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G = C42.276D4order 128 = 27

258th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.276D4, C42.404C23, C4.1102+ 1+4, C42Q1627C2, C8.2D411C2, C4⋊C8.63C22, (C2×C8).66C23, Q8⋊D4.3C2, D4.D410C2, C4⋊C4.157C23, (C2×C4).416C24, C22⋊Q1622C2, C23.696(C2×D4), (C22×C4).505D4, C4⋊Q8.306C22, C42.6C414C2, C8⋊C4.20C22, (C2×D4).165C23, (C4×D4).107C22, C22⋊C8.51C22, (C2×Q8).153C23, (C2×Q16).71C22, (C4×Q8).104C22, C4⋊D4.194C22, C4.120(C8.C22), (C2×C42).883C22, Q8⋊C4.46C22, (C2×SD16).35C22, C22.676(C22×D4), C22⋊Q8.199C22, C42.30C221C2, (C22×C4).1087C23, C4.4D4.156C22, C22.21(C8.C22), C42.C2.127C22, (C22×Q8).324C22, C2.87(C22.29C24), C23.36C23.24C2, (C2×C4⋊Q8)⋊41C2, (C2×C4).545(C2×D4), C2.58(C2×C8.C22), SmallGroup(128,1950)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.276D4
C1C2C4C2×C4C22×C4C22×Q8C2×C4⋊Q8 — C42.276D4
C1C2C2×C4 — C42.276D4
C1C22C2×C42 — C42.276D4
C1C2C2C2×C4 — C42.276D4

Generators and relations for C42.276D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, cac-1=ab2, dad=a-1b2, cbc-1=a2b, dbd=a2b-1, dcd=c3 >

Subgroups: 372 in 193 conjugacy classes, 88 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×4], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×6], C2×C4 [×19], D4 [×4], Q8 [×14], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×12], C2×C8 [×4], SD16 [×4], Q16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×6], C8⋊C4 [×2], C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C2×C42, C2×C4⋊C4 [×2], C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C4⋊Q8 [×4], C4⋊Q8 [×2], C2×SD16 [×4], C2×Q16 [×4], C22×Q8 [×2], C42.6C4, Q8⋊D4 [×2], C22⋊Q16 [×2], D4.D4 [×2], C42Q16 [×2], C42.30C22 [×2], C8.2D4 [×2], C23.36C23, C2×C4⋊Q8, C42.276D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×4], C22×D4, 2+ 1+4 [×2], C22.29C24, C2×C8.C22 [×2], C42.276D4

Character table of C42.276D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11112282222444488888888888
ρ111111111111111111111111111    trivial
ρ21111-1-111111-11-1-1-1111-1-1-1-111-1    linear of order 2
ρ31111-1-11-1-1111-1-111-11-1-1-11-11-11    linear of order 2
ρ41111111-1-111-1-11-1-1-11-111-111-1-1    linear of order 2
ρ5111111-1111111111-1111-1-1-1-1-1-1    linear of order 2
ρ61111-1-1-11111-11-1-1-1-111-1111-1-11    linear of order 2
ρ71111-1-1-1-1-1111-1-11111-1-11-11-11-1    linear of order 2
ρ8111111-1-1-111-1-11-1-111-11-11-1-111    linear of order 2
ρ91111-1-11-1-1111-1-11-1-1-111-111-11-1    linear of order 2
ρ101111111-1-111-1-11-11-1-11-11-1-1-111    linear of order 2
ρ11111111111111111-11-1-1-111-1-1-1-1    linear of order 2
ρ121111-1-111111-11-1-111-1-11-1-11-1-11    linear of order 2
ρ131111-1-1-1-1-1111-1-11-11-1111-1-11-11    linear of order 2
ρ14111111-1-1-111-1-11-111-11-1-1111-1-1    linear of order 2
ρ15111111-111111111-1-1-1-1-1-1-11111    linear of order 2
ρ161111-1-1-11111-11-1-11-1-1-1111-111-1    linear of order 2
ρ172222-2-20-2-2-2-2-222200000000000    orthogonal lifted from D4
ρ182222220-2-2-2-222-2-200000000000    orthogonal lifted from D4
ρ19222222022-2-2-2-2-2200000000000    orthogonal lifted from D4
ρ202222-2-2022-2-22-22-200000000000    orthogonal lifted from D4
ρ214-44-4000004-4000000000000000    orthogonal lifted from 2+ 1+4
ρ224-44-400000-44000000000000000    orthogonal lifted from 2+ 1+4
ρ2344-4-4000-4400000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ244-4-444-400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-4-44-4400000000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-40004-400000000000000000    symplectic lifted from C8.C22, Schur index 2

Smallest permutation representation of C42.276D4
On 64 points
Generators in S64
(1 63 29 52)(2 60 30 49)(3 57 31 54)(4 62 32 51)(5 59 25 56)(6 64 26 53)(7 61 27 50)(8 58 28 55)(9 22 35 42)(10 19 36 47)(11 24 37 44)(12 21 38 41)(13 18 39 46)(14 23 40 43)(15 20 33 48)(16 17 34 45)
(1 36 5 40)(2 11 6 15)(3 38 7 34)(4 13 8 9)(10 25 14 29)(12 27 16 31)(17 54 21 50)(18 58 22 62)(19 56 23 52)(20 60 24 64)(26 33 30 37)(28 35 32 39)(41 61 45 57)(42 51 46 55)(43 63 47 59)(44 53 48 49)
(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)
(2 4)(3 7)(6 8)(9 37)(10 40)(11 35)(12 38)(13 33)(14 36)(15 39)(16 34)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(41 45)(42 48)(44 46)(49 58)(50 61)(51 64)(52 59)(53 62)(54 57)(55 60)(56 63)

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

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

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

Matrix representation of C42.276D4 in GL8(𝔽17)

001250000
00550000
512000000
1212000000
000005140
000050014
000030012
000003120
,
01000000
160000000
000160000
00100000
000001600
000016000
000000016
000000160
,
00010000
00100000
160000000
01000000
000000160
00000001
00001000
000001600
,
10000000
016000000
00010000
00100000
00001000
000001600
000000160
00000001

G:=sub<GL(8,GF(17))| [0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,12,5,0,0,0,0,0,0,5,5,0,0,0,0,0,0,0,0,0,0,0,5,3,0,0,0,0,0,5,0,0,3,0,0,0,0,14,0,0,12,0,0,0,0,0,14,12,0],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0],[0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1] >;

C42.276D4 in GAP, Magma, Sage, TeX

C_4^2._{276}D_4
% in TeX

G:=Group("C4^2.276D4");
// GroupNames label

G:=SmallGroup(128,1950);
// by ID

G=gap.SmallGroup(128,1950);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,568,758,891,352,675,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=d^2=1,c^4=b^2,a*b=b*a,c*a*c^-1=a*b^2,d*a*d=a^-1*b^2,c*b*c^-1=a^2*b,d*b*d=a^2*b^-1,d*c*d=c^3>;
// generators/relations

Export

Character table of C42.276D4 in TeX

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