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G = C42.56C23  order 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), C8⋊8D4⋊54C2, C8⋊2D4⋊26C2, C8⋊9D4⋊23C2, C4⋊C4.370D4, D4⋊D4⋊46C2, Q8⋊5D4⋊10C2, Q8⋊Q8⋊18C2, Q8⋊D4⋊22C2, (C2×D4).174D4, D4.2D4⋊43C2, 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⋊C4⋊38C2, C8⋊C4.48C22, C4.Q8.59C22, C2.79(D4○SD16), (C4×D4).164C22, (C2×D4).239C23, C4⋊D4.88C22, C22⋊C8.85C22, (C4×Q8).162C22, (C2×Q8).224C23, C2.149(D4⋊5D4), C2.D8.122C22, C22⋊Q8.87C22, D4⋊C4.75C22, C23.38D4⋊15C2, C23.24D4⋊33C2, C23.19D4⋊37C2, C23.20D4⋊38C2, (C22×C8).366C22, (C2×SD16).59C22, C4.4D4.69C22, C22.773(C22×D4), C2.89(D8⋊C22), C22.49C24⋊6C2, (C22×C4).1157C23, Q8⋊C4.182C22, (C22×Q8).346C22, C42.28C22⋊19C2, C42⋊C2.193C22, (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

Derived series
C1 — C2×C4 — C42.56C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8⋊5D4 — C42.56C23
Lower central
C1 — C2 — C2×C4 — C42.56C23
Upper central
C1 — C22 — C4×D4 — C42.56C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.56C23

Subgroups: 400 in 196 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×12], C8 [×4], C2×C4 [×5], C2×C4 [×16], D4 [×11], Q8 [×2], Q8 [×6], C23 [×2], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×5], C4⋊C4 [×3], C2×C8 [×4], C2×C8, M4(2), D8, SD16 [×3], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×5], C4○D4 [×3], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×6], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8, C4⋊D4 [×3], C4⋊D4 [×2], C22⋊Q8, C22⋊Q8, C4.4D4, C4.4D4 [×3], C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×SD16 [×2], C22×Q8, C2×C4○D4, C23.24D4, C23.38D4, C8⋊9D4, SD16⋊C4, Q8⋊D4, D4⋊D4, D4.2D4, C8⋊8D4, C8⋊2D4, Q8⋊Q8, C23.19D4, C23.20D4, C42.28C22, Q8⋊5D4, C22.49C24, C42.56C23

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○SD16, C42.56C23

Generators and relations
 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 >

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

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

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

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

Matrix representation ►G ⊆ 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] >;
 

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
ρ212-22-200002-20000-22i02i200002i002i00    complex lifted from C4○D4
ρ222-22-200002-2000022i02i-200002i002i00    complex lifted from C4○D4
ρ232-22-200002-2000022i02i-200002i002i00    complex lifted from C4○D4
ρ242-22-200002-20000-22i02i200002i002i00    complex lifted from C4○D4
ρ254-44-40000-440000000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ274-4-440000004i4i00000000000000000    complex lifted from D8⋊C22
ρ2844-4-4000000000000000000002√-22√-2000    complex lifted from D4○SD16
ρ2944-4-4000000000000000000002√-22√-2000    complex lifted from D4○SD16

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
 

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