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G = C42.57C23  order 128 = 27

57th non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.57C23, C4.782- (1+4), C8⋊Q8⋊23C2, C8⋊9(C4○D4), C8⋊D4⋊50C2, C8⋊9D4⋊24C2, C8⋊8D4⋊21C2, C4⋊C4.165D4, D4.Q8⋊40C2, C4.Q16⋊38C2, (C2×D4).329D4, C2.57(Q8○D8), D4⋊6D4.10C2, C4⋊C8.116C22, C4⋊C4.248C23, (C2×C4).535C24, (C2×C8).200C23, C22⋊C4.175D4, C23.480(C2×D4), C4⋊Q8.167C22, SD16⋊C4⋊39C2, C2.88(D4⋊6D4), C8⋊C4.49C22, C4.Q8.65C22, (C4×D4).175C22, (C2×D4).254C23, C22.D8⋊31C2, C22⋊C8.94C22, (C2×Q8).239C23, (C4×Q8).176C22, M4(2)⋊C4⋊30C2, C2.D8.128C22, D4⋊C4.78C22, C23.47D4⋊20C2, C23.48D4⋊31C2, C4⋊D4.103C22, C23.19D4⋊42C2, C22.10(C8⋊C22), (C22×C8).286C22, (C22×C4).339C23, Q8⋊C4.76C22, (C2×SD16).63C22, C22.795(C22×D4), C22⋊Q8.102C22, C42.C2.48C22, C22.46C24⋊9C2, C42⋊C2.206C22, (C2×M4(2)).128C22, (C2×C2.D8)⋊42C2, C4.117(C2×C4○D4), (C2×C4).619(C2×D4), C2.82(C2×C8⋊C22), (C2×C4⋊C4).684C22, SmallGroup(128,2075)

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

Derived series
C1 — C2×C4 — C42.57C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4⋊C4 — C22.46C24 — C42.57C23
Lower central
C1 — C2 — C2×C4 — C42.57C23
Upper central
C1 — C22 — C4×D4 — C42.57C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.57C23

Subgroups: 352 in 188 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×8], C8 [×2], C8 [×3], C2×C4 [×5], C2×C4 [×19], D4 [×8], Q8 [×4], C23 [×2], C23, C42, C42 [×2], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×7], C4⋊C4 [×9], C2×C8 [×4], C2×C8 [×2], M4(2) [×2], SD16 [×2], C22×C4 [×2], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8, C4○D4 [×4], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8, C4.Q8 [×3], C2.D8 [×6], C2×C4⋊C4 [×3], C42⋊C2, C42⋊C2, C4×D4 [×2], C4×Q8, C4⋊D4 [×2], C22⋊Q8 [×2], C22⋊Q8, C22.D4 [×3], C42.C2, C42.C2, C42⋊2C2, C4⋊Q8, C22×C8, C2×M4(2), C2×SD16, C2×C4○D4, C2×C2.D8, M4(2)⋊C4, C8⋊9D4, SD16⋊C4, C8⋊8D4, C8⋊D4, C4.Q16, D4.Q8, C22.D8, C23.19D4, C23.47D4, C23.48D4, C8⋊Q8, D4⋊6D4, C22.46C24, C42.57C23

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, Q8○D8, C42.57C23

Generators and relations
 G = < a,b,c,d,e | a4=b4=e2=1, c2=d2=a2, 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 >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

00010000
00100000
016000000
160000000
00005005
000005120
0000012120
000050012
,
160000000
016000000
001600000
000160000
00000100
000016000
00000001
000000160
,
001600000
00010000
10000000
016000000
00000550
000050012
00005005
000001250
,
013000000
130000000
000130000
001300000
00002151515
00001515152
00001515215
00001521515
,
00100000
00010000
10000000
01000000
0000012120
000050012
00005005
000005120

G:=sub<GL(8,GF(17))| [0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,5,0,0,12],[16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0],[0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,5,0,0,12,0,0,0,0,5,0,0,5,0,0,0,0,0,12,5,0],[0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,0,2,15,15,15,0,0,0,0,15,15,15,2,0,0,0,0,15,15,2,15,0,0,0,0,15,2,15,15],[0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,5,5,0,0,0,0,0,12,0,0,5,0,0,0,0,12,0,0,12,0,0,0,0,0,12,5,0] >;
 

Character table of C42.57C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11112248224444444488888444488
ρ111111111111111111111111111111    trivial
ρ21111111111111-1111-1-111-1-1-1-1-1-1-1-1    linear of order 2
ρ31111-1-11-1111-1-1-11-11-11-11-111-1-11-11    linear of order 2
ρ41111-1-11-1111-1-111-111-1-111-1-111-11-1    linear of order 2
ρ51111-1-11111-1-1-111-1-11-11-11-11-1-11-11    linear of order 2
ρ61111-1-11111-1-1-1-11-1-1-111-1-11-111-11-1    linear of order 2
ρ71111111-111-111-111-1-1-1-1-1-1-1111111    linear of order 2
ρ81111111-111-111111-111-1-111-1-1-1-1-1-1    linear of order 2
ρ91111-1-1-111111-1-1-111-11-1-11-11-1-111-1    linear of order 2
ρ101111-1-1-111111-11-1111-1-1-1-11-111-1-11    linear of order 2
ρ11111111-1-1111-111-1-11111-1-1-11111-1-1    linear of order 2
ρ12111111-1-1111-11-1-1-11-1-11-111-1-1-1-111    linear of order 2
ρ13111111-1111-1-11-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ14111111-1111-1-111-1-1-111-11-1-1-1-1-1-111    linear of order 2
ρ151111-1-1-1-111-11-11-11-11-111-111-1-111-1    linear of order 2
ρ161111-1-1-1-111-11-1-1-11-1-11111-1-111-1-11    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
ρ212-22-200002-22i002i002i2i0000002-2000    complex lifted from C4○D4
ρ222-22-200002-22i002i002i2i0000002-2000    complex lifted from C4○D4
ρ232-22-200002-22i002i002i2i000000-22000    complex lifted from C4○D4
ρ242-22-200002-22i002i002i2i000000-22000    complex lifted from C4○D4
ρ254-4-444-400000000000000000000000    orthogonal lifted from C8⋊C22
ρ264-4-44-4400000000000000000000000    orthogonal lifted from C8⋊C22
ρ274-44-40000-440000000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2844-4-400000000000000000002√2002√200    symplectic lifted from Q8○D8, Schur index 2
ρ2944-4-400000000000000000002√2002√200    symplectic lifted from Q8○D8, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{57}C_2^3
 
% in TeX
 
G:=Group("C4^2.57C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2075);
 
// by ID
 
G=gap.SmallGroup(128,2075);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,758,723,100,346,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=d^2=a^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
 

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