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G = C42.531C23order 128 = 27

392nd non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.531C23, C4.1522+ 1+4, C4⋊C4.188D4, C84Q823C2, C83D4.3C2, D8⋊C431C2, C4⋊SD1628C2, (C4×SD16)⋊26C2, C8.22(C4○D4), C8.2D422C2, (C2×Q8).143D4, Q16⋊C431C2, D4.D429C2, D4.2D450C2, C8.12D425C2, C4⋊C4.452C23, C4⋊C8.154C22, (C2×C8).122C23, (C4×C8).211C22, (C2×C4).593C24, Q8.D449C2, (C2×D8).97C22, C4⋊Q8.219C22, C8⋊C4.80C22, C2.47(Q86D4), (C2×D4).287C23, (C4×D4).226C22, (C2×Q16).96C22, (C2×Q8).272C23, (C4×Q8).216C22, C4.Q8.143C22, C2.115(D4○SD16), C41D4.111C22, C4.4D4.93C22, C22.853(C22×D4), D4⋊C4.101C22, C22.53C249C2, Q8⋊C4.192C22, (C2×SD16).127C22, C2.108(D8⋊C22), C22.50C2416C2, C4.171(C2×C4○D4), (C2×C4).657(C2×D4), SmallGroup(128,2133)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.531C23
C1C2C4C2×C4C42C4×Q8C22.50C24 — C42.531C23
C1C2C2×C4 — C42.531C23
C1C22C4×Q8 — C42.531C23
C1C2C2C2×C4 — C42.531C23

Generators and relations for C42.531C23
 G = < a,b,c,d,e | a4=b4=c2=1, d2=e2=a2b2, ab=ba, ac=ca, dad-1=a-1b2, ae=ea, cbc=ebe-1=b-1, bd=db, dcd-1=a2b2c, ece-1=bc, ede-1=b2d >

Subgroups: 368 in 184 conjugacy classes, 86 normal (84 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×7], C2×C4 [×9], D4 [×9], Q8 [×7], C23 [×3], C42 [×3], C42 [×3], C22⋊C4 [×11], C4⋊C4 [×5], C4⋊C4 [×5], C2×C8 [×4], D8 [×3], SD16 [×6], Q16 [×3], C22×C4 [×3], C2×D4 [×3], C2×D4 [×2], C2×Q8 [×4], C4×C8, C8⋊C4 [×2], D4⋊C4 [×3], Q8⋊C4 [×3], C4⋊C8 [×3], C4.Q8, C42⋊C2, C4×D4 [×3], C4×D4, C4×Q8 [×4], C22⋊Q8, C22.D4 [×2], C4.4D4 [×4], C4.4D4, C422C2 [×2], C41D4, C4⋊Q8, C2×D8 [×2], C2×SD16 [×5], C2×Q16 [×2], C4×SD16, Q16⋊C4, D8⋊C4, C84Q8, C4⋊SD16, D4.D4, D4.2D4 [×2], Q8.D4 [×2], C8.12D4, C83D4, C8.2D4, C22.50C24, C22.53C24, C42.531C23
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, Q86D4, D8⋊C22, D4○SD16, C42.531C23

