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G = C42.56C23order 128 = 27

56th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.56C23, C4.662+ 1+4, C88D454C2, C82D426C2, C89D423C2, C4⋊C4.370D4, D4⋊D446C2, Q8⋊D422C2, Q85D410C2, Q8⋊Q818C2, (C2×D4).174D4, D4.2D443C2, C4⋊C8.107C22, C4⋊C4.413C23, (C2×C8).355C23, (C2×C4).513C24, Q8.27(C4○D4), C22⋊C4.170D4, (C2×D8).86C22, C23.330(C2×D4), C4⋊Q8.154C22, SD16⋊C438C2, C4.Q8.59C22, C8⋊C4.48C22, C2.79(D4○SD16), (C4×D4).164C22, (C2×D4).239C23, C4⋊D4.88C22, C22⋊C8.85C22, (C4×Q8).162C22, (C2×Q8).224C23, C2.149(D45D4), C2.D8.122C22, C22⋊Q8.87C22, D4⋊C4.75C22, C23.20D438C2, C23.38D415C2, C23.19D437C2, C23.24D433C2, (C22×C8).366C22, (C2×SD16).59C22, C4.4D4.69C22, C22.773(C22×D4), C2.89(D8⋊C22), C22.49C246C2, (C22×C4).1157C23, Q8⋊C4.182C22, (C22×Q8).346C22, C42⋊C2.193C22, C42.28C2219C2, (C2×M4(2)).119C22, C4.238(C2×C4○D4), (C2×C4).610(C2×D4), (C2×C4○D4).215C22, SmallGroup(128,2053)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.56C23
C1C2C4C2×C4C22×C4C2×C4○D4Q85D4 — C42.56C23
C1C2C2×C4 — C42.56C23
C1C22C4×D4 — C42.56C23
C1C2C2C2×C4 — C42.56C23

Generators and relations for C42.56C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2b2, d2=e2=b2, ab=ba, cac-1=eae-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, ede-1=b2d >

Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2, C2, C4, C4, C22, C22, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), D8, SD16, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C8⋊C4, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊C8, C4.Q8, C2.D8, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C4⋊D4, C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16, C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C89D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C88D4, C82D4, Q8⋊Q8, C23.19D4, C23.20D4, C42.28C22, Q85D4, C22.49C24, C42.56C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, D45D4, D8⋊C22, D4○SD16, C42.56C23

Character table of C42.56C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-111-1-11-11-11-1111-11-111-11-1    linear of order 2
ρ31111111-11111111-11-11-1-111-1-1-1-1-1-1    linear of order 2
ρ41111-1-1-1111-1-11-111111-1-1-111-1-11-11    linear of order 2
ρ51111-111-111-1-1-11-11-11-1-11-11-111-1-11    linear of order 2
ρ611111-1-111111-1-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ71111-111111-1-1-11-1-1-1-1-11-1-111-1-111-1    linear of order 2
ρ811111-1-1-11111-1-1-11-11-11-111-1-1-1-111    linear of order 2
ρ91111-1-11-111-1-11-1-1-11-1-1111-11-1-11-11    linear of order 2
ρ10111111-11111111-1111-111-1-1-1-1-1-1-1-1    linear of order 2
ρ111111-1-11111-1-11-1-1111-1-1-11-1-111-11-1    linear of order 2
ρ12111111-1-1111111-1-11-1-1-1-1-1-1111111    linear of order 2
ρ1311111-1111111-1-11-1-1-11-11-1-1-1-1-1-111    linear of order 2
ρ141111-11-1-111-1-1-1111-111-111-11-1-111-1    linear of order 2
ρ1511111-11-11111-1-111-1111-1-1-11111-1-1    linear of order 2
ρ161111-11-1111-1-1-111-1-1-111-11-1-111-1-11    linear of order 2
ρ1722222-200-2-2-2-2-22002000000000000    orthogonal lifted from D4
ρ182222-2-200-2-2222200-2000000000000    orthogonal lifted from D4
ρ1922222200-2-2-2-22-200-2000000000000    orthogonal lifted from D4
ρ202222-2200-2-222-2-2002000000000000    orthogonal lifted from D4
ρ212-22-200002-20000-22i0-2i200002i00-2i00    complex lifted from C4○D4
ρ222-22-200002-2000022i0-2i-20000-2i002i00    complex lifted from C4○D4
ρ232-22-200002-200002-2i02i-200002i00-2i00    complex lifted from C4○D4
ρ242-22-200002-20000-2-2i02i20000-2i002i00    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-440000004i-4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-44000000-4i4i00000000000000000    complex lifted from D8⋊C22
ρ2844-4-400000000000000000000-2-22-2000    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002-2-2-2000    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.56C23 in GL6(𝔽17)

110000
15160000
0010010
0001001
001070
000107
,
100000
010000
000100
0016000
000001
0000160
,
400000
9130000
0000125
000055
0051200
00121200
,
1600000
0160000
0000160
000001
001000
0001600
,
16160000
010000
00160100
00016010
0010010
0001001

G:=sub<GL(6,GF(17))| [1,15,0,0,0,0,1,16,0,0,0,0,0,0,10,0,1,0,0,0,0,10,0,1,0,0,1,0,7,0,0,0,0,1,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0],[4,9,0,0,0,0,0,13,0,0,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,12,5,0,0,0,0,5,5,0,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,16,0,0,0,0,0,0,1,0,0],[16,0,0,0,0,0,16,1,0,0,0,0,0,0,16,0,10,0,0,0,0,16,0,10,0,0,10,0,1,0,0,0,0,10,0,1] >;

C42.56C23 in GAP, Magma, Sage, TeX

C_4^2._{56}C_2^3
% in TeX

G:=Group("C4^2.56C2^3");
// GroupNames label

G:=SmallGroup(128,2053);
// by ID

G=gap.SmallGroup(128,2053);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,352,346,248,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.56C23 in TeX

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