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G = C42.257D4order 128 = 27

239th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.257D4, C42.392C23, C8⋊Q810C2, C88(C4○D4), C83D48C2, C87D428C2, C88D416C2, D8⋊C416C2, C4⋊C4.119C23, (C2×C8).280C23, (C2×C4).378C24, (C2×D8).65C22, (C4×D4).98C22, C23.397(C2×D4), (C22×C4).477D4, C4⋊Q8.294C22, SD16⋊C423C2, (C4×Q8).95C22, C2.D8.97C22, C4.Q8.30C22, (C2×D4).132C23, C22.1(C8⋊C22), (C2×Q8).120C23, C8⋊C4.135C22, C41D4.157C22, C4⋊D4.175C22, (C2×C42).864C22, (C22×C8).280C22, (C2×SD16).26C22, C22.638(C22×D4), C22⋊Q8.180C22, D4⋊C4.136C22, C2.44(D8⋊C22), (C22×C4).1574C23, C23.36C237C2, C22.26C2414C2, Q8⋊C4.129C22, C4.4D4.146C22, C42.C2.123C22, C42.28C2235C2, C42.29C2222C2, C2.75(C22.26C24), (C2×C8⋊C4)⋊11C2, C4.63(C2×C4○D4), (C2×C4).702(C2×D4), C2.46(C2×C8⋊C22), SmallGroup(128,1912)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.257D4
C1C2C4C2×C4C42C8⋊C4C2×C8⋊C4 — C42.257D4
C1C2C2×C4 — C42.257D4
C1C22C2×C42 — C42.257D4
C1C2C2C2×C4 — C42.257D4

Generators and relations for C42.257D4
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, ac=ca, dad-1=ab2, cbc-1=a2b, bd=db, dcd-1=c3 >

Subgroups: 412 in 205 conjugacy classes, 90 normal (42 characteristic)
C1, C2 [×3], C2 [×5], C4 [×2], C4 [×10], C22, C22 [×2], C22 [×11], 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], C2×C8 [×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], C8⋊C4 [×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, C22×C8 [×2], C2×D8 [×2], C2×SD16 [×2], C2×C4○D4, C2×C8⋊C4, SD16⋊C4 [×2], D8⋊C4 [×2], C88D4 [×2], C87D4 [×2], C42.28C22, C42.29C22, C83D4, C8⋊Q8, C23.36C23, C22.26C24, C42.257D4
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, C42.257D4

Smallest permutation representation of C42.257D4
On 64 points
Generators in S64
(1 45 5 41)(2 46 6 42)(3 47 7 43)(4 48 8 44)(9 50 13 54)(10 51 14 55)(11 52 15 56)(12 53 16 49)(17 39 21 35)(18 40 22 36)(19 33 23 37)(20 34 24 38)(25 59 29 63)(26 60 30 64)(27 61 31 57)(28 62 32 58)
(1 33 27 9)(2 38 28 14)(3 35 29 11)(4 40 30 16)(5 37 31 13)(6 34 32 10)(7 39 25 15)(8 36 26 12)(17 63 52 47)(18 60 53 44)(19 57 54 41)(20 62 55 46)(21 59 56 43)(22 64 49 48)(23 61 50 45)(24 58 51 42)
(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 60 13 64)(10 63 14 59)(11 58 15 62)(12 61 16 57)(17 28 21 32)(18 31 22 27)(19 26 23 30)(20 29 24 25)(33 44 37 48)(34 47 38 43)(35 42 39 46)(36 45 40 41)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G2H4A···4H4I4J4K···4O8A···8H
order1222222224···4444···48···8
size1111228882···2448···84···4

32 irreducible representations

dim11111111111122244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C8⋊C22D8⋊C22
kernelC42.257D4C2×C8⋊C4SD16⋊C4D8⋊C4C88D4C87D4C42.28C22C42.29C22C83D4C8⋊Q8C23.36C23C22.26C24C42C22×C4C8C22C2
# reps11222211111122822

Matrix representation of C42.257D4 in GL6(𝔽17)

480000
13130000
004000
000400
000040
000004
,
1300000
0130000
0000115
0000116
0016200
0016100
,
16150000
110000
00152710
00160120
00710215
0012010
,
1600000
110000
0001108
0014040
000806
004030

G:=sub<GL(6,GF(17))| [4,13,0,0,0,0,8,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,16,16,0,0,0,0,2,1,0,0,1,1,0,0,0,0,15,16,0,0],[16,1,0,0,0,0,15,1,0,0,0,0,0,0,15,16,7,12,0,0,2,0,10,0,0,0,7,12,2,1,0,0,10,0,15,0],[16,1,0,0,0,0,0,1,0,0,0,0,0,0,0,14,0,4,0,0,11,0,8,0,0,0,0,4,0,3,0,0,8,0,6,0] >;

C42.257D4 in GAP, Magma, Sage, TeX

C_4^2._{257}D_4
% in TeX

G:=Group("C4^2.257D4");
// GroupNames label

G:=SmallGroup(128,1912);
// by ID

G=gap.SmallGroup(128,1912);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,184,521,80,4037,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,a*c=c*a,d*a*d^-1=a*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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