C2^4.124D4

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

Aliases: C₂⁴.₁₂₄D₄, C₄.₆2₊⁽¹⁺⁴⁾, C₈⋊₇D₄⋊₄C₂, D₄⋊D₄⋊₂C₂, C₈⋊₈D₄⋊₂₆C₂, (C₂×D₈)⋊₄C₂², C₂.D₈⋊₇C₂², D₄.₇D₄⋊₂C₂, C₈.₁₈D₄⋊₄C₂, (C₂×Q₁₆)⋊₄C₂², C₄.Q₈⋊₃₄C₂², C₄⋊C₄.₁₃₀C₂³, (C₂×C₄).₃₈₉C₂⁴, (C₂×C₈).₁₅₃C₂³, C₂².₄(C₄○D₈), (C₂²×C₄).₁₇₁D₄, C₂³.₂₇₃(C₂×D₄), D₄⋊C₄⋊₄₄C₂², Q₈⋊C₄⋊₄₇C₂², (C₂×SD₁₆)⋊₄₁C₂², (C₂×D₄).₁₄₁C₂³, C₂³.₂₀D₄⋊₂C₂, C₂³.₁₉D₄⋊₂C₂, (C₂×Q₈).₁₂₉C₂³, C₂².₁₉C₂⁴⋊₁₀C₂, C₄⋊D₄.₁₈₂C₂², C₂.₇₀(C₂³⋊₃D₄), C₂²⋊C₈.₁₉₁C₂², (C₂³×C₄).₅₆₉C₂², (C₂²×C₈).₁₅₀C₂², C₂².₆₄₉(C₂²×D₄), C₂²⋊Q₈.₁₈₇C₂², C₂.₅₀(D₈⋊C₂²), (C₂²×C₄).₁₀₆₇C₂³, C₄²⋊C₂.₁₅₁C₂², C₂.₄₀(C₂×C₄○D₈), (C₂×C₂²⋊C₈)⋊₃₀C₂, (C₂×C₄).₇₀₆(C₂×D₄), (C₂×C₄○D₄).₁₆₂C₂², SmallGroup(128,1923)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₂₄D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂×C₄○D₄ — C₂².₁₉C₂⁴ — C₂⁴.₁₂₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₂₄D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₂₄D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₂₄D₄

Subgroups: 452 in 212 conjugacy classes, 86 normal (26 characteristic)
C₁, C₂^[×3], C₂^[×6], C₄^[×2], C₄^[×9], C₂², C₂²^[×2], C₂²^[×18], C₈^[×4], C₂×C₄^[×2], C₂×C₄^[×2], C₂×C₄^[×19], D₄^[×16], Q₈^[×4], C₂³, C₂³^[×2], C₂³^[×6], C₄²^[×2], C₂²⋊C₄^[×10], C₄⋊C₄^[×4], C₄⋊C₄^[×4], C₂×C₈^[×2], C₂×C₈^[×2], C₂×C₈^[×2], D₈, SD₁₆^[×2], Q₁₆, C₂²×C₄^[×2], C₂²×C₄^[×4], C₂²×C₄^[×4], C₂×D₄^[×2], C₂×D₄^[×6], C₂×Q₈^[×2], C₄○D₄^[×4], C₂⁴, C₂²⋊C₈^[×4], D₄⋊C₄^[×4], Q₈⋊C₄^[×4], C₄.Q₈^[×2], C₂.D₈^[×2], C₄²⋊C₂^[×2], C₄×D₄^[×4], C₂²≀C₂^[×2], C₄⋊D₄^[×4], C₂²⋊Q₈^[×4], C₂².D₄^[×2], C₂²×C₈^[×2], C₂×D₈, C₂×SD₁₆^[×2], C₂×Q₁₆, C₂³×C₄, C₂×C₄○D₄^[×2], C₂×C₂²⋊C₈, D₄⋊D₄^[×2], D₄.₇D₄^[×2], C₈⋊₈D₄^[×2], C₈⋊₇D₄, C₈.₁₈D₄, C₂³.₁₉D₄^[×2], C₂³.₂₀D₄^[×2], C₂².₁₉C₂⁴^[×2], C₂⁴.₁₂₄D₄

Quotients:
C₁, C₂^[×15], C₂²^[×35], D₄^[×4], C₂³^[×15], C₂×D₄^[×6], C₂⁴, C₄○D₈^[×2], C₂²×D₄, 2₊⁽¹⁺⁴⁾^[×2], C₂³⋊₃D₄, C₂×C₄○D₈, D₈⋊C₂², C₂⁴.₁₂₄D₄

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

Smallest permutation representation
►On 32 points

Generators in S₃₂

(9 20)(10 21)(11 22)(12 23)(13 24)(14 17)(15 18)(16 19)

(1 5)(2 27)(3 7)(4 29)(6 31)(8 25)(9 20)(11 22)(13 24)(15 18)(26 30)(28 32)

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₁₇)

16	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	0	0	1

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	1	16	0
0	0	1	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

8	0	0	0	0	0
0	15	0	0	0	0
0	0	1	0	0	15
0	0	0	1	15	0
0	0	0	1	16	0
0	0	1	0	0	16

0	2	0	0	0	0
9	0	0	0	0	0
0	0	0	1	15	0
0	0	1	0	0	15
0	0	0	0	0	16
0	0	0	0	16	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[8,0,0,0,0,0,0,15,0,0,0,0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,0,15,16,0,0,0,15,0,0,16],[0,9,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,15,0,0,16,0,0,0,15,16,0] >;

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₈

2₊⁽¹⁺⁴⁾

D₈⋊C₂²

kernel

C₂⁴.₁₂₄D₄

C₂×C₂²⋊C₈

D₄⋊D₄

D₄.₇D₄

C₈⋊₈D₄

C₈⋊₇D₄

C₈.₁₈D₄

C₂³.₁₉D₄

C₂³.₂₀D₄

C₂².₁₉C₂⁴

C₂²×C₄

C₂⁴

C₂²

C₄

C₂

# reps

In GAP, Magma, Sage, TeX

C_2^4._{124}D_4

% in TeX

G:=Group("C2^4.124D4");

// GroupNames label

G:=SmallGroup(128,1923);

// by ID

G=gap.SmallGroup(128,1923);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,219,675,248,4037,1027,124]);

// 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,f*a*f=a*c=c*a,a*d=d*a,a*e=e*a,e*b*e^-1=b*c=c*b,b*d=d*b,f*b*f=b*c*d,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 = C₂⁴.₁₂₄D₄ order 128 = 2⁷

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

G = C24.124D4 order 128 = 27

79th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₂₄D₄ order 128 = 2⁷

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