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G = C42.259D4order 128 = 27

241st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.259D4, C42.725C23, C82(C4○D4), C85D47C2, C8⋊D410C2, C82Q830C2, C4.4D843C2, (C4×M4(2))⋊9C2, C4⋊C4.121C23, C4.23(C8⋊C22), (C2×C8).462C23, (C2×C4).380C24, (C4×C8).185C22, C4.SD1644C2, (C22×C4).479D4, C23.267(C2×D4), C4⋊Q8.296C22, SD16⋊C425C2, (C4×Q8).97C22, C2.D8.99C22, (C2×D4).134C23, (C4×D4).100C22, C4.23(C8.C22), (C2×Q8).122C23, C8⋊C4.137C22, C4⋊D4.177C22, C41D4.158C22, (C2×C42).866C22, (C2×SD16).28C22, C22.640(C22×D4), C22⋊Q8.182C22, D4⋊C4.138C22, (C22×C4).1058C23, Q8⋊C4.131C22, C23.37C2315C2, (C2×M4(2)).288C22, C22.26C24.39C2, C2.77(C22.26C24), C4.65(C2×C4○D4), C2.47(C2×C8⋊C22), (C2×C4).1224(C2×D4), C2.47(C2×C8.C22), SmallGroup(128,1914)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.259D4
C1C2C4C2×C4C42C8⋊C4C4×M4(2) — C42.259D4
C1C2C2×C4 — C42.259D4
C1C22C2×C42 — C42.259D4
C1C2C2C2×C4 — C42.259D4

Generators and relations for C42.259D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=a-1b2, cbc-1=a2b, bd=db, dcd-1=c3 >

Subgroups: 388 in 201 conjugacy classes, 92 normal (34 characteristic)
C1, C2 [×3], C2 [×3], C4 [×6], C4 [×9], C22, C22 [×9], C8 [×4], C8 [×2], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×4], SD16 [×8], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C2.D8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C41D4, C4⋊Q8 [×3], C2×M4(2) [×2], C2×SD16 [×4], C2×C4○D4, C4×M4(2), SD16⋊C4 [×4], C8⋊D4 [×4], C4.4D8, C4.SD16, C85D4, C82Q8, C22.26C24, C23.37C23, C42.259D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, C2×C4○D4 [×2], C22.26C24, C2×C8⋊C22, C2×C8.C22, C42.259D4

Smallest permutation representation of C42.259D4
On 64 points
Generators in S64
(1 42 5 46)(2 47 6 43)(3 44 7 48)(4 41 8 45)(9 53 13 49)(10 50 14 54)(11 55 15 51)(12 52 16 56)(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 37 28 9)(3 34 29 14)(4 39 30 11)(5 36 31 16)(6 33 32 13)(7 38 25 10)(8 35 26 15)(17 63 50 48)(18 60 51 45)(19 57 52 42)(20 62 53 47)(21 59 54 44)(22 64 55 41)(23 61 56 46)(24 58 49 43)
(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 4 5 8)(2 7 6 3)(9 10 13 14)(11 16 15 12)(17 53 21 49)(18 56 22 52)(19 51 23 55)(20 54 24 50)(25 32 29 28)(26 27 30 31)(33 34 37 38)(35 40 39 36)(41 57 45 61)(42 60 46 64)(43 63 47 59)(44 58 48 62)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A···4J4K4L···4Q8A···8H
order12222224···444···48···8
size11114882···248···84···4

32 irreducible representations

dim111111111122244
type+++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22C8.C22
kernelC42.259D4C4×M4(2)SD16⋊C4C8⋊D4C4.4D8C4.SD16C85D4C82Q8C22.26C24C23.37C23C42C22×C4C8C4C4
# reps114411111122822

Matrix representation of C42.259D4 in GL6(𝔽17)

490000
4130000
0000160
0000016
001000
000100
,
400000
040000
000001
0000160
0001600
001000
,
1150000
1160000
00314413
003344
00413143
00441414
,
1620000
010000
00314413
0014141313
00413143
00131333

G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,9,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,16,0,0,0,0,0,0,16,0,0],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,1,0,0,0],[1,1,0,0,0,0,15,16,0,0,0,0,0,0,3,3,4,4,0,0,14,3,13,4,0,0,4,4,14,14,0,0,13,4,3,14],[16,0,0,0,0,0,2,1,0,0,0,0,0,0,3,14,4,13,0,0,14,14,13,13,0,0,4,13,14,3,0,0,13,13,3,3] >;

C42.259D4 in GAP, Magma, Sage, TeX

C_4^2._{259}D_4
% in TeX

G:=Group("C4^2.259D4");
// GroupNames label

G:=SmallGroup(128,1914);
// by ID

G=gap.SmallGroup(128,1914);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,723,184,521,80,4037,124]);
// 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,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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