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G = C24.86D4order 128 = 27

41st non-split extension by C24 of D4 acting via D4/C2=C22

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

Aliases: C24.86D4, C23.8Q16, C4⋊C4.91D4, (C2×C8).45D4, (C2×Q8).91D4, C2.18(C8⋊D4), (C22×C4).148D4, C23.912(C2×D4), C2.32(D4⋊D4), C22.55(C2×Q16), C4.145(C4⋊D4), C22.4Q1624C2, C4.39(C4.4D4), (C22×C8).72C22, C22.218C22≀C2, C2.13(C8.18D4), C22.110(C4○D8), (C23×C4).274C22, C2.21(C22⋊Q16), C23.7Q8.19C2, (C22×Q8).64C22, C22.229(C4⋊D4), C22.137(C8⋊C22), (C22×C4).1446C23, C4.20(C22.D4), C2.9(C23.20D4), C2.6(C23.48D4), C2.9(C23.10D4), C22.126(C8.C22), C22.115(C22.D4), (C2×C2.D8)⋊8C2, (C2×Q8⋊C4)⋊14C2, (C2×C4).1038(C2×D4), (C2×C22⋊C8).27C2, (C2×C22⋊Q8).14C2, (C2×C4).773(C4○D4), (C2×C4⋊C4).121C22, SmallGroup(128,768)

Series: Derived Chief Lower central Upper central Jennings

C1C22×C4 — C24.86D4
C1C2C22C2×C4C22×C4C2×C4⋊C4C23.7Q8 — C24.86D4
C1C2C22×C4 — C24.86D4
C1C23C23×C4 — C24.86D4
C1C2C2C22×C4 — C24.86D4

Generators and relations for C24.86D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=f2=d, faf-1=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-1=bde3 >

Subgroups: 336 in 158 conjugacy classes, 52 normal (44 characteristic)
C1, C2 [×7], C2 [×2], C4 [×4], C4 [×7], C22 [×7], C22 [×10], C8 [×3], C2×C4 [×6], C2×C4 [×21], Q8 [×6], C23, C23 [×2], C23 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×9], C2×C8 [×2], C2×C8 [×5], C22×C4 [×2], C22×C4 [×10], C2×Q8 [×2], C2×Q8 [×5], C24, C2.C42, C22⋊C8 [×2], Q8⋊C4 [×4], C2.D8 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×3], C2×C4⋊C4, C22⋊Q8 [×4], C22×C8 [×2], C23×C4, C22×Q8, C22.4Q16, C23.7Q8, C2×C22⋊C8, C2×Q8⋊C4 [×2], C2×C2.D8, C2×C22⋊Q8, C24.86D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×8], C23, Q16 [×2], C2×D4 [×4], C4○D4 [×3], C22≀C2, C4⋊D4 [×3], C22.D4 [×2], C4.4D4, C2×Q16, C4○D8, C8⋊C22, C8.C22, C23.10D4, D4⋊D4, C22⋊Q16, C8.18D4, C8⋊D4, C23.48D4, C23.20D4, C24.86D4

Smallest permutation representation of C24.86D4
On 64 points
Generators in S64
(2 9)(4 11)(6 13)(8 15)(17 47)(18 35)(19 41)(20 37)(21 43)(22 39)(23 45)(24 33)(25 57)(27 59)(29 61)(31 63)(34 49)(36 51)(38 53)(40 55)(42 52)(44 54)(46 56)(48 50)
(1 32)(2 25)(3 26)(4 27)(5 28)(6 29)(7 30)(8 31)(9 57)(10 58)(11 59)(12 60)(13 61)(14 62)(15 63)(16 64)(17 34)(18 35)(19 36)(20 37)(21 38)(22 39)(23 40)(24 33)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 49)(48 50)
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 25)(10 26)(11 27)(12 28)(13 29)(14 30)(15 31)(16 32)(17 47)(18 48)(19 41)(20 42)(21 43)(22 44)(23 45)(24 46)(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)
(1 37 5 33)(2 19 6 23)(3 35 7 39)(4 17 8 21)(9 51 13 55)(10 48 14 44)(11 49 15 53)(12 46 16 42)(18 30 22 26)(20 28 24 32)(25 36 29 40)(27 34 31 38)(41 61 45 57)(43 59 47 63)(50 62 54 58)(52 60 56 64)

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

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

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

32 conjugacy classes

class 1 2A···2G2H2I4A4B4C4D4E4F4G···4N8A···8H
order12···2224444444···48···8
size11···1442222448···84···4

32 irreducible representations

dim11111112222222244
type++++++++++++-+-
imageC1C2C2C2C2C2C2D4D4D4D4D4C4○D4Q16C4○D8C8⋊C22C8.C22
kernelC24.86D4C22.4Q16C23.7Q8C2×C22⋊C8C2×Q8⋊C4C2×C2.D8C2×C22⋊Q8C4⋊C4C2×C8C22×C4C2×Q8C24C2×C4C23C22C22C22
# reps11112112212164411

Matrix representation of C24.86D4 in GL6(𝔽17)

100000
16160000
001000
0001600
000010
00001116
,
1600000
0160000
001000
000100
0000160
0000016
,
1600000
0160000
0016000
0001600
000010
000001
,
1600000
0160000
001000
000100
000010
000001
,
1500000
1490000
000100
0016000
00001416
0000103
,
8160000
1490000
0016000
000100
0000513
0000612

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

C24.86D4 in GAP, Magma, Sage, TeX

C_2^4._{86}D_4
% in TeX

G:=Group("C2^4.86D4");
// GroupNames label

G:=SmallGroup(128,768);
// by ID

G=gap.SmallGroup(128,768);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,448,141,422,387,394,2804,718,172]);
// Polycyclic

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

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