C2^4.84D4 - GroupNames

G = C₂⁴.₈₄D₄ order 128 = 2⁷

39^th 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(C₈⋊₈D₄), C₂.₁₃(C₈⋊₂D₄), C₂³.₉₁₀(C₂×D₄), (C₂²×C₄).₁₄₆D₄, C₂.₃₁(D₄⋊D₄), C₄.₁₄₃(C₄⋊D₄), C₂².₄Q₁₆⋊₂₂C₂, C₄.₃₇(C₄.₄D₄), (C₂²×C₈).₇₀C₂², C₂³.₇Q₈⋊₁₀C₂, 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

Derived series

C₁ — C₂²×C₄ — C₂⁴.₈₄D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂³.₇Q₈ — C₂⁴.₈₄D₄

Lower central

C₁ — C₂ — C₂²×C₄ — C₂⁴.₈₄D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂⁴.₈₄D₄

Jennings

C₁ — C₂ — C₂ — C₂²×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, C₂, C₄, C₄, C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂²⋊C₈, D₄⋊C₄, C₄.Q₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₂²×C₈, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂².₄Q₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂×C₄.Q₈, C₂×C₄⋊D₄, C₂⁴.₈₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₄○D₄, C₂²≀C₂, C₄⋊D₄, C₂².D₄, C₄.₄D₄, C₂×SD₁₆, C₄○D₈, C₈⋊C₂², C₂³.₁₀D₄, D₄⋊D₄, C₂²⋊SD₁₆, C₈⋊₈D₄, C₈⋊₂D₄, 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 47)(10 55)(11 41)(12 49)(13 43)(14 51)(15 45)(16 53)(17 46)(18 54)(19 48)(20 56)(21 42)(22 50)(23 44)(24 52)(26 34)(28 36)(30 38)(32 40)(33 64)(35 58)(37 60)(39 62)

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

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

(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 44)(3 7)(4 42)(6 48)(8 46)(9 25)(10 16)(11 31)(12 14)(13 29)(15 27)(17 19)(18 64)(20 62)(21 23)(22 60)(24 58)(26 32)(28 30)(33 37)(34 49)(36 55)(38 53)(40 51)(43 47)(50 54)(57 63)(59 61)

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

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

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

32 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	4F	4G	···	4L	8A	···	8H
order	1	2	···	2	2	2	2	2	4	4	4	4	4	4	4	···	4	8	···	8
size	1	1	···	1	4	4	8	8	2	2	2	2	4	4	8	···	8	4	···	4

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

SD₁₆

C₄○D₈

C₈⋊C₂²

kernel

C₂⁴.₈₄D₄

C₂².₄Q₁₆

C₂³.₇Q₈

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂×C₄.Q₈

C₂×C₄⋊D₄

C₄⋊C₄

C₂×C₈

C₂²×C₄

C₂×D₄

C₂⁴

C₂×C₄

C₂³

C₂²

# reps

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

0	4	0	0	0	0
13	0	0	0	0	0
0	0	16	2	0	0
0	0	0	1	0	0
0	0	0	0	16	15
0	0	0	0	0	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	3	0	0	0	0
14	14	0	0	0	0
0	0	16	2	0	0
0	0	0	1	0	0
0	0	0	0	13	0
0	0	0	0	4	4

1	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	1	16	0	0
0	0	0	0	1	0
0	0	0	0	16	16

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

G = C24.84D4 order 128 = 27

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

G = C₂⁴.₈₄D₄ order 128 = 2⁷

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