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G = C42.104D4order 128 = 27

86th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.104D4, C8⋊C4.15C4, (C22×C4).45Q8, C23.82(C2×Q8), C42.143(C2×C4), C4.83(C4.4D4), C2.11(C428C4), C4.61(C42⋊C2), C4⋊M4(2).28C2, C4.C42.10C2, (C22×C8).386C22, (C2×C42).256C22, C2.9(M4(2).C4), C22.2(C42.C2), (C22×C4).1342C23, (C2×M4(2)).168C22, (C2×C4).46(C4⋊C4), (C2×C8).145(C2×C4), (C2×C8⋊C4).29C2, (C2×C4).1522(C2×D4), C22.100(C2×C4⋊C4), (C2×C4).560(C4○D4), (C2×C4).540(C22×C4), SmallGroup(128,570)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.104D4
C1C2C4C2×C4C22×C4C22×C8C2×C8⋊C4 — C42.104D4
C1C2C2×C4 — C42.104D4
C1C2×C4C2×C42 — C42.104D4
C1C2C2C22×C4 — C42.104D4

Generators and relations for C42.104D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=a2b-1, ab=ba, cac-1=ab2, dad-1=a-1, bc=cb, bd=db, dcd-1=a2c3 >

Subgroups: 140 in 90 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×4], C2×C8 [×8], M4(2) [×8], C22×C4, C22×C4 [×2], C8⋊C4 [×4], C4⋊C8 [×4], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C4.C42 [×4], C2×C8⋊C4, C4⋊M4(2) [×2], C42.104D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], Q8 [×2], C23, C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4○D4 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4.4D4 [×2], C42.C2 [×2], C428C4, M4(2).C4 [×2], C42.104D4

Smallest permutation representation of C42.104D4
On 64 points
Generators in S64
(1 38 31 15)(2 35 32 12)(3 40 25 9)(4 37 26 14)(5 34 27 11)(6 39 28 16)(7 36 29 13)(8 33 30 10)(17 64 48 55)(18 61 41 52)(19 58 42 49)(20 63 43 54)(21 60 44 51)(22 57 45 56)(23 62 46 53)(24 59 47 50)
(1 25 5 29)(2 26 6 30)(3 27 7 31)(4 28 8 32)(9 34 13 38)(10 35 14 39)(11 36 15 40)(12 37 16 33)(17 46 21 42)(18 47 22 43)(19 48 23 44)(20 41 24 45)(49 64 53 60)(50 57 54 61)(51 58 55 62)(52 59 56 63)
(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 58 7 60 5 62 3 64)(2 52 8 54 6 56 4 50)(9 48 15 42 13 44 11 46)(10 20 16 22 14 24 12 18)(17 38 19 36 21 34 23 40)(25 55 31 49 29 51 27 53)(26 59 32 61 30 63 28 57)(33 43 39 45 37 47 35 41)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H4I4J8A···8H8I···8P
order12222244444444448···88···8
size11112211112244444···48···8

32 irreducible representations

dim111112224
type+++++-
imageC1C2C2C2C4D4Q8C4○D4M4(2).C4
kernelC42.104D4C4.C42C2×C8⋊C4C4⋊M4(2)C8⋊C4C42C22×C4C2×C4C2
# reps141282284

Matrix representation of C42.104D4 in GL6(𝔽17)

1340000
040000
00121400
003500
00441210
00138115
,
1600000
0160000
0013000
0001300
0000130
0000013
,
1300000
0130000
0061000
0071100
00126103
0091657
,
4130000
8130000
0021336
00811210
00311415
00111000

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,4,4,0,0,0,0,0,0,12,3,4,13,0,0,14,5,4,8,0,0,0,0,12,11,0,0,0,0,10,5],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13],[13,0,0,0,0,0,0,13,0,0,0,0,0,0,6,7,12,9,0,0,10,11,6,16,0,0,0,0,10,5,0,0,0,0,3,7],[4,8,0,0,0,0,13,13,0,0,0,0,0,0,2,8,3,11,0,0,13,1,1,10,0,0,3,12,14,0,0,0,6,10,15,0] >;

C42.104D4 in GAP, Magma, Sage, TeX

C_4^2._{104}D_4
% in TeX

G:=Group("C4^2.104D4");
// GroupNames label

G:=SmallGroup(128,570);
// by ID

G=gap.SmallGroup(128,570);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,120,422,723,58,2019,248,2804,172,124]);
// Polycyclic

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

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