C2^4.121D4

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

Aliases: C₂⁴.₁₂₁D₄, C₄.₃2₊⁽¹⁺⁴⁾, C₈⋊₈D₄⋊₁C₂, D₄⋊D₄⋊₁C₂, C₂²⋊D₈⋊₅C₂, C₈⋊₇D₄⋊₁₇C₂, (C₂×D₈)⋊₃C₂², C₂.D₈⋊₄C₂², C₄.Q₈⋊₃₂C₂², C₂²⋊SD₁₆⋊₂₉C₂, C₄⋊D₄⋊₅₇C₂², C₄⋊C₄.₁₂₇C₂³, C₂²⋊C₈⋊₆₀C₂², (C₂×C₄).₃₈₆C₂⁴, (C₂×C₈).₃₂₀C₂³, (C₂²×C₈)⋊₁₆C₂², Q₈⋊C₄⋊₁C₂², C₂³.₂₇₁(C₂×D₄), (C₂²×C₄).₄₈₄D₄, C₂²⋊Q₈⋊₆₉C₂², D₄⋊C₄⋊₄₃C₂², C₂².₁₉C₂⁴⋊₉C₂, (C₂×SD₁₆)⋊₃₉C₂², (C₂×D₄).₁₃₉C₂³, C₂².₃₁(C₄○D₈), C₂³.₄₈D₄⋊₄C₂, C₂³.₁₉D₄⋊₁C₂, C₂².₂(C₈⋊C₂²), (C₂×Q₈).₁₂₆C₂³, C₄²⋊C₂⋊₁₆C₂², C₂³.₄₆D₄⋊₂₉C₂, C₂.₆₇(C₂³⋊₃D₄), (C₂³×C₄).₅₆₆C₂², C₂².₆₄₆(C₂²×D₄), (C₂²×C₄).₁₀₆₄C₂³, (C₂²×D₄).₃₈₁C₂², C₂.₃₈(C₂×C₄○D₈), (C₂×C₄⋊D₄)⋊₅₀C₂, (C₂×C₂²⋊C₈)⋊₂₈C₂, (C₂×C₄).₇₀₄(C₂×D₄), (C₂×C₄○D₄)⋊₈C₂², C₂.₄₉(C₂×C₈⋊C₂²), (C₂×C₄⋊C₄).₆₃₆C₂², SmallGroup(128,1920)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₂₁D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×D₄ — C₂×C₄⋊D₄ — C₂⁴.₁₂₁D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₂₁D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₂₁D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₂₁D₄

Subgroups: 556 in 236 conjugacy classes, 88 normal (42 characteristic)
C₁, C₂^[×3], C₂^[×8], C₄^[×2], C₄^[×8], C₂², C₂²^[×4], C₂²^[×24], C₈^[×4], C₂×C₄^[×4], C₂×C₄^[×18], D₄^[×22], Q₈^[×2], C₂³^[×3], C₂³^[×14], C₄², C₂²⋊C₄^[×9], C₄⋊C₄^[×2], C₄⋊C₄^[×2], C₄⋊C₄^[×3], C₂×C₈^[×4], C₂×C₈^[×2], D₈^[×2], SD₁₆^[×2], C₂²×C₄^[×6], C₂²×C₄^[×4], C₂×D₄, C₂×D₄^[×2], C₂×D₄^[×14], C₂×Q₈, C₄○D₄^[×2], C₂⁴, C₂⁴, C₂²⋊C₈^[×2], C₂²⋊C₈^[×2], D₄⋊C₄^[×6], Q₈⋊C₄^[×2], C₄.Q₈^[×2], C₂.D₈^[×2], C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄^[×2], C₂²≀C₂, C₄⋊D₄^[×2], C₄⋊D₄^[×4], C₄⋊D₄^[×2], C₂²⋊Q₈^[×2], C₂².D₄, C₂²×C₈^[×2], C₂×D₈^[×2], C₂×SD₁₆^[×2], C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂×C₄○D₄, C₂×C₂²⋊C₈, C₂²⋊D₈, D₄⋊D₄^[×2], C₂²⋊SD₁₆, C₈⋊₈D₄^[×2], C₈⋊₇D₄^[×2], C₂³.₄₆D₄, C₂³.₁₉D₄^[×2], C₂³.₄₈D₄, C₂×C₄⋊D₄, C₂².₁₉C₂⁴, C₂⁴.₁₂₁D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₂⁴, C₄○D₈^[×2], C₈⋊C₂²^[×2], C₂²×D₄, 2₊⁽¹⁺⁴⁾^[×2], C₂³⋊₃D₄, C₂×C₄○D₈, C₂×C₈⋊C₂², C₂⁴.₁₂₁D₄

