Copied to
clipboard

G = C24.84D4order 128 = 27

39th non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.84D4, C23.17SD16, (C2×C8)⋊21D4, C4⋊C4.89D4, (C2×D4).100D4, C2.17(C88D4), C2.13(C82D4), C23.910(C2×D4), (C22×C4).146D4, C2.31(D4⋊D4), C4.143(C4⋊D4), C22.4Q1622C2, C4.37(C4.4D4), (C22×C8).70C22, C23.7Q810C2, C22.95(C2×SD16), C22.216C22≀C2, C2.21(C22⋊SD16), C22.108(C4○D8), (C23×C4).272C22, (C22×D4).77C22, C22.227(C4⋊D4), C22.136(C8⋊C22), (C22×C4).1444C23, C4.18(C22.D4), C2.9(C23.19D4), C2.6(C23.46D4), C2.7(C23.10D4), C22.113(C22.D4), (C2×C4.Q8)⋊20C2, (C2×C22⋊C8)⋊32C2, (C2×D4⋊C4)⋊13C2, (C2×C4⋊D4).14C2, (C2×C4).1036(C2×D4), (C2×C4).771(C4○D4), (C2×C4⋊C4).119C22, SmallGroup(128,766)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.84D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.84D4
C1C2C22×C4 — C24.84D4
C1C23C23×C4 — C24.84D4
C1C2C2C22×C4 — C24.84D4

Generators and relations for C24.84D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=f2=1, e4=d, faf=ab=ba, ac=ca, ad=da, eae-1=abc, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=be3 >

Subgroups: 464 in 184 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×5], C22 [×7], C22 [×20], C8 [×3], C2×C4 [×6], C2×C4 [×17], D4 [×14], C23, C23 [×2], C23 [×14], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×9], C2×D4 [×2], C2×D4 [×13], C24, C24, C2.C42, C22⋊C8 [×2], D4⋊C4 [×4], C4.Q8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C4⋊D4 [×4], C22×C8 [×2], C23×C4, C22×D4, C22×D4, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×D4⋊C4 [×2], C2×C4.Q8, C2×C4⋊D4, C24.84D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, SD16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×SD16, C4○D8, C8⋊C22 [×2], C23.10D4, D4⋊D4, C22⋊SD16, C88D4, C82D4, C23.46D4, C23.19D4, C24.84D4

Smallest permutation representation of C24.84D4
On 64 points
Generators in S64
(1 31)(2 63)(3 25)(4 57)(5 27)(6 59)(7 29)(8 61)(9 51)(10 41)(11 53)(12 43)(13 55)(14 45)(15 49)(16 47)(17 42)(18 54)(19 44)(20 56)(21 46)(22 50)(23 48)(24 52)(26 34)(28 36)(30 38)(32 40)(33 64)(35 58)(37 60)(39 62)
(1 54)(2 55)(3 56)(4 49)(5 50)(6 51)(7 52)(8 53)(9 59)(10 60)(11 61)(12 62)(13 63)(14 64)(15 57)(16 58)(17 30)(18 31)(19 32)(20 25)(21 26)(22 27)(23 28)(24 29)(33 45)(34 46)(35 47)(36 48)(37 41)(38 42)(39 43)(40 44)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 41)(8 42)(9 28)(10 29)(11 30)(12 31)(13 32)(14 25)(15 26)(16 27)(17 61)(18 62)(19 63)(20 64)(21 57)(22 58)(23 59)(24 60)(33 56)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 21)(18 22)(19 23)(20 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(41 45)(42 46)(43 47)(44 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 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)
(2 49)(3 7)(4 55)(6 53)(8 51)(9 11)(10 64)(12 62)(13 15)(14 60)(16 58)(17 23)(18 31)(19 21)(20 29)(22 27)(24 25)(26 32)(28 30)(33 37)(34 44)(36 42)(38 48)(40 46)(41 45)(52 56)(57 63)(59 61)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E4F4G···4L8A···8H
order12···222224444444···48···8
size11···144882222448···84···4

32 irreducible representations

dim1111111222222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D4D4D4D4C4○D4SD16C4○D8C8⋊C22
kernelC24.84D4C22.4Q16C23.7Q8C2×C22⋊C8C2×D4⋊C4C2×C4.Q8C2×C4⋊D4C4⋊C4C2×C8C22×C4C2×D4C24C2×C4C23C22C22
# reps1111211221216442

Matrix representation of C24.84D4 in GL6(𝔽17)

040000
1300000
0016200
000100
00001615
000001
,
1600000
0160000
0016000
0001600
0000160
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
1430000
14140000
0016200
000100
0000130
000044
,
100000
0160000
001000
0011600
000010
00001616

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

C24.84D4 in GAP, Magma, Sage, TeX

C_2^4._{84}D_4
% in TeX

G:=Group("C2^4.84D4");
// GroupNames label

G:=SmallGroup(128,766);
// by ID

G=gap.SmallGroup(128,766);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,141,422,387,394,2804,718,172]);
// Polycyclic

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

׿
×
𝔽