Copied to
clipboard

G = C42.229D4order 128 = 27

211st non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.229D4, C42.345C23, D8⋊C46C2, Q8.Q817C2, D4.Q817C2, Q16⋊C45C2, D4.6(C4○D4), D4⋊D4.1C2, C4⋊C4.64C23, C4⋊C8.48C22, (C2×C8).38C23, Q8.5(C4○D4), SD16⋊C46C2, D4.2D418C2, D4.7D419C2, C8⋊C4.5C22, (C2×C4).309C24, C42.6C43C2, Q8.D418C2, (C4×D4).76C22, (C2×D8).58C22, C23.252(C2×D4), (C22×C4).449D4, (C4×Q8).73C22, C2.D8.86C22, C4.Q8.15C22, (C2×D4).404C23, C22⋊C8.22C22, (C2×Q16).56C22, (C2×Q8).376C23, D4⋊C4.30C22, C4⋊D4.165C22, C23.19D417C2, C23.20D417C2, (C2×C42).836C22, Q8⋊C4.30C22, (C2×SD16).11C22, C22.569(C22×D4), C22⋊Q8.170C22, C2.30(D8⋊C22), C23.36C233C2, (C22×C4).1025C23, C4.4D4.130C22, C42.C2.107C22, C42⋊C2.322C22, C2.110(C22.19C24), (C4×C4○D4)⋊11C2, C4.194(C2×C4○D4), (C2×C4).1219(C2×D4), (C2×C4○D4).313C22, SmallGroup(128,1843)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.229D4
C1C2C4C2×C4C42C4×D4C4×C4○D4 — C42.229D4
C1C2C2×C4 — C42.229D4
C1C22C2×C42 — C42.229D4
C1C2C2C2×C4 — C42.229D4

Generators and relations for C42.229D4
 G = < a,b,c,d | a4=b4=d2=1, c4=a2, ab=ba, cac-1=dad=ab2, cbc-1=a2b, bd=db, dcd=a2c3 >

Subgroups: 356 in 196 conjugacy classes, 88 normal (84 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×12], C22, C22 [×10], C8 [×4], C2×C4 [×6], C2×C4 [×20], D4 [×2], D4 [×9], Q8 [×2], Q8 [×3], C23, C23 [×2], C42 [×4], C42 [×4], C22⋊C4 [×8], C4⋊C4 [×4], C4⋊C4 [×6], C2×C8 [×4], D8 [×2], SD16 [×4], Q16 [×2], C22×C4 [×3], C22×C4 [×4], C2×D4 [×2], C2×D4 [×2], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×2], C22⋊C8 [×2], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×2], C4.Q8 [×2], C2.D8 [×2], C2×C42, C2×C42, C42⋊C2, C42⋊C2 [×2], C4×D4 [×3], C4×D4 [×3], C4×Q8 [×3], C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, C2×D8, C2×SD16 [×2], C2×Q16, C2×C4○D4, C42.6C4, SD16⋊C4 [×2], Q16⋊C4, D8⋊C4, D4⋊D4, D4.7D4, D4.2D4, Q8.D4, D4.Q8, Q8.Q8, C23.19D4, C23.20D4, C4×C4○D4, C23.36C23, C42.229D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], C22.19C24, D8⋊C22 [×2], C42.229D4

Smallest permutation representation of C42.229D4
On 64 points
Generators in S64
(1 57 5 61)(2 55 6 51)(3 59 7 63)(4 49 8 53)(9 47 13 43)(10 18 14 22)(11 41 15 45)(12 20 16 24)(17 34 21 38)(19 36 23 40)(25 50 29 54)(26 62 30 58)(27 52 31 56)(28 64 32 60)(33 42 37 46)(35 44 39 48)
(1 39 29 10)(2 36 30 15)(3 33 31 12)(4 38 32 9)(5 35 25 14)(6 40 26 11)(7 37 27 16)(8 34 28 13)(17 60 47 49)(18 57 48 54)(19 62 41 51)(20 59 42 56)(21 64 43 53)(22 61 44 50)(23 58 45 55)(24 63 46 52)
(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 8)(3 7)(4 6)(9 11)(12 16)(13 15)(17 41)(18 48)(19 47)(20 46)(21 45)(22 44)(23 43)(24 42)(26 32)(27 31)(28 30)(33 37)(34 36)(38 40)(49 62)(50 61)(51 60)(52 59)(53 58)(54 57)(55 64)(56 63)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4J4K···4Q4R4S4T8A8B8C8D
order122222224···44···44448888
size111144482···24···48888888

32 irreducible representations

dim11111111111111122224
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4C4○D4D8⋊C22
kernelC42.229D4C42.6C4SD16⋊C4Q16⋊C4D8⋊C4D4⋊D4D4.7D4D4.2D4Q8.D4D4.Q8Q8.Q8C23.19D4C23.20D4C4×C4○D4C23.36C23C42C22×C4D4Q8C2
# reps11211111111111122444

Matrix representation of C42.229D4 in GL6(𝔽17)

1620000
010000
0013000
0001300
0000130
0000013
,
400000
040000
000010
000001
001000
000100
,
100000
1160000
000206
00162146
00011015
00311115
,
100000
1160000
001000
0011600
000010
0000116

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

C42.229D4 in GAP, Magma, Sage, TeX

C_4^2._{229}D_4
% in TeX

G:=Group("C4^2.229D4");
// GroupNames label

G:=SmallGroup(128,1843);
// by ID

G=gap.SmallGroup(128,1843);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,521,80,4037,1027,124]);
// Polycyclic

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

׿
×
𝔽