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, ac=ca, ad=da, ae=ea, faf=acd, ebe^-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=de³ >

Smallest permutation representation
►On 32 points

Generators in S₃₂

(1 24)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 32)(10 25)(11 26)(12 27)(13 28)(14 29)(15 30)(16 31)

(2 30)(4 32)(6 26)(8 28)(9 19)(11 21)(13 23)(15 17)

(1 29)(2 30)(3 31)(4 32)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)

(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)

(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)

(1 2)(3 8)(4 7)(5 6)(9 18)(10 17)(11 24)(12 23)(13 22)(14 21)(15 20)(16 19)(25 26)(27 32)(28 31)(29 30)

G:=sub<Sym(32)| (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (2,30)(4,32)(6,26)(8,28)(9,19)(11,21)(13,23)(15,17), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (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), (1,2)(3,8)(4,7)(5,6)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,26)(27,32)(28,31)(29,30)>;

G:=Group( (1,24)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,32)(10,25)(11,26)(12,27)(13,28)(14,29)(15,30)(16,31), (2,30)(4,32)(6,26)(8,28)(9,19)(11,21)(13,23)(15,17), (1,29)(2,30)(3,31)(4,32)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,21)(12,22)(13,23)(14,24)(15,17)(16,18), (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), (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), (1,2)(3,8)(4,7)(5,6)(9,18)(10,17)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(25,26)(27,32)(28,31)(29,30) );

G=PermutationGroup([(1,24),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,32),(10,25),(11,26),(12,27),(13,28),(14,29),(15,30),(16,31)], [(2,30),(4,32),(6,26),(8,28),(9,19),(11,21),(13,23),(15,17)], [(1,29),(2,30),(3,31),(4,32),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,21),(12,22),(13,23),(14,24),(15,17),(16,18)], [(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)], [(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)], [(1,2),(3,8),(4,7),(5,6),(9,18),(10,17),(11,24),(12,23),(13,22),(14,21),(15,20),(16,19),(25,26),(27,32),(28,31),(29,30)])

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

4	9	0	0	0	0
4	13	0	0	0	0
0	0	1	2	0	0
0	0	0	16	0	0
0	0	2	2	1	2
0	0	15	15	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	15	15	0	16

1	0	0	0	0	0
0	1	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

0	6	0	0	0	0
14	6	0	0	0	0
0	0	0	0	1	0
0	0	1	1	0	1
0	0	1	0	0	0
0	0	16	0	16	16

11	6	0	0	0	0
14	6	0	0	0	0
0	0	0	0	1	0
0	0	16	16	16	16
0	0	1	0	0	0
0	0	0	0	0	1

G:=sub<GL(6,GF(17))| [4,4,0,0,0,0,9,13,0,0,0,0,0,0,1,0,2,15,0,0,2,16,2,15,0,0,0,0,1,0,0,0,0,0,2,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,15,0,0,0,1,0,15,0,0,0,0,16,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,14,0,0,0,0,6,6,0,0,0,0,0,0,0,1,1,16,0,0,0,1,0,0,0,0,1,0,0,16,0,0,0,1,0,16],[11,14,0,0,0,0,6,6,0,0,0,0,0,0,0,16,1,0,0,0,0,16,0,0,0,0,1,16,0,0,0,0,0,16,0,1] >;

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₈

2₊⁽¹⁺⁴⁾

C₈⋊C₂²

kernel

C₂⁴.₁₂₁D₄

C₂×C₂²⋊C₈

C₂²⋊D₈

D₄⋊D₄

C₂²⋊SD₁₆

C₈⋊₈D₄

C₈⋊₇D₄

C₂³.₄₆D₄

C₂³.₁₉D₄

C₂³.₄₈D₄

C₂×C₄⋊D₄

C₂².₁₉C₂⁴

C₂²×C₄

C₂⁴

C₂²

C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{121}D_4

% in TeX

G:=Group("C2^4.121D4");

// GroupNames label

G:=SmallGroup(128,1920);

// by ID

G=gap.SmallGroup(128,1920);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,675,4037,1027,124]);

// Polycyclic

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

// generators/relations

G = C₂⁴.₁₂₁D₄ order 128 = 2⁷

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

G = C24.121D4 order 128 = 27

76th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₂₁D₄ order 128 = 2⁷

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