C2^4.183D4

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

Aliases: C₂⁴.₁₈₃D₄, C₄.Q₈⋊₇C₂², C₄⋊C₄.₅₂C₂³, (C₂×C₈).₃₂C₂³, C₂.D₈⋊₁₈C₂², C₂⁴.₄C₄⋊₇C₂, (C₂×C₄).₂₉₀C₂⁴, (C₂×D₄).₇₉C₂³, (C₂²×C₄).₄₄₁D₄, C₂³.₆₆₅(C₂×D₄), (C₂×Q₈).₆₇C₂³, D₄⋊C₄⋊₁₈C₂², Q₈⋊C₄⋊₂₀C₂², C₂³.₃₆D₄⋊₉C₂, C₂².D₈⋊₁₁C₂, C₂³.₄₇D₄⋊₁C₂, C₂³.₄₆D₄⋊₁C₂, C₂²⋊C₈.₁₄C₂², M₄(2)⋊C₄⋊₂₂C₂, C₂³.₄₈D₄⋊₁₁C₂, C₄⋊D₄.₁₅₅C₂², C₂².₂₉(C₈⋊C₂²), (C₂³×C₄).₅₆₀C₂², C₂².₅₅₀(C₂²×D₄), C₂²⋊Q₈.₁₆₀C₂², C₂².₁₉C₂⁴.₁₈C₂, (C₂²×C₄).₁₀₀₆C₂³, C₂².₁₈(C₈.C₂²), (C₂×M₄(2)).₇₂C₂², C₄²⋊C₂.₁₂₃C₂², C₄.₁₂₆(C₂².D₄), C₂².₄₀(C₂².D₄), (C₂²×C₄⋊C₄)⋊₃₅C₂, C₄.₁₀₀(C₂×C₄○D₄), (C₂×C₄).₄₈₆(C₂×D₄), C₂.₂₈(C₂×C₈⋊C₂²), C₂.₂₈(C₂×C₈.C₂²), (C₂×C₄).₄₈₅(C₄○D₄), (C₂×C₄⋊C₄).₉₂₇C₂², (C₂×C₄○D₄).₁₃₇C₂², C₂.₅₅(C₂×C₂².D₄), SmallGroup(128,1824)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₈₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄⋊C₄ — C₂²×C₄⋊C₄ — C₂⁴.₁₈₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₈₃D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₈₃D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₈₃D₄

Subgroups: 436 in 229 conjugacy classes, 96 normal (34 characteristic)
C₁, C₂^[×3], C₂^[×7], C₄^[×4], C₄^[×9], C₂², C₂²^[×6], C₂²^[×17], C₈^[×4], C₂×C₄^[×4], C₂×C₄^[×4], C₂×C₄^[×31], D₄^[×8], Q₈^[×2], C₂³^[×3], C₂³^[×7], C₄², C₂²⋊C₄^[×5], C₄⋊C₄^[×2], C₄⋊C₄^[×4], C₄⋊C₄^[×8], C₂×C₈^[×4], M₄(2)^[×4], C₂²×C₄^[×6], C₂²×C₄^[×15], 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×C₄⋊C₄^[×6], C₂×C₄⋊C₄^[×3], C₄²⋊C₂, C₄×D₄^[×2], C₂²≀C₂, C₄⋊D₄^[×2], C₂²⋊Q₈^[×2], C₂².D₄, C₂×M₄(2)^[×2], C₂³×C₄, C₂³×C₄, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₆D₄^[×2], M₄(2)⋊C₄^[×2], C₂².D₈^[×2], C₂³.₄₆D₄^[×2], C₂³.₄₇D₄^[×2], C₂³.₄₈D₄^[×2], C₂²×C₄⋊C₄, C₂².₁₉C₂⁴, C₂⁴.₁₈₃D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₄○D₄^[×4], C₂⁴, C₂².D₄^[×4], C₈⋊C₂²^[×2], C₈.C₂²^[×2], C₂²×D₄, C₂×C₄○D₄^[×2], C₂×C₂².D₄, C₂×C₈⋊C₂², C₂×C₈.C₂², C₂⁴.₁₈₃D₄

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=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 22)(11 24)(13 18)(15 20)

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

(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)(9,22)(11,24)(13,18)(15,20), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (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)(9,22)(11,24)(13,18)(15,20), (1,26)(2,31)(3,28)(4,25)(5,30)(6,27)(7,32)(8,29)(9,22)(10,19)(11,24)(12,21)(13,18)(14,23)(15,20)(16,17), (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),(9,22),(11,24),(13,18),(15,20)], [(1,26),(2,31),(3,28),(4,25),(5,30),(6,27),(7,32),(8,29),(9,22),(10,19),(11,24),(12,21),(13,18),(14,23),(15,20),(16,17)], [(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	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	15	9	1	0
0	0	15	9	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

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
13	0	0	0	0	0
0	0	4	16	5	8
0	0	12	14	7	15
0	0	6	15	16	0
0	0	2	8	16	0

0	1	0	0	0	0
1	0	0	0	0	0
0	0	2	8	15	0
0	0	0	0	16	1
0	0	10	8	11	4
0	0	10	9	11	4

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,15,15,0,0,0,16,9,9,0,0,0,0,1,0,0,0,0,0,0,1],[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],[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],[0,13,0,0,0,0,4,0,0,0,0,0,0,0,4,12,6,2,0,0,16,14,15,8,0,0,5,7,16,16,0,0,8,15,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,2,0,10,10,0,0,8,0,8,9,0,0,15,16,11,11,0,0,0,1,4,4] >;

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₄

C₈⋊C₂²

C₈.C₂²

kernel

C₂⁴.₁₈₃D₄

C₂⁴.₄C₄

C₂³.₃₆D₄

M₄(2)⋊C₄

C₂².D₈

C₂³.₄₆D₄

C₂³.₄₇D₄

C₂³.₄₈D₄

C₂²×C₄⋊C₄

C₂².₁₉C₂⁴

C₂²×C₄

C₂⁴

C₂×C₄

C₂²

# reps

In GAP, Magma, Sage, TeX

C_2^4._{183}D_4

% in TeX

G:=Group("C2^4.183D4");

// GroupNames label

G:=SmallGroup(128,1824);

// by ID

G=gap.SmallGroup(128,1824);

# by ID

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

// 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=f*a*f=a*c=c*a,a*d=d*a,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 = C₂⁴.₁₈₃D₄ order 128 = 2⁷

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

G = C24.183D4 order 128 = 27

38th non-split extension by C24 of D4 acting via D4/C22=C2

G = C₂⁴.₁₈₃D₄ order 128 = 2⁷

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