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G = C42.59C23  order 128 = 27

59th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.59C23, C4.802- (1+4), C8⋊Q8⋊25C2, C8⋊10(C4○D4), C8⋊9D4⋊25C2, C8⋊7D4⋊31C2, C8⋊2D4⋊31C2, C4⋊C4.167D4, D4.Q8⋊41C2, D4⋊6D4⋊15C2, D8⋊C4⋊25C2, D4⋊2Q8⋊21C2, (C2×D4).331D4, C22⋊C4.62D4, C4⋊C4.250C23, C4⋊C8.118C22, (C2×C4).537C24, (C2×C8).365C23, (C2×D8).89C22, C23.343(C2×D4), C4⋊Q8.169C22, C2.90(D4⋊6D4), C8⋊C4.51C22, C2.90(D4○SD16), (C4×D4).177C22, (C2×D4).255C23, C22.D8⋊32C2, C22⋊C8.96C22, M4(2)⋊C4⋊32C2, C2.D8.130C22, C4.Q8.137C22, D4⋊C4.79C22, C4⋊D4.104C22, C23.46D4⋊20C2, C23.19D4⋊43C2, C22.13(C8⋊C22), (C22×C4).341C23, (C22×C8).288C22, C22.797(C22×D4), C42.C2.50C22, C22.47C24⋊9C2, C42⋊C2.208C22, (C2×M4(2)).130C22, (C2×C4.Q8)⋊13C2, C4.119(C2×C4○D4), (C2×C4).621(C2×D4), C2.83(C2×C8⋊C22), (C2×C4⋊C4).686C22, SmallGroup(128,2077)

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

Derived series
C1 — C2×C4 — C42.59C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — D4⋊6D4 — C42.59C23
Lower central
C1 — C2 — C2×C4 — C42.59C23
Upper central
C1 — C22 — C4×D4 — C42.59C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.59C23

Subgroups: 392 in 194 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×18], D4 [×13], Q8 [×2], C23 [×2], C23 [×2], C42, C42, C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×7], C4⋊C4 [×6], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], D8 [×2], C22×C4 [×2], C22×C4 [×5], C2×D4 [×3], C2×D4 [×4], C2×Q8, C4○D4 [×4], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], C4⋊C8, C4.Q8 [×6], C2.D8 [×3], C2×C4⋊C4 [×3], C42⋊C2, C4×D4 [×3], C4×D4, C4⋊D4 [×4], C4⋊D4, C22⋊Q8, C22.D4 [×3], C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×D8, C2×C4○D4, C2×C4.Q8, M4(2)⋊C4, C8⋊9D4, D8⋊C4, C8⋊7D4, C8⋊2D4, D4⋊2Q8, D4.Q8, C22.D8, C23.46D4 [×2], C23.19D4, C8⋊Q8, D4⋊6D4, C22.47C24, C42.59C23

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), D4⋊6D4, C2×C8⋊C22, D4○SD16, C42.59C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=a2b2, e2=b2, ab=ba, cac-1=eae-1=a-1b2, dad-1=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2b2c, 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 52 28 31)(2 49 25 32)(3 50 26 29)(4 51 27 30)(5 60 61 54)(6 57 62 55)(7 58 63 56)(8 59 64 53)(9 33 14 40)(10 34 15 37)(11 35 16 38)(12 36 13 39)(17 45 24 41)(18 46 21 42)(19 47 22 43)(20 48 23 44)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

001600000
00110000
10000000
1616000000
00000040
00001313139
000013000
00004404
,
160000000
016000000
001600000
000160000
00000100
000016000
000016161615
00001011
,
130000000
013000000
00400000
00040000
000014141111
00001414011
000014300
00003036
,
00100000
00010000
160000000
016000000
00000040
00004448
00004000
00001301313
,
0016150000
00110000
12000000
1616000000
000013000
000001300
00000040
00004404

G:=sub<GL(8,GF(17))| [0,0,1,16,0,0,0,0,0,0,0,16,0,0,0,0,16,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,13,13,4,0,0,0,0,0,13,0,4,0,0,0,0,4,13,0,0,0,0,0,0,0,9,0,4],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,16,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,16,1,0,0,0,0,0,0,15,1],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,14,14,14,3,0,0,0,0,14,14,3,0,0,0,0,0,11,0,0,3,0,0,0,0,11,11,0,6],[0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,4,13,0,0,0,0,0,4,0,0,0,0,0,0,4,4,0,13,0,0,0,0,0,8,0,13],[0,0,1,16,0,0,0,0,0,0,2,16,0,0,0,0,16,1,0,0,0,0,0,0,15,1,0,0,0,0,0,0,0,0,0,0,13,0,0,4,0,0,0,0,0,13,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
 

Character table of C42.59C23

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N8A8B8C8D8E8F
 size 11112248822444444448888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-1-1111-1-1-1111-11-1-1111-11-1-11    linear of order 2
ρ31111-1-11-1-1111-11-1-1111-111-1-11-11-11    linear of order 2
ρ4111111-11-111-11-1-1-11-111-11-1-1-1-1-111    linear of order 2
ρ51111111-1-11111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ61111-1-1-11-111-1-1111-11-111-1-11-11-1-11    linear of order 2
ρ71111-1-1111111-1-1-1-1-1-1-11-1-11-11-11-11    linear of order 2
ρ8111111-1-1111-111-1-1-11-1-11-11-1-1-1-111    linear of order 2
ρ9111111-11111-11-1-1-1-1-1-1-111-11111-1-1    linear of order 2
ρ101111-1-11-11111-11-1-1-11-11-11-11-11-11-1    linear of order 2
ρ111111-1-1-1-1-111-1-1-111-1-1-11111-11-111-1    linear of order 2
ρ1211111111-11111111-11-1-1-111-1-1-1-1-1-1    linear of order 2
ρ13111111-1-1-111-111-1-11111-1-111111-1-1    linear of order 2
ρ141111-1-111-1111-1-1-1-11-11-11-111-11-11-1    linear of order 2
ρ151111-1-1-11111-1-1111111-1-1-1-1-11-111-1    linear of order 2
ρ161111111-111111-1111-1111-1-1-1-1-1-1-1-1    linear of order 2
ρ17222222-200-2-22-202-20000000000000    orthogonal lifted from D4
ρ182222-2-2-200-2-2220-220000000000000    orthogonal lifted from D4
ρ192222-2-2200-2-2-2202-20000000000000    orthogonal lifted from D4
ρ20222222200-2-2-2-20-220000000000000    orthogonal lifted from D4
ρ212-22-200000-22002i002i2i2i0000020-200    complex lifted from C4○D4
ρ222-22-200000-22002i002i2i2i0000020-200    complex lifted from C4○D4
ρ232-22-200000-22002i002i2i2i00000-20200    complex lifted from C4○D4
ρ242-22-200000-22002i002i2i2i00000-20200    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-4000004-4000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2844-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002√-202√-2000    complex lifted from D4○SD16

In GAP, Magma, Sage, TeX 

C_4^2._{59}C_2^3
 
% in TeX
 
G:=Group("C4^2.59C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2077);
 
// by ID
 
G=gap.SmallGroup(128,2077);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,456,758,723,100,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^2=a^2*b^2,e^2=b^2,a*b=b*a,c*a*c^-1=e*a*e^-1=a^-1*b^2,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*b^2*c,e*d*e^-1=b^2*d>;
 
// generators/relations
 

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