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G = C42.52C23order 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, C88D452C2, C8⋊D439C2, C89D419C2, C4⋊C4.368D4, Q8.Q836C2, C4⋊SD1622C2, (C2×D4).172D4, Q86D4.6C2, C22⋊C4.51D4, Q16⋊C424C2, D4.7D447C2, C4⋊C8.105C22, C4⋊C4.411C23, (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, C41D4.88C22, C22⋊C8.83C22, (C2×Q8).222C23, (C2×Q16).86C22, (C4×Q8).160C22, C2.145(D45D4), C2.D8.120C22, C4.Q8.105C22, C22⋊Q8.84C22, D4⋊C4.74C22, C23.24D432C2, C23.36D420C2, C23.19D434C2, C23.46D416C2, (C22×C8).364C22, C22.769(C22×D4), C42.C2.41C22, C2.87(D8⋊C22), (C22×C4).1153C23, C22.46C245C2, Q8⋊C4.181C22, (C2×SD16).101C22, C42.29C2210C2, C42⋊C2.191C22, (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

C1C2×C4 — C42.52C23
C1C2C4C2×C4C22×C4C2×C4○D4Q86D4 — C42.52C23
C1C2C2×C4 — C42.52C23
C1C22C4×D4 — C42.52C23
C1C2C2C2×C4 — C42.52C23

Generators and relations for C42.52C23
 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 >

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, C422C2, C41D4, C41D4, C22×C8, C2×M4(2), C2×SD16 [×2], C2×Q16, C2×C4○D4 [×2], C23.24D4, C23.36D4, C89D4, Q16⋊C4, D4.7D4 [×2], C4⋊SD16, C88D4, C8⋊D4, Q8.Q8, C23.46D4, C23.19D4, C42.29C22, Q86D4, 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, D45D4, D8⋊C22, D4○SD16, C42.52C23

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
ρ212-22-20000020-220-200-2i2i00000-2i02i00    complex lifted from C4○D4
ρ222-22-20000020-2-20200-2i2i000002i0-2i00    complex lifted from C4○D4
ρ232-22-20000020-2-202002i-2i00000-2i02i00    complex lifted from C4○D4
ρ242-22-20000020-220-2002i-2i000002i0-2i00    complex lifted from C4○D4
ρ254-44-400000-40400000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-4400004i0-4i000000000000000000    complex lifted from D8⋊C22
ρ274-4-440000-4i04i000000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000-2-202-2000    complex lifted from D4○SD16
ρ2944-4-400000000000000000002-20-2-2000    complex lifted from D4○SD16

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

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

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

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

Matrix representation of C42.52C23 in 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] >;

C42.52C23 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

Export

Character table of C42.52C23 in TeX

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