C2^4.117D4

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

Aliases: C₂⁴.₁₁₇D₄, C₄.Q₈⋊₉C₂², (C₂×C₈).₃₄C₂³, C₂.D₈⋊₂₀C₂², C₄⋊C₄.₃₉₂C₂³, C₂²⋊C₈⋊₁₁C₂², C₂⁴.₄C₄⋊₉C₂, (C₂×C₄).₂₉₂C₂⁴, (C₂×D₄).₈₁C₂³, (C₂²×C₄).₄₄₃D₄, C₂³.₂₄₃(C₂×D₄), D₄⋊C₄⋊₂₀C₂², C₂³.₄₆D₄⋊₂C₂, C₂².D₈⋊₁₂C₂, M₄(2)⋊C₄⋊₂₄C₂, C₄²⋊C₂⋊₇₉C₂², C₂³.₃₇D₄⋊₁₀C₂, C₂³.₁₉D₄⋊₁₃C₂, C₄⋊D₄.₁₅₇C₂², C₂².₄₆(C₈⋊C₂²), (C₂³×C₄).₅₆₂C₂², C₂².₅₅₂(C₂²×D₄), C₂.₂₃(D₈⋊C₂²), (C₂²×C₄).₁₀₀₈C₂³, C₄.₉₉(C₂².D₄), (C₂²×D₄).₃₅₉C₂², (C₂×M₄(2)).₇₄C₂², C₂².₆₄(C₂².D₄), C₄.₁₀₂(C₂×C₄○D₄), (C₂×C₄).₄₈₇(C₂×D₄), C₂.₂₉(C₂×C₈⋊C₂²), (C₂×C₄⋊C₄)⋊₁₁₈C₂², (C₂×C₄⋊D₄).₅₈C₂, (C₂×C₄²⋊C₂)⋊₄₇C₂, (C₂×C₄).₄₈₇(C₄○D₄), C₂.₅₇(C₂×C₂².D₄), SmallGroup(128,1826)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₁₇D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₄²⋊C₂ — C₂×C₄²⋊C₂ — C₂⁴.₁₁₇D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₁₇D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₁₇D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₁₇D₄

Subgroups: 492 in 231 conjugacy classes, 94 normal (28 characteristic)
C₁, C₂^[×3], C₂^[×7], C₄^[×4], C₄^[×8], C₂², C₂²^[×4], C₂²^[×21], C₈^[×4], C₂×C₄^[×4], C₂×C₄^[×4], C₂×C₄^[×22], D₄^[×14], C₂³^[×3], C₂³^[×13], C₄²^[×4], C₂²⋊C₄^[×8], C₄⋊C₄^[×6], C₄⋊C₄^[×3], C₂×C₈^[×4], M₄(2)^[×4], C₂²×C₄^[×6], C₂²×C₄^[×7], C₂×D₄^[×2], C₂×D₄^[×13], C₂⁴, C₂⁴, C₂²⋊C₈^[×4], D₄⋊C₄^[×8], C₄.Q₈^[×4], C₂.D₈^[×4], C₂×C₄², C₂×C₂²⋊C₄^[×2], C₂×C₄⋊C₄^[×3], C₄²⋊C₂^[×4], C₄²⋊C₂^[×2], C₄⋊D₄^[×4], C₄⋊D₄^[×2], C₂×M₄(2)^[×2], C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂⁴.₄C₄, C₂³.₃₇D₄^[×2], M₄(2)⋊C₄^[×2], C₂².D₈^[×2], C₂³.₄₆D₄^[×2], C₂³.₁₉D₄^[×4], C₂×C₄²⋊C₂, C₂×C₄⋊D₄, C₂⁴.₁₁₇D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂².D₄^[×4], C₈⋊C₂²^[×2], C₂²×D₄, C₂×C₄○D₄^[×2], C₂×C₂².D₄, C₂×C₈⋊C₂², D₈⋊C₂², C₂⁴.₁₁₇D₄

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

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

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

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

(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)

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

13	15	0	0	0	0
16	4	0	0	0	0
0	0	13	4	0	0
0	0	9	4	0	0
0	0	0	0	13	4
0	0	0	0	9	4

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	1	0
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

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

4	0	0	0	0	0
1	13	0	0	0	0
0	0	0	0	16	0
0	0	0	0	15	1
0	0	16	1	0	0
0	0	0	1	0	0

1	0	0	0	0	0
13	16	0	0	0	0
0	0	1	16	0	0
0	0	0	16	0	0
0	0	0	0	16	0
0	0	0	0	15	1

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

C₈⋊C₂²

D₈⋊C₂²

kernel

C₂⁴.₁₁₇D₄

C₂⁴.₄C₄

C₂³.₃₇D₄

M₄(2)⋊C₄

C₂².D₈

C₂³.₄₆D₄

C₂³.₁₉D₄

C₂×C₄²⋊C₂

C₂×C₄⋊D₄

C₂²×C₄

C₂⁴

C₂×C₄

C₂²

C₂

# reps

In GAP, Magma, Sage, TeX

C_2^4._{117}D_4

% in TeX

G:=Group("C2^4.117D4");

// GroupNames label

G:=SmallGroup(128,1826);

// by ID

G=gap.SmallGroup(128,1826);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,100,2019,248,4037,1027,124]);

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

// generators/relations

G = C₂⁴.₁₁₇D₄ order 128 = 2⁷

72^nd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

G = C24.117D4 order 128 = 27

72nd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₁₇D₄ order 128 = 2⁷

72^nd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²