Copied to
clipboard

G = C42.106D4order 128 = 27

88th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.106D4, C8.5(C4⋊C4), (C2×C8).11Q8, C8⋊C4.6C4, (C2×C8).109D4, C4.47(C4⋊Q8), C22.2(C4⋊Q8), C23.84(C2×Q8), (C22×C4).46Q8, C4.47(C41D4), C42.145(C2×C4), C2.10(C429C4), C4⋊M4(2).30C2, (C2×C42).260C22, (C22×C8).221C22, (C22×C4).1348C23, C2.10(M4(2).C4), (C2×M4(2)).170C22, C4.38(C2×C4⋊C4), (C2×C8).64(C2×C4), (C2×C8⋊C4).5C2, (C2×C4).49(C4⋊C4), (C2×C4).733(C2×D4), (C2×C4).198(C2×Q8), C22.107(C2×C4⋊C4), (C2×C8.C4).12C2, (C2×C4).547(C22×C4), SmallGroup(128,581)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.106D4
C1C2C4C2×C4C22×C4C2×C42C2×C8⋊C4 — C42.106D4
C1C2C2×C4 — C42.106D4
C1C2×C4C2×C42 — C42.106D4
C1C2C2C22×C4 — C42.106D4

Generators and relations for C42.106D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b-1, ab=ba, cac-1=ab2, dad-1=a-1b2, bc=cb, bd=db, dcd-1=c3 >

Subgroups: 156 in 106 conjugacy classes, 68 normal (12 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C4 [×4], C22, C22 [×2], C22 [×2], C8 [×8], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×2], C23, C42 [×4], C2×C8 [×12], C2×C8 [×4], M4(2) [×8], C22×C4, C22×C4 [×2], C8⋊C4 [×4], C4⋊C8 [×4], C8.C4 [×8], C2×C42, C22×C8 [×2], C2×M4(2) [×4], C2×C8⋊C4, C4⋊M4(2) [×2], C2×C8.C4 [×4], C42.106D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×6], C23, C4⋊C4 [×12], C22×C4, C2×D4 [×3], C2×Q8 [×3], C2×C4⋊C4 [×3], C41D4, C4⋊Q8 [×3], C429C4, M4(2).C4 [×2], C42.106D4

Smallest permutation representation of C42.106D4
On 64 points
Generators in S64
(1 40 27 16)(2 37 28 13)(3 34 29 10)(4 39 30 15)(5 36 31 12)(6 33 32 9)(7 38 25 14)(8 35 26 11)(17 43 55 58)(18 48 56 63)(19 45 49 60)(20 42 50 57)(21 47 51 62)(22 44 52 59)(23 41 53 64)(24 46 54 61)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 43 45 47)(42 44 46 48)(49 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 64)
(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 47 3 45 5 43 7 41)(2 42 4 48 6 46 8 44)(9 50 11 56 13 54 15 52)(10 53 12 51 14 49 16 55)(17 34 23 36 21 38 19 40)(18 37 24 39 22 33 20 35)(25 64 27 62 29 60 31 58)(26 59 28 57 30 63 32 61)

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim1111122224
type++++++--
imageC1C2C2C2C4D4D4Q8Q8M4(2).C4
kernelC42.106D4C2×C8⋊C4C4⋊M4(2)C2×C8.C4C8⋊C4C42C2×C8C2×C8C22×C4C2
# reps1124824424

Matrix representation of C42.106D4 in GL6(𝔽17)

0160000
100000
000100
0016000
0000016
000010
,
1600000
0160000
0013000
0001300
0000130
0000013
,
1600000
0160000
000200
002000
000008
000080
,
1670000
710000
000008
000080
000900
009000

G:=sub<GL(6,GF(17))| [0,1,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0],[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],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,8,0,0,0,0,8,0],[16,7,0,0,0,0,7,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,9,0,0,0,0,8,0,0,0,0,8,0,0,0] >;

C42.106D4 in GAP, Magma, Sage, TeX

C_4^2._{106}D_4
% in TeX

G:=Group("C4^2.106D4");
// GroupNames label

G:=SmallGroup(128,581);
// by ID

G=gap.SmallGroup(128,581);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,64,422,723,100,2019,248,124]);
// Polycyclic

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

׿
×
𝔽