Copied to
clipboard

G = C42.279D4order 128 = 27

261st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.279D4, C42.413C23, C4.512- 1+4, C83Q820C2, D42Q835C2, C4.4D828C2, (C2×C4).57SD16, C4.72(C2×SD16), C4⋊C4.166C23, C4⋊C8.337C22, C4.28(C8⋊C22), (C4×C8).267C22, (C2×C4).425C24, (C2×C8).331C23, (C22×C4).508D4, C23.698(C2×D4), C4⋊Q8.309C22, C4.Q8.83C22, (C4×D4).112C22, (C2×D4).174C23, C2.24(C22×SD16), C22.39(C2×SD16), C42.12C444C2, C4⋊D4.197C22, C41D4.170C22, C23.46D430C2, C22⋊C8.220C22, (C2×C42).886C22, C22.685(C22×D4), D4⋊C4.110C22, (C22×C4).1090C23, C22.26C24.44C2, C2.73(C23.38C23), (C2×C4⋊Q8)⋊43C2, (C2×C4).869(C2×D4), C2.59(C2×C8⋊C22), (C2×C4⋊C4).646C22, SmallGroup(128,1959)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.279D4
C1C2C4C2×C4C22×C4C2×C4⋊C4C2×C4⋊Q8 — C42.279D4
C1C2C2×C4 — C42.279D4
C1C22C2×C42 — C42.279D4
C1C2C2C2×C4 — C42.279D4

Generators and relations for C42.279D4
 G = < a,b,c,d | a4=b4=d2=1, c4=b2, ab=ba, ac=ca, dad=a-1, cbc-1=a2b-1, dbd=a2b, dcd=a2b2c3 >

Subgroups: 412 in 202 conjugacy classes, 96 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×8], C22, C22 [×2], C22 [×8], C8 [×4], C2×C4 [×6], C2×C4 [×4], C2×C4 [×18], D4 [×12], Q8 [×10], C23, C23 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×6], C4⋊C4 [×7], C2×C8 [×4], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×9], C4○D4 [×4], C4×C8 [×2], C22⋊C8 [×2], D4⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×8], C2×C42, C2×C4⋊C4 [×2], C2×C4⋊C4, C4×D4 [×2], C4×D4, C4⋊D4 [×2], C4⋊D4, C4.4D4, C41D4, C4⋊Q8, C4⋊Q8 [×4], C4⋊Q8 [×2], C22×Q8, C2×C4○D4, C42.12C4, D42Q8 [×4], C23.46D4 [×4], C4.4D8 [×2], C83Q8 [×2], C2×C4⋊Q8, C22.26C24, C42.279D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], SD16 [×4], C2×D4 [×6], C24, C2×SD16 [×6], C8⋊C22 [×2], C22×D4, 2- 1+4 [×2], C23.38C23, C22×SD16, C2×C8⋊C22, C42.279D4

Smallest permutation representation of C42.279D4
On 64 points
Generators in S64
(1 56 25 59)(2 49 26 60)(3 50 27 61)(4 51 28 62)(5 52 29 63)(6 53 30 64)(7 54 31 57)(8 55 32 58)(9 43 38 24)(10 44 39 17)(11 45 40 18)(12 46 33 19)(13 47 34 20)(14 48 35 21)(15 41 36 22)(16 42 37 23)
(1 12 5 16)(2 38 6 34)(3 14 7 10)(4 40 8 36)(9 30 13 26)(11 32 15 28)(17 61 21 57)(18 55 22 51)(19 63 23 59)(20 49 24 53)(25 33 29 37)(27 35 31 39)(41 62 45 58)(42 56 46 52)(43 64 47 60)(44 50 48 54)
(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)
(2 32)(3 7)(4 30)(6 28)(8 26)(9 15)(10 35)(11 13)(12 33)(14 39)(16 37)(17 21)(18 47)(20 45)(22 43)(24 41)(27 31)(34 40)(36 38)(44 48)(49 55)(50 57)(51 53)(52 63)(54 61)(56 59)(58 60)(62 64)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K···4P8A···8H
order122222224···4444···48···8
size111122882···2448···84···4

32 irreducible representations

dim1111111122244
type+++++++++++-
imageC1C2C2C2C2C2C2C2D4D4SD16C8⋊C222- 1+4
kernelC42.279D4C42.12C4D42Q8C23.46D4C4.4D8C83Q8C2×C4⋊Q8C22.26C24C42C22×C4C2×C4C4C4
# reps1144221122822

Matrix representation of C42.279D4 in GL6(𝔽17)

010000
1600000
0016000
0001600
0000160
0000016
,
0160000
100000
0001015
0016020
0000016
000010
,
5120000
550000
0041122
0064152
00411136
00641113
,
100000
0160000
001000
0001600
0010160
0001601

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

C42.279D4 in GAP, Magma, Sage, TeX

C_4^2._{279}D_4
% in TeX

G:=Group("C4^2.279D4");
// GroupNames label

G:=SmallGroup(128,1959);
// by ID

G=gap.SmallGroup(128,1959);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,120,758,219,436,675,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