Copied to
clipboard

G = C42.200D4order 128 = 27

182nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C42.200D4, C23.721C24, C24.104C23, C22.3772- 1+4, C22.4942+ 1+4, C425C438C2, C23⋊Q8.31C2, (C22×C4).232C23, (C2×C42).733C22, C22.453(C22×D4), C23.4Q8.31C2, C23.11D4.60C2, (C22×Q8).236C22, C23.78C2365C2, C2.73(C22.29C24), C23.81C23134C2, C23.83C23132C2, C2.C42.424C22, C2.70(C23.38C23), C2.60(C22.57C24), (C2×C4⋊Q8)⋊25C2, (C2×C4).438(C2×D4), (C2×C4⋊C4).530C22, (C2×C422C2).18C2, (C2×C22⋊C4).340C22, SmallGroup(128,1553)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.200D4
C1C2C22C23C22×C4C22×Q8C23⋊Q8 — C42.200D4
C1C23 — C42.200D4
C1C23 — C42.200D4
C1C23 — C42.200D4

Generators and relations for C42.200D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=a2b, dbd=a2b-1, dcd=b2c-1 >

Subgroups: 420 in 215 conjugacy classes, 92 normal (24 characteristic)
C1, C2 [×3], C2 [×4], C2, C4 [×17], C22, C22 [×6], C22 [×7], C2×C4 [×6], C2×C4 [×39], Q8 [×8], C23, C23 [×7], C42 [×4], C22⋊C4 [×10], C4⋊C4 [×18], C22×C4 [×6], C22×C4 [×8], C2×Q8 [×8], C24, C2.C42 [×14], C2×C42, C2×C22⋊C4 [×3], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×8], C422C2 [×4], C4⋊Q8 [×4], C22×Q8 [×2], C425C4, C23⋊Q8 [×2], C23.78C23 [×2], C23.11D4 [×2], C23.81C23 [×2], C23.4Q8 [×2], C23.83C23 [×2], C2×C422C2, C2×C4⋊Q8, C42.200D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2+ 1+4 [×2], 2- 1+4 [×4], C22.29C24, C23.38C23 [×2], C22.57C24 [×4], C42.200D4

Character table of C42.200D4

 class 12A2B2C2D2E2F2G2H4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q
 size 11111111844444488888888888
ρ111111111111111111111111111    trivial
ρ211111111-1-1-1-1-11111-11-111-1-11-1    linear of order 2
ρ311111111-1-1-111-1-1-1-111-1-11111-1    linear of order 2
ρ411111111111-1-1-1-1-1-1-111-11-1-111    linear of order 2
ρ511111111111-1-1-1-11-11-1-11-11-11-1    linear of order 2
ρ611111111-1-1-111-1-11-1-1-111-1-1111    linear of order 2
ρ711111111-1-1-1-1-111-111-11-1-11-111    linear of order 2
ρ8111111111111111-11-1-1-1-1-1-111-1    linear of order 2
ρ911111111-1111111-1-1-1-11111-1-1-1    linear of order 2
ρ10111111111-1-1-1-111-1-11-1-111-11-11    linear of order 2
ρ11111111111-1-111-1-111-1-1-1-111-1-11    linear of order 2
ρ1211111111-111-1-1-1-1111-11-11-11-1-1    linear of order 2
ρ1311111111-111-1-1-1-1-11-11-11-111-11    linear of order 2
ρ14111111111-1-111-1-1-111111-1-1-1-1-1    linear of order 2
ρ15111111111-1-1-1-1111-1-111-1-111-1-1    linear of order 2
ρ1611111111-11111111-111-1-1-1-1-1-11    linear of order 2
ρ172-22-22-22-202-22-22-200000000000    orthogonal lifted from D4
ρ182-22-22-22-20-222-2-2200000000000    orthogonal lifted from D4
ρ192-22-22-22-20-22-222-200000000000    orthogonal lifted from D4
ρ202-22-22-22-202-2-22-2200000000000    orthogonal lifted from D4
ρ214-4-4-444-44000000000000000000    orthogonal lifted from 2+ 1+4
ρ2244-444-4-4-4000000000000000000    orthogonal lifted from 2+ 1+4
ρ234-444-4-4-44000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ24444-4-44-4-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ254-4-44-444-4000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2644-4-4-4-444000000000000000000    symplectic lifted from 2- 1+4, Schur index 2

Smallest permutation representation of C42.200D4
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 43 22 10)(2 44 23 11)(3 41 24 12)(4 42 21 9)(5 25 38 54)(6 26 39 55)(7 27 40 56)(8 28 37 53)(13 62 46 34)(14 63 47 35)(15 64 48 36)(16 61 45 33)(17 29 50 58)(18 30 51 59)(19 31 52 60)(20 32 49 57)
(1 30 39 14)(2 58 40 46)(3 32 37 16)(4 60 38 48)(5 15 21 31)(6 47 22 59)(7 13 23 29)(8 45 24 57)(9 50 25 62)(10 20 26 33)(11 52 27 64)(12 18 28 35)(17 54 34 42)(19 56 36 44)(41 51 53 63)(43 49 55 61)
(2 23)(4 21)(5 38)(7 40)(9 11)(10 41)(12 43)(13 29)(14 59)(15 31)(16 57)(17 64)(18 33)(19 62)(20 35)(25 27)(26 53)(28 55)(30 47)(32 45)(34 52)(36 50)(42 44)(46 58)(48 60)(49 63)(51 61)(54 56)

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

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

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

Matrix representation of C42.200D4 in GL12(𝔽5)

004000000000
000400000000
400000000000
040000000000
000001200000
000042230000
000014410000
000000040000
000000000010
000000000001
000000004000
000000000400
,
040000000000
100000000000
000400000000
001000000000
000001000000
000040000000
000001140000
000041240000
000000000100
000000001000
000000000001
000000000010
,
030000000000
200000000000
000200000000
003000000000
000020000000
000002000000
000003300000
000032030000
000000000010
000000000004
000000001000
000000000400
,
100000000000
040000000000
004000000000
000100000000
000010000000
000004000000
000000400000
000003310000
000000001000
000000000400
000000000010
000000000004

G:=sub<GL(12,GF(5))| [0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,0,0,1,2,4,0,0,0,0,0,0,0,0,0,2,2,4,0,0,0,0,0,0,0,0,0,0,3,1,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0],[0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,4,0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0],[0,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,3,0,0,0,0,0,0,0,0,0,2,3,2,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0],[1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,0,0,0,0,0,0,4,3,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,4] >;

C42.200D4 in GAP, Magma, Sage, TeX

C_4^2._{200}D_4
% in TeX

G:=Group("C4^2.200D4");
// GroupNames label

G:=SmallGroup(128,1553);
// by ID

G=gap.SmallGroup(128,1553);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,100,794,185,80]);
// Polycyclic

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

Export

Character table of C42.200D4 in TeX

׿
×
𝔽