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G = C42.51C23order 128 = 27

51st non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.51C23, C4.612+ 1+4, C89D418C2, C88D451C2, C4⋊C4.158D4, D43Q84C2, Q8⋊Q817C2, (C2×D4).318D4, C8.D426C2, (C2×C8).99C23, Q85D4.4C2, Q16⋊C423C2, D4.7D446C2, C4⋊C8.104C22, C4⋊C4.236C23, (C2×C4).508C24, Q8.24(C4○D4), Q8.D443C2, C22⋊Q1632C2, C22⋊C4.168D4, C23.476(C2×D4), C4⋊Q8.152C22, C8⋊C4.45C22, C4.Q8.58C22, C2.76(D4○SD16), (C2×D4).234C23, (C4×D4).161C22, C4⋊D4.85C22, C22⋊C8.82C22, (C2×Q8).221C23, (C4×Q8).159C22, (C2×Q16).85C22, C2.144(D45D4), C22⋊Q8.83C22, D4⋊C4.73C22, C23.47D417C2, C23.46D415C2, C23.36D419C2, (C22×C8).363C22, Q8⋊C4.72C22, C4.4D4.67C22, C22.768(C22×D4), C22.7(C8.C22), (C22×C4).1152C23, (C2×SD16).100C22, (C22×Q8).344C22, C42.28C2217C2, (C2×M4(2)).115C22, C4.233(C2×C4○D4), (C2×C4).605(C2×D4), (C2×Q8⋊C4)⋊43C2, C2.76(C2×C8.C22), (C2×C4⋊C4).669C22, (C2×C4○D4).212C22, SmallGroup(128,2048)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.51C23
C1C2C4C2×C4C22×C4C22×Q8Q85D4 — C42.51C23
C1C2C2×C4 — C42.51C23
C1C22C4×D4 — C42.51C23
C1C2C2C2×C4 — C42.51C23

Generators and relations for C42.51C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=a2b2, d2=b2, ab=ba, cac-1=eae=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, ede=b2d >

Subgroups: 376 in 194 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×5], C2×C4 [×17], D4 [×8], Q8 [×2], Q8 [×8], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×5], C4⋊C4 [×8], C2×C8 [×4], C2×C8, M4(2), SD16, Q16 [×3], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×3], C2×Q8 [×5], C4○D4 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×7], C4⋊C8, C4.Q8 [×3], C2×C4⋊C4 [×2], C4×D4, C4×D4 [×2], C4×Q8 [×2], C4⋊D4, C4⋊D4, C22⋊Q8 [×3], C22⋊Q8 [×3], C4.4D4, C4.4D4, C42.C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×Q16 [×2], C22×Q8, C2×C4○D4, C2×Q8⋊C4, C23.36D4, C89D4, Q16⋊C4, C22⋊Q16, D4.7D4, Q8.D4, C88D4, C8.D4, Q8⋊Q8, C23.46D4, C23.47D4, C42.28C22, Q85D4, D43Q8, C42.51C23
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, D45D4, C2×C8.C22, D4○SD16, C42.51C23

Character table of C42.51C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11112248224444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-111111-1-111-111-1-11-1-1-111-11-1    linear of order 2
ρ31111-1-11-1111-1-1-11-11-11-11-111-1-11-11    linear of order 2
ρ41111111-111111-1111-1-1111-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-11111-1-1-111-1-11-11-11-11-1-11-11    linear of order 2
ρ61111111111-111111-111-1-1-11-1-1-1-1-1-1    linear of order 2
ρ71111111-111-111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111-1-11-111-1-1-1-11-1-1-111-111-111-11-1    linear of order 2
ρ91111-1-1-111111-1-1-111-11-1-11-11-1-111-1    linear of order 2
ρ10111111-11111-11-1-1-11-1-11-1-11-1-1-1-111    linear of order 2
ρ11111111-1-1111-111-1-11111-1-1-11111-1-1    linear of order 2
ρ121111-1-1-1-11111-11-1111-1-1-111-111-1-11    linear of order 2
ρ13111111-1111-1-11-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ141111-1-1-1111-11-1-1-11-1-1111-1-1-111-1-11    linear of order 2
ρ151111-1-1-1-111-11-11-11-11-111-111-1-111-1    linear of order 2
ρ16111111-1-111-1-111-1-1-111-111-1-1-1-1-111    linear of order 2
ρ172222-2-2-20-2-202202-20000000000000    orthogonal lifted from D4
ρ1822222220-2-202-20-2-20000000000000    orthogonal lifted from D4
ρ19222222-20-2-20-2-20220000000000000    orthogonal lifted from D4
ρ202222-2-220-2-20-220-220000000000000    orthogonal lifted from D4
ρ212-22-200002-2-2i002002i-2000000-2i2i000    complex lifted from C4○D4
ρ222-22-200002-22i00-200-2i2000000-2i2i000    complex lifted from C4○D4
ρ232-22-200002-22i00200-2i-20000002i-2i000    complex lifted from C4○D4
ρ242-22-200002-2-2i00-2002i20000002i-2i000    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-444-400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ274-4-44-4400000000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2844-4-40000000000000000000-2-2002-200    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-200-2-200    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.51C23 in GL8(𝔽17)

1611620000
00100000
016000000
1601610000
00000010
000000016
00001000
000001600
,
160000000
016000000
001600000
000160000
00000100
000016000
000000016
00000010
,
00100000
1161150000
160000000
161010000
0000152152
0000221515
000015151515
0000215152
,
00100000
1161150000
10000000
00010000
00000010
00000001
000016000
000001600
,
00100000
1611620000
10000000
101160000
00000010
000000016
00001000
000001600

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

C42.51C23 in GAP, Magma, Sage, TeX

C_4^2._{51}C_2^3
% in TeX

G:=Group("C4^2.51C2^3");
// GroupNames label

G:=SmallGroup(128,2048);
// by ID

G=gap.SmallGroup(128,2048);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,352,346,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2*b^2,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,b*e=e*b,d*c*d^-1=b*c,e*c*e=a^2*c,e*d*e=b^2*d>;
// generators/relations

Export

Character table of C42.51C23 in TeX

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