005005
000001250
,
82240000
1090150000
006150000
0010110000
00005005
000005120
0000012120
000050012
,
1601500000
0160150000
00100000
00010000
00002151515
00001515152
00001515215
00001521515

G:=sub<GL(8,GF(17))| [16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[6,10,11,7,0,0,0,0,14,9,3,8,0,0,0,0,15,0,11,7,0,0,0,0,0,15,3,8,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,5,0,0,12,0,0,0,0,5,0,0,5,0,0,0,0,0,12,5,0],[8,10,0,0,0,0,0,0,2,9,0,0,0,0,0,0,2,0,6,10,0,0,0,0,4,15,15,11,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,5,0,0,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,15,0,1,0,0,0,0,0,0,0,0,2,15,15,15,0,0,0,0,15,15,15,2,0,0,0,0,15,15,2,15,0,0,0,0,15,2,15,15] >;

C42.518C23 in GAP, Magma, Sage, TeX

C_4^2._{518}C_2^3
% in TeX

G:=Group("C4^2.518C2^3");
// GroupNames label

G:=SmallGroup(128,2109);
// by ID

G=gap.SmallGroup(128,2109);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,723,352,346,304,4037,1027,124]);
// 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*b^2,a*e=e*a,c*b*c^-1=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations

Export

Character table of C42.518C23 in TeX

׿
×
𝔽