C2^4.104D4 - GroupNames

G = C₂⁴.₁₀₄D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₀₄D₄, (C₂×D₄)⋊₅₀D₄, (C₂×Q₈)⋊₃₇D₄, D₄.₄₁(C₂×D₄), C₄⋊C₄.₇C₂³, (C₂×C₈).₈C₂³, Q₈.₄₁(C₂×D₄), D₄⋊D₄⋊₁₃C₂, C₄.₈₃C₂²≀C₂, (C₂×D₈)⋊₁₄C₂², C₂²⋊C₈⋊₄C₂², C₄.₄₂(C₂²×D₄), D₄.₇D₄⋊₁₃C₂, D₄⋊C₄⋊₉C₂², C₄⋊D₄⋊₅₁C₂², C₂⁴.₄C₄⋊₄C₂, (C₂×C₄).₂₂₄C₂⁴, (C₂×Q₁₆)⋊₁₃C₂², (C₂×SD₁₆)⋊₃C₂², (C₂×D₄).₂₇C₂³, C₂³.₂₃₀(C₂×D₄), (C₂²×C₄).₇₁₅D₄, C₂²⋊Q₈⋊₆₃C₂², (C₂×Q₈).₂₀C₂³, C₂².₁₉C₂⁴⋊₄C₂, Q₈⋊C₄⋊₁₂C₂², C₂².₅₉C₂²≀C₂, C₂³.₃₈D₄⋊₃C₂, C₂³.₃₇D₄⋊₃C₂, C₂.₇(D₈⋊C₂²), (C₂²×C₄).₉₆₂C₂³, (C₂³×C₄).₅₄₄C₂², C₂².₄₈₄(C₂²×D₄), C₄²⋊C₂.₉₆C₂², (C₂²×D₄).₅₆₄C₂², (C₂×M₄(2)).₄₀C₂², (C₂²×Q₈).₄₆₈C₂², (C₂×C₈⋊C₂²)⋊₆C₂, (C₂²×C₄○D₄)⋊₉C₂, (C₂×C₈.C₂²)⋊₆C₂, C₂.₄₂(C₂×C₂²≀C₂), (C₂×C₄).₁₂₁₃(C₂×D₄), (C₂×C₄○D₄).₁₀₀C₂², SmallGroup(128,1737)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₀₄D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂⁴.₁₀₄D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₀₄D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₀₄D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₀₄D₄

Generators and relations for C₂⁴.₁₀₄D₄
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=de³ >

Subgroups: 740 in 379 conjugacy classes, 108 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), D₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂×M₄(2), C₂×D₈, C₂×SD₁₆, C₂×Q₁₆, C₈⋊C₂², C₈.C₂², C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄○D₄, C₂⁴.₄C₄, C₂³.₃₇D₄, C₂³.₃₈D₄, D₄⋊D₄, D₄.₇D₄, C₂².₁₉C₂⁴, C₂×C₈⋊C₂², C₂×C₈.C₂², C₂²×C₄○D₄, C₂⁴.₁₀₄D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₂²≀C₂, C₂²×D₄, C₂×C₂²≀C₂, D₈⋊C₂², C₂⁴.₁₀₄D₄

Smallest permutation representation of C₂⁴.₁₀₄D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	···	2K	2L	4A	···	4H	4I	4J	4K	4L	4M	4N	4O	8A	8B	8C	8D
order	1	2	2	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	4	···	4	8	2	···	2	4	4	4	4	8	8	8	8	8	8	8

32 irreducible representations

dim

type

image

C₁

C₂

D₄

D₈⋊C₂²

kernel

C₂⁴.₁₀₄D₄

C₂⁴.₄C₄

C₂³.₃₇D₄

C₂³.₃₈D₄

D₄⋊D₄

D₄.₇D₄

C₂².₁₉C₂⁴

C₂×C₈⋊C₂²

C₂×C₈.C₂²

C₂²×C₄○D₄

C₂²×C₄

C₂×D₄

C₂×Q₈

C₂⁴

C₂

# reps

Matrix representation of C₂⁴.₁₀₄D₄ ►in GL₆(𝔽₁₇)

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

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

0	16	0	0	0	0
1	0	0	0	0	0
0	0	4	0	0	10
0	0	0	13	6	0
0	0	0	8	4	0
0	0	9	0	0	13

0	16	0	0	0	0
16	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

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

C₂⁴.₁₀₄D₄ in GAP, Magma, Sage, TeX

C_2^4._{104}D_4

% in TeX

G:=Group("C2^4.104D4");

// GroupNames label

G:=SmallGroup(128,1737);

// by ID

G=gap.SmallGroup(128,1737);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,758,352,2019,2804,1411,172]);

// 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=d*e^3>;

// generators/relations

G = C24.104D4 order 128 = 27

59th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₀₄D₄ order 128 = 2⁷

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