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G = C42.54D4order 128 = 27

36th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.54D4, C42.618C23, Q8⋊C837C2, C4⋊Q8.13C4, C4.6(C8○D4), C22⋊Q8.8C4, C42.68(C2×C4), (C4×C8).10C22, (C4×Q8).8C22, C4⋊C8.202C22, (C22×C4).208D4, C4.130(C8.C22), C23.50(C22⋊C4), C42.6C4.16C2, (C2×C42).171C22, C42.12C4.19C2, C2.6(C23.38D4), C2.14(C42⋊C22), C23.37C23.4C2, C4⋊C4.58(C2×C4), (C2×Q8).50(C2×C4), (C2×C4).1454(C2×D4), (C2×C4).84(C22⋊C4), (C22×C4).193(C2×C4), (C2×C4).323(C22×C4), C22.173(C2×C22⋊C4), C2.23((C22×C8)⋊C2), SmallGroup(128,229)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.54D4
C1C2C22C2×C4C42C2×C42C23.37C23 — C42.54D4
C1C2C2×C4 — C42.54D4
C1C2×C4C2×C42 — C42.54D4
C1C22C22C42 — C42.54D4

Generators and relations for C42.54D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=a-1, dad-1=ab2, bc=cb, bd=db, dcd-1=b-1c3 >

Subgroups: 180 in 101 conjugacy classes, 46 normal (26 characteristic)
C1, C2 [×3], C2, C4 [×4], C4 [×8], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×9], Q8 [×6], C23, C42 [×4], C42 [×2], C22⋊C4 [×2], C4⋊C4 [×2], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C2×Q8 [×2], C2×Q8, C4×C8 [×2], C8⋊C4, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C42⋊C2, C4×Q8 [×2], C4×Q8, C22⋊Q8 [×2], C22⋊Q8, C42.C2, C4⋊Q8 [×2], Q8⋊C8 [×4], C42.12C4, C42.6C4, C23.37C23, C42.54D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], C8.C22 [×2], (C22×C8)⋊C2, C23.38D4, C42⋊C22, C42.54D4

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

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

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

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

32 conjugacy classes

class 1 2A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A···8H8I8J8K8L
order122224444444444444448···88888
size111141111222244488884···48888

32 irreducible representations

dim111111122244
type+++++++-
imageC1C2C2C2C2C4C4D4D4C8○D4C8.C22C42⋊C22
kernelC42.54D4Q8⋊C8C42.12C4C42.6C4C23.37C23C22⋊Q8C4⋊Q8C42C22×C4C4C4C2
# reps141114422822

Matrix representation of C42.54D4 in GL6(𝔽17)

100000
0160000
0001600
001000
0055016
0051210
,
1300000
0130000
004000
000400
000040
000004
,
800000
080000
001010013
00710130
0020710
0001577
,
080000
800000
002000
0001500
007702
0010720

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,0,1,5,5,0,0,16,0,5,12,0,0,0,0,0,1,0,0,0,0,16,0],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,10,7,2,0,0,0,10,10,0,15,0,0,0,13,7,7,0,0,13,0,10,7],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,2,0,7,10,0,0,0,15,7,7,0,0,0,0,0,2,0,0,0,0,2,0] >;

C42.54D4 in GAP, Magma, Sage, TeX

C_4^2._{54}D_4
% in TeX

G:=Group("C4^2.54D4");
// GroupNames label

G:=SmallGroup(128,229);
// by ID

G=gap.SmallGroup(128,229);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1430,1059,184,1123,570,136,172]);
// Polycyclic

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

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