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G = C24.159D4order 128 = 27

14th non-split extension by C24 of D4 acting via D4/C22=C2

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

Aliases: C24.159D4, C23.34SD16, C22⋊C89C4, C4⋊C4.298D4, C4.137(C4×D4), C221(C4.Q8), C4.2(C22⋊Q8), (C22×C4).49Q8, C23.72(C4⋊C4), C2.2(Q8⋊D4), C23.758(C2×D4), (C22×C4).283D4, C22.4Q1639C2, C2.2(C22⋊SD16), C22.78C22≀C2, C22.53(C2×SD16), C22.68(C8⋊C22), (C22×C8).311C22, (C23×C4).249C22, C23.7Q8.13C2, C2.9(C23.8Q8), (C22×C4).1350C23, C2.2(C23.47D4), C2.2(C23.46D4), C22.57(C8.C22), C2.11(M4(2)⋊C4), C22.82(C22.D4), (C2×C8)⋊18(C2×C4), C2.9(C2×C4.Q8), (C2×C4.Q8)⋊15C2, (C2×C4).52(C4⋊C4), (C2×C4).980(C2×D4), (C2×C4).200(C2×Q8), (C22×C4⋊C4).16C2, (C2×C22⋊C8).36C2, (C2×C4⋊C4).52C22, C22.110(C2×C4⋊C4), (C2×C4).746(C4○D4), (C2×C4).549(C22×C4), (C22×C4).272(C2×C4), SmallGroup(128,585)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C24.159D4
C1C2C22C23C22×C4C23×C4C22×C4⋊C4 — C24.159D4
C1C2C2×C4 — C24.159D4
C1C23C23×C4 — C24.159D4
C1C2C2C22×C4 — C24.159D4

Generators and relations for C24.159D4
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e4=d, f2=c, eae-1=faf-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 356 in 180 conjugacy classes, 72 normal (28 characteristic)
C1, C2 [×7], C2 [×4], C4 [×4], C4 [×8], C22 [×7], C22 [×4], C22 [×12], C8 [×4], C2×C4 [×2], C2×C4 [×6], C2×C4 [×32], C23, C23 [×6], C23 [×4], C22⋊C4 [×2], C4⋊C4 [×4], C4⋊C4 [×10], C2×C8 [×4], C2×C8 [×4], C22×C4 [×2], C22×C4 [×4], C22×C4 [×16], C24, C2.C42, C22⋊C8 [×4], C4.Q8 [×4], C2×C22⋊C4, C2×C4⋊C4 [×2], C2×C4⋊C4 [×2], C2×C4⋊C4 [×5], C22×C8 [×2], C23×C4, C23×C4, C22.4Q16 [×2], C23.7Q8, C2×C22⋊C8, C2×C4.Q8 [×2], C22×C4⋊C4, C24.159D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C4⋊C4 [×4], SD16 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C4.Q8 [×4], C2×C4⋊C4, C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C2×SD16 [×2], C8⋊C22, C8.C22, C23.8Q8, C2×C4.Q8, M4(2)⋊C4, Q8⋊D4, C22⋊SD16, C23.46D4, C23.47D4, C24.159D4

Smallest permutation representation of C24.159D4
On 64 points
Generators in S64
(1 40)(2 58)(3 34)(4 60)(5 36)(6 62)(7 38)(8 64)(9 31)(10 23)(11 25)(12 17)(13 27)(14 19)(15 29)(16 21)(18 41)(20 43)(22 45)(24 47)(26 48)(28 42)(30 44)(32 46)(33 52)(35 54)(37 56)(39 50)(49 63)(51 57)(53 59)(55 61)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 49)(8 50)(9 45)(10 46)(11 47)(12 48)(13 41)(14 42)(15 43)(16 44)(17 26)(18 27)(19 28)(20 29)(21 30)(22 31)(23 32)(24 25)(33 58)(34 59)(35 60)(36 61)(37 62)(38 63)(39 64)(40 57)
(1 40)(2 33)(3 34)(4 35)(5 36)(6 37)(7 38)(8 39)(9 31)(10 32)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 48)(18 41)(19 42)(20 43)(21 44)(22 45)(23 46)(24 47)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(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 48 40 17)(2 43 33 20)(3 46 34 23)(4 41 35 18)(5 44 36 21)(6 47 37 24)(7 42 38 19)(8 45 39 22)(9 64 31 50)(10 59 32 53)(11 62 25 56)(12 57 26 51)(13 60 27 54)(14 63 28 49)(15 58 29 52)(16 61 30 55)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A4B4C4D4E···4N4O4P4Q4R8A···8H
order12···2222244444···444448···8
size11···1222222224···488884···4

38 irreducible representations

dim111111122222244
type++++++++-++-
imageC1C2C2C2C2C2C4D4D4Q8D4C4○D4SD16C8⋊C22C8.C22
kernelC24.159D4C22.4Q16C23.7Q8C2×C22⋊C8C2×C4.Q8C22×C4⋊C4C22⋊C8C4⋊C4C22×C4C22×C4C24C2×C4C23C22C22
# reps121121841214811

Matrix representation of C24.159D4 in GL5(𝔽17)

10000
016000
04100
000160
000016
,
10000
016000
001600
00010
00001
,
160000
016000
001600
000160
000016
,
10000
01000
00100
000160
000016
,
10000
01900
0131600
00007
00057
,
40000
0131500
00400
00040
000413

G:=sub<GL(5,GF(17))| [1,0,0,0,0,0,16,4,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,1,0,0,0,0,0,1],[16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,1,13,0,0,0,9,16,0,0,0,0,0,0,5,0,0,0,7,7],[4,0,0,0,0,0,13,0,0,0,0,15,4,0,0,0,0,0,4,4,0,0,0,0,13] >;

C24.159D4 in GAP, Magma, Sage, TeX

C_2^4._{159}D_4
% in TeX

G:=Group("C2^4.159D4");
// GroupNames label

G:=SmallGroup(128,585);
// by ID

G=gap.SmallGroup(128,585);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,680,422,2019,1018,248]);
// Polycyclic

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

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