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G = C42.52C23  order 128 = 27

52nd non-split extension by C42 of C23 acting faithfully

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

Aliases: C42.52C23, C4.622+ (1+4), C8⋊9D4⋊19C2, C8⋊D4⋊39C2, C8⋊8D4⋊52C2, C4⋊C4.368D4, Q8.Q8⋊36C2, C4⋊SD16⋊22C2, (C2×D4).172D4, Q8⋊6D4.6C2, C22⋊C4.51D4, Q16⋊C4⋊24C2, D4.7D4⋊47C2, C4⋊C4.411C23, C4⋊C8.105C22, (C2×C8).354C23, (C2×C4).509C24, Q8.25(C4○D4), C23.326(C2×D4), C8⋊C4.46C22, C2.77(D4○SD16), (C2×D4).235C23, (C4×D4).162C22, C4⋊D4.86C22, C4⋊1D4.88C22, C22⋊C8.83C22, (C2×Q16).86C22, (C2×Q8).222C23, (C4×Q8).160C22, C2.145(D4⋊5D4), C4.Q8.105C22, C2.D8.120C22, C22⋊Q8.84C22, D4⋊C4.74C22, C23.36D4⋊20C2, C23.24D4⋊32C2, C23.46D4⋊16C2, C23.19D4⋊34C2, (C22×C8).364C22, C22.769(C22×D4), C42.C2.41C22, C2.87(D8⋊C22), (C22×C4).1153C23, C22.46C24⋊5C2, Q8⋊C4.181C22, (C2×SD16).101C22, C42⋊C2.191C22, C42.29C22⋊10C2, (C2×M4(2)).116C22, C4.234(C2×C4○D4), (C2×C4).606(C2×D4), (C2×C4⋊C4).670C22, (C2×C4○D4).213C22, SmallGroup(128,2049)

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

Derived series
C1 — C2×C4 — C42.52C23
Chief series
C1 — C2 — C4 — C2×C4 — C22×C4 — C2×C4○D4 — Q8⋊6D4 — C42.52C23
Lower central
C1 — C2 — C2×C4 — C42.52C23
Upper central
C1 — C22 — C4×D4 — C42.52C23
Jennings
C1 — C2 — C2 — C2×C4 — C42.52C23

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 [×14], Q8 [×2], Q8 [×3], C23 [×2], C23 [×2], C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×5], C4⋊C4 [×7], C2×C8 [×4], C2×C8, M4(2), SD16 [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×D4 [×6], C2×Q8 [×2], C4○D4 [×6], C8⋊C4, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×4], C4⋊C8, C4.Q8 [×2], C2.D8, C2×C4⋊C4, C42⋊C2, C42⋊C2, C4×D4, C4×D4, C4×Q8 [×2], C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4, C42.C2, C42.C2, C42⋊2C2, C4⋊1D4, C4⋊1D4, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16, C2×C4○D4 [×2], C23.24D4, C23.36D4, C8⋊9D4, Q16⋊C4, D4.7D4 [×2], C4⋊SD16, C8⋊8D4, C8⋊D4, Q8.Q8, C23.46D4, C23.19D4, C42.29C22, Q8⋊6D4, C22.46C24, C42.52C23

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, C42.52C23

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

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

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

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

Matrix representation ►G ⊆ GL6(𝔽17)

1620000
1610000
00161500
001100
00001615
000011
,
100000
010000
00161500
001100
000012
00001616
,
1300000
1340000
000010
000001
0016000
0001600
,
100000
010000
0013000
004400
0000139
000004
,
1620000
010000
004800
00131300
000048
00001313

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

Character table of C42.52C23

 class 12A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114488222244444448888444488
ρ111111111111111111111111111111    trivial
ρ211111111111111111-1-1-1-1-11-1-1-1-1-1-1    linear of order 2
ρ311111-11-1-11-11-1-1-11-1-1-111-11-11-111-1    linear of order 2
ρ411111-11-1-11-11-1-1-11-111-1-1111-11-1-11    linear of order 2
ρ51111-111-111111-11-1-1-1-1-111-1-1-1-1-111    linear of order 2
ρ61111-111-111111-11-1-1111-1-1-11111-1-1    linear of order 2
ρ71111-1-111-11-11-11-1-1111-11-1-11-11-11-1    linear of order 2
ρ81111-1-111-11-11-11-1-11-1-11-11-1-11-11-11    linear of order 2
ρ9111111-1-11111-11-111-1-1-1-1-1-1111111    linear of order 2
ρ10111111-1-11111-11-11111111-1-1-1-1-1-1-1    linear of order 2
ρ1111111-1-11-11-111-111-111-1-11-1-11-111-1    linear of order 2
ρ1211111-1-11-11-111-111-1-1-111-1-11-11-1-11    linear of order 2
ρ131111-11-111111-1-1-1-1-1111-1-11-1-1-1-111    linear of order 2
ρ141111-11-111111-1-1-1-1-1-1-1-11111111-1-1    linear of order 2
ρ151111-1-1-1-1-11-11111-11-1-11-1111-11-11-1    linear of order 2
ρ161111-1-1-1-1-11-11111-1111-11-11-11-11-11    linear of order 2
ρ1722222-2002-22-2020-2-2000000000000    orthogonal lifted from D4
ρ182222-2-2002-22-20-2022000000000000    orthogonal lifted from D4
ρ192222-2200-2-2-2-20202-2000000000000    orthogonal lifted from D4
ρ2022222200-2-2-2-20-20-22000000000000    orthogonal lifted from D4
ρ212-22-20000020-220-2002i2i000002i02i00    complex lifted from C4○D4
ρ222-22-20000020-2-202002i2i000002i02i00    complex lifted from C4○D4
ρ232-22-20000020-2-202002i2i000002i02i00    complex lifted from C4○D4
ρ242-22-20000020-220-2002i2i000002i02i00    complex lifted from C4○D4
ρ254-44-400000-40400000000000000000    orthogonal lifted from 2+ (1+4)
ρ264-4-4400004i04i000000000000000000    complex lifted from D8⋊C22
ρ274-4-4400004i04i000000000000000000    complex lifted from D8⋊C22
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._{52}C_2^3
 
% in TeX
 
G:=Group("C4^2.52C2^3");
 
// GroupNames label
 
G:=SmallGroup(128,2049);
 
// by ID
 
G=gap.SmallGroup(128,2049);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,560,253,758,723,352,346,248,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^4=e^2=1,c^2=a^2,d^2=b^2,a*b=b*a,c*a*c^-1=e*a*e=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=a^2*b^2*c,e*d*e=b^2*d>;
 
// generators/relations
 

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