C2^4.116D4

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

Aliases: C₂⁴.₁₁₆D₄, C₄.Q₈⋊₈C₂², (C₂×C₈).₃₃C₂³, C₂.D₈⋊₁₉C₂², C₄⋊C₄.₃₉₁C₂³, (C₂×C₄).₂₉₁C₂⁴, C₂⁴.₄C₄⋊₈C₂, (C₂×D₄).₈₀C₂³, C₂³.₂₄₂(C₂×D₄), (C₂²×C₄).₄₄₂D₄, (C₂×Q₈).₆₈C₂³, D₄⋊C₄⋊₁₉C₂², Q₈⋊C₄⋊₂₁C₂², C₂²⋊C₈.₁₅C₂², M₄(2)⋊C₄⋊₂₃C₂, C₄⋊D₄.₁₅₆C₂², C₂³.₂₀D₄⋊₁₂C₂, C₂³.₁₉D₄⋊₁₂C₂, C₂³.₃₆D₄⋊₁₀C₂, (C₂³×C₄).₅₆₁C₂², C₂².₅₅₁(C₂²×D₄), C₂²⋊Q₈.₁₆₁C₂², C₂.₂₂(D₈⋊C₂²), (C₂²×C₄).₁₀₀₇C₂³, C₂².₁₉C₂⁴.₁₉C₂, (C₂×M₄(2)).₇₃C₂², C₄²⋊C₂.₃₁₇C₂², C₄.₁₂₇(C₂².D₄), C₂².₄₁(C₂².D₄), C₄.₁₀₁(C₂×C₄○D₄), (C₂×C₄).₁₂₁₈(C₂×D₄), (C₂×C₄²⋊C₂)⋊₄₆C₂, (C₂×C₄).₄₈₆(C₄○D₄), (C₂×C₄⋊C₄).₉₂₈C₂², (C₂×C₄○D₄).₁₃₈C₂², C₂.₅₆(C₂×C₂².D₄), SmallGroup(128,1825)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₁₆D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂×C₄²⋊C₂ — C₂⁴.₁₁₆D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₁₆D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₁₆D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₁₆D₄

Subgroups: 388 in 207 conjugacy classes, 92 normal (18 characteristic)
C₁, C₂, C₂^[×2], C₂^[×5], C₄^[×4], C₄^[×9], C₂², C₂²^[×2], C₂²^[×15], C₈^[×4], C₂×C₄^[×2], C₂×C₄^[×6], C₂×C₄^[×23], D₄^[×8], Q₈^[×2], C₂³, C₂³^[×2], C₂³^[×5], C₄²^[×5], C₂²⋊C₄^[×9], C₄⋊C₄^[×6], C₄⋊C₄^[×4], C₂×C₈^[×4], M₄(2)^[×4], C₂²×C₄^[×2], C₂²×C₄^[×4], C₂²×C₄^[×7], C₂×D₄, C₂×D₄^[×3], C₂×Q₈, C₄○D₄^[×4], C₂⁴, C₂²⋊C₈^[×4], D₄⋊C₄^[×4], Q₈⋊C₄^[×4], C₄.Q₈^[×4], C₂.D₈^[×4], C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄^[×2], C₄²⋊C₂, C₄²⋊C₂^[×4], C₄²⋊C₂^[×2], C₄×D₄^[×2], C₂²≀C₂, C₄⋊D₄^[×2], C₂²⋊Q₈^[×2], C₂².D₄, C₂×M₄(2)^[×2], C₂³×C₄, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₆D₄^[×2], M₄(2)⋊C₄^[×2], C₂³.₁₉D₄^[×4], C₂³.₂₀D₄^[×4], C₂×C₄²⋊C₂, C₂².₁₉C₂⁴, C₂⁴.₁₁₆D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂².D₄^[×4], C₂²×D₄, C₂×C₄○D₄^[×2], C₂×C₂².D₄, D₈⋊C₂²^[×2], C₂⁴.₁₁₆D₄

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, eae^-1=ac=ca, ad=da, faf=acd, bc=cb, ebe^-1=fbf=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=cde³ >

Smallest permutation representation
►On 32 points

Generators in S₃₂

(2 31)(4 25)(6 27)(8 29)(9 18)(10 14)(11 20)(12 16)(13 22)(15 24)(17 21)(19 23)

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

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

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

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

0	4	0	0	0	0
4	0	0	0	0	0
0	0	4	0	0	10
0	0	13	0	1	7
0	0	4	13	0	4
0	0	9	0	0	13

0	16	0	0	0	0
16	0	0	0	0	0
0	0	0	1	1	0
0	0	0	16	0	0
0	0	1	1	0	0
0	0	0	15	0	1

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,1,16,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,1,16,0,16],[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],[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],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,13,4,9,0,0,0,0,13,0,0,0,0,1,0,0,0,0,10,7,4,13],[0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,0,1,0,0,0,1,16,1,15,0,0,1,0,0,0,0,0,0,0,0,1] >;

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	···	4H	4I	···	4P	4Q	4R	4S	8A	8B	8C	8D
order	1	2	2	2	2	2	2	2	2	4	···	4	4	···	4	4	4	4	8	8	8	8
size	1	1	1	1	2	2	4	4	8	2	···	2	4	···	4	8	8	8	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

D₈⋊C₂²

kernel

C₂⁴.₁₁₆D₄

C₂⁴.₄C₄

C₂³.₃₆D₄

M₄(2)⋊C₄

C₂³.₁₉D₄

C₂³.₂₀D₄

C₂×C₄²⋊C₂

C₂².₁₉C₂⁴

C₂²×C₄

C₂⁴

C₂×C₄

C₂

# reps

In GAP, Magma, Sage, TeX

C_2^4._{116}D_4

% in TeX

G:=Group("C2^4.116D4");

// GroupNames label

G:=SmallGroup(128,1825);

// by ID

G=gap.SmallGroup(128,1825);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,100,2019,4037,1027,124]);

// 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,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f=a*c*d,b*c=c*b,e*b*e^-1=f*b*f=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=c*d*e^3>;

// generators/relations

G = C₂⁴.₁₁₆D₄ order 128 = 2⁷

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

G = C24.116D4 order 128 = 27

71st non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₁₆D₄ order 128 = 2⁷

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