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G = C42.255D4order 128 = 27

237th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.255D4, C42.383C23, C42(C8⋊Q8), C8⋊Q837C2, (C4×D8)⋊29C2, C81(C4○D4), C42(C83D4), C42(C8⋊D4), C42(C82D4), C8⋊D458C2, C82D436C2, C83D425C2, (C4×SD16)⋊15C2, (C4×M4(2))⋊7C2, C4⋊C4.110C23, (C2×C4).369C24, (C2×C8).276C23, (C4×C8).181C22, (C4×D4).90C22, (C22×C4).475D4, C23.265(C2×D4), C4⋊Q8.292C22, (C4×Q8).87C22, C4.150(C8⋊C22), (C2×D4).124C23, (C2×D8).163C22, (C2×Q8).112C23, C2.D8.218C22, C4.Q8.134C22, C8⋊C4.126C22, C4⋊D4.173C22, C41D4.156C22, (C2×C42).862C22, C22.629(C22×D4), C22⋊Q8.178C22, D4⋊C4.202C22, C2.42(D8⋊C22), C22.26C2413C2, (C22×C4).1049C23, C23.36C236C2, Q8⋊C4.204C22, (C2×SD16).116C22, C4.4D4.144C22, C42.C2.121C22, C42(C42.28C22), C42(C42.29C22), C42.28C2238C2, C42.29C2224C2, (C2×M4(2)).279C22, C2.66(C22.26C24), C4.54(C2×C4○D4), (C2×C4).522(C2×D4), C2.45(C2×C8⋊C22), SmallGroup(128,1903)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.255D4
C1C2C4C2×C4C42C4×C8C4×M4(2) — C42.255D4
C1C2C2×C4 — C42.255D4
C1C2×C4C2×C42 — C42.255D4
C1C2C2C2×C4 — C42.255D4

Generators and relations for C42.255D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, bc=cb, dbd-1=a2b, dcd-1=c3 >

Subgroups: 412 in 204 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×4], C4 [×4], C4 [×9], C22, C22 [×12], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×16], Q8 [×4], C23, C23 [×3], C42 [×4], C42, C22⋊C4 [×9], C4⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×4], M4(2) [×4], D8 [×4], SD16 [×4], C22×C4 [×3], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×D4 [×5], C2×Q8, C2×Q8, C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C42⋊C2, C4×D4, C4×D4 [×2], C4×D4 [×2], C4×Q8, C4⋊D4, C4⋊D4 [×2], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C4.4D4, C42.C2, C422C2, C41D4, C4⋊Q8, C2×M4(2) [×2], C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C4×M4(2), C4×D8 [×2], C4×SD16 [×2], C8⋊D4 [×2], C82D4 [×2], C42.28C22, C42.29C22, C83D4, C8⋊Q8, C23.36C23, C22.26C24, C42.255D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22, D8⋊C22, C42.255D4

Smallest permutation representation of C42.255D4
On 64 points
Generators in S64
(1 48 5 44)(2 45 6 41)(3 42 7 46)(4 47 8 43)(9 51 13 55)(10 56 14 52)(11 53 15 49)(12 50 16 54)(17 34 21 38)(18 39 22 35)(19 36 23 40)(20 33 24 37)(25 63 29 59)(26 60 30 64)(27 57 31 61)(28 62 32 58)
(1 40 27 12)(2 33 28 13)(3 34 29 14)(4 35 30 15)(5 36 31 16)(6 37 32 9)(7 38 25 10)(8 39 26 11)(17 63 56 46)(18 64 49 47)(19 57 50 48)(20 58 51 41)(21 59 52 42)(22 60 53 43)(23 61 54 44)(24 62 55 45)
(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 53 5 49)(2 56 6 52)(3 51 7 55)(4 54 8 50)(9 63 13 59)(10 58 14 62)(11 61 15 57)(12 64 16 60)(17 32 21 28)(18 27 22 31)(19 30 23 26)(20 25 24 29)(33 42 37 46)(34 45 38 41)(35 48 39 44)(36 43 40 47)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G4H4I4J4K4L···4P8A···8H
order12222222444444444444···48···8
size11114888111122224448···84···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22D8⋊C22
kernelC42.255D4C4×M4(2)C4×D8C4×SD16C8⋊D4C82D4C42.28C22C42.29C22C83D4C8⋊Q8C23.36C23C22.26C24C42C22×C4C8C4C2
# reps11222211111122822

Matrix representation of C42.255D4 in GL6(𝔽17)

010000
100000
00501616
000590
00016120
0091012
,
400000
040000
004800
00131300
000048
00001313
,
1600000
0160000
000010
000001
00161500
001100
,
0130000
400000
00441411
00151303
0036013
00014150

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,5,0,0,9,0,0,0,5,16,1,0,0,16,9,12,0,0,0,16,0,0,12],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,13,0,0,0,0,8,13],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,1,0,0,0,0,15,1,0,0,1,0,0,0,0,0,0,1,0,0],[0,4,0,0,0,0,13,0,0,0,0,0,0,0,4,15,3,0,0,0,4,13,6,14,0,0,14,0,0,15,0,0,11,3,13,0] >;

C42.255D4 in GAP, Magma, Sage, TeX

C_4^2._{255}D_4
% in TeX

G:=Group("C4^2.255D4");
// GroupNames label

G:=SmallGroup(128,1903);
// by ID

G=gap.SmallGroup(128,1903);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,80,4037,124]);
// Polycyclic

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

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