Character table of C42.531C23

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P8A8B8C8D8E8F
 size 11118882222444444444888444488
ρ111111111111111111111111111111    trivial
ρ21111-1-1-111111-11-111-1-11-1-1-1111111    linear of order 2
ρ311111-111111111-1111-11-1-11-1-1-1-1-1-1    linear of order 2
ρ41111-11-111111-11111-11111-1-1-1-1-1-1-1    linear of order 2
ρ51111-1-111111-1111-1-111-11-1-11111-1-1    linear of order 2
ρ6111111-11111-1-11-1-1-1-1-1-1-1111111-1-1    linear of order 2
ρ71111-1111111-111-1-1-11-1-1-11-1-1-1-1-111    linear of order 2
ρ811111-1-11111-1-111-1-1-11-11-11-1-1-1-111    linear of order 2
ρ911111-11-11-111-1-111-1-11-1-11-1-1-111-11    linear of order 2
ρ101111-11-1-11-1111-1-11-11-1-11-11-1-111-11    linear of order 2
ρ111111111-11-111-1-1-11-1-1-1-11-1-111-1-11-1    linear of order 2
ρ121111-1-1-1-11-1111-111-111-1-11111-1-11-1    linear of order 2
ρ131111-111-11-11-1-1-11-11-111-1-11-1-1111-1    linear of order 2
ρ1411111-1-1-11-11-11-1-1-111-1111-1-1-1111-1    linear of order 2
ρ151111-1-11-11-11-1-1-1-1-11-1-1111111-1-1-11    linear of order 2
ρ16111111-1-11-11-11-11-11111-1-1-111-1-1-11    linear of order 2
ρ1722220002-22-2-20-202200-2000000000    orthogonal lifted from D4
ρ182222000-2-2-2-2-20202-2002000000000    orthogonal lifted from D4
ρ1922220002-22-220-20-2-2002000000000    orthogonal lifted from D4
ρ202222000-2-2-2-22020-2200-2000000000    orthogonal lifted from D4
ρ212-22-2000020-202i02i00-2i-2i0000-220000    complex lifted from C4○D4
ρ222-22-2000020-202i0-2i00-2i2i00002-20000    complex lifted from C4○D4
ρ232-22-2000020-20-2i0-2i002i2i0000-220000    complex lifted from C4○D4
ρ242-22-2000020-20-2i02i002i-2i00002-20000    complex lifted from C4○D4
ρ254-44-40000-404000000000000000000    orthogonal lifted from 2+ 1+4
ρ264-4-44000-4i04i0000000000000000000    complex lifted from D8⋊C22
ρ274-4-440004i0-4i0000000000000000000    complex lifted from D8⋊C22
ρ2844-4-40000000000000000000002-2-2-200    complex lifted from D4○SD16
ρ2944-4-4000000000000000000000-2-22-200    complex lifted from D4○SD16

Smallest permutation representation of C42.531C23
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 28 20 23)(2 25 17 24)(3 26 18 21)(4 27 19 22)(5 12 15 63)(6 9 16 64)(7 10 13 61)(8 11 14 62)(29 36 37 41)(30 33 38 42)(31 34 39 43)(32 35 40 44)(45 51 56 60)(46 52 53 57)(47 49 54 58)(48 50 55 59)
(1 47)(2 48)(3 45)(4 46)(5 40)(6 37)(7 38)(8 39)(9 36)(10 33)(11 34)(12 35)(13 30)(14 31)(15 32)(16 29)(17 55)(18 56)(19 53)(20 54)(21 51)(22 52)(23 49)(24 50)(25 59)(26 60)(27 57)(28 58)(41 64)(42 61)(43 62)(44 63)
(1 56 18 47)(2 48 19 53)(3 54 20 45)(4 46 17 55)(5 35 13 42)(6 43 14 36)(7 33 15 44)(8 41 16 34)(9 31 62 37)(10 38 63 32)(11 29 64 39)(12 40 61 30)(21 49 28 60)(22 57 25 50)(23 51 26 58)(24 59 27 52)
(1 38 18 32)(2 39 19 29)(3 40 20 30)(4 37 17 31)(5 49 13 60)(6 50 14 57)(7 51 15 58)(8 52 16 59)(9 48 62 53)(10 45 63 54)(11 46 64 55)(12 47 61 56)(21 44 28 33)(22 41 25 34)(23 42 26 35)(24 43 27 36)

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

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

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

Matrix representation of C42.531C23 in GL6(𝔽17)

0160000
100000
004000
000400
000040
000004
,
100000
010000
000100
0016000
0000016
000010
,
040000
1300000
00130413
000444
001313013
00413130
,
0130000
1300000
00130413
000131313
0013440
004404
,
1300000
0130000
000010
000001
001000
000100

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

C42.531C23 in GAP, Magma, Sage, TeX

C_4^2._{531}C_2^3
% in TeX

G:=Group("C4^2.531C2^3");
// GroupNames label

G:=SmallGroup(128,2133);
// by ID

G=gap.SmallGroup(128,2133);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,723,100,346,304,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=c^2=1,d^2=e^2=a^2*b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1*b^2,a*e=e*a,c*b*c=e*b*e^-1=b^-1,b*d=d*b,d*c*d^-1=a^2*b^2*c,e*c*e^-1=b*c,e*d*e^-1=b^2*d>;
// generators/relations

Export

Character table of C42.531C23 in TeX

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