C2^4.123D4 - GroupNames

G = C₂⁴.₁₂₃D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₁₂₃D₄, C₄.₅2₊¹⁺⁴, C₈⋊₈D₄⋊₂C₂, C₂.D₈⋊₆C₂², Q₈⋊D₄⋊₂₉C₂, D₄.₇D₄⋊₁C₂, C₂²⋊Q₁₆⋊₅C₂, (C₂×Q₁₆)⋊₃C₂², C₄.Q₈⋊₃₃C₂², C₈.₁₈D₄⋊₁₇C₂, D₄⋊C₄⋊₂C₂², C₄⋊C₄.₁₂₉C₂³, (C₂×C₄).₃₈₈C₂⁴, (C₂×C₈).₁₅₂C₂³, Q₈⋊C₄⋊₃C₂², C₂³.₂₇₂(C₂×D₄), (C₂²×C₄).₄₈₆D₄, C₂².D₈⋊₅C₂, (C₂×SD₁₆)⋊₄₀C₂², (C₂×D₄).₁₄₀C₂³, C₂².₃₂(C₄○D₈), C₂³.₂₀D₄⋊₁C₂, (C₂×Q₈).₁₂₈C₂³, C₄⋊D₄.₁₈₁C₂², C₂³.₄₇D₄⋊₂₉C₂, C₂.₆₉(C₂³⋊₃D₄), C₂²⋊C₈.₂₁₇C₂², (C₂³×C₄).₅₆₈C₂², (C₂²×C₈).₁₈₆C₂², C₂².₆₄₈(C₂²×D₄), C₂².₂(C₈.C₂²), C₂²⋊Q₈.₁₈₆C₂², (C₂²×C₄).₁₀₆₆C₂³, C₂².₁₉C₂⁴.₂₀C₂, (C₂²×Q₈).₃₁₄C₂², C₄²⋊C₂.₁₅₀C₂², C₂.₃₉(C₂×C₄○D₈), (C₂×C₂²⋊C₈)⋊₂₉C₂, (C₂×C₄).₇₀₅(C₂×D₄), (C₂×C₂²⋊Q₈)⋊₅₈C₂, C₂.₄₉(C₂×C₈.C₂²), (C₂×C₄⋊C₄).₆₃₈C₂², (C₂×C₄○D₄).₁₆₁C₂², SmallGroup(128,1922)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₁₂₃D₄

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂×C₂²⋊Q₈ — C₂⁴.₁₂₃D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₁₂₃D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₁₂₃D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₁₂₃D₄

Generators and relations for C₂⁴.₁₂₃D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=f²=1, e⁴=d, ab=ba, ac=ca, ad=da, ae=ea, faf=acd, ebe^-1=fbf=bc=cb, bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e³ >

Subgroups: 428 in 210 conjugacy classes, 88 normal (42 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, Q₁₆, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂×Q₈, C₄○D₄, C₂⁴, C₂²⋊C₈, C₂²⋊C₈, D₄⋊C₄, Q₈⋊C₄, C₄.Q₈, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²⋊Q₈, C₂².D₄, C₂²×C₈, C₂×SD₁₆, C₂×Q₁₆, C₂³×C₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₂²⋊C₈, Q₈⋊D₄, C₂²⋊Q₁₆, D₄.₇D₄, C₈⋊₈D₄, C₈.₁₈D₄, C₂².D₈, C₂³.₄₇D₄, C₂³.₂₀D₄, C₂×C₂²⋊Q₈, C₂².₁₉C₂⁴, C₂⁴.₁₂₃D₄
Quotients: C₁, C₂, C₂², D₄, C₂³, C₂×D₄, C₂⁴, C₄○D₈, C₈.C₂², C₂²×D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂×C₄○D₈, C₂×C₈.C₂², C₂⁴.₁₂₃D₄

Smallest permutation representation of C₂⁴.₁₂₃D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim

type

image

C₁

C₂

D₄

C₄○D₈

2₊¹⁺⁴

C₈.C₂²

kernel

C₂⁴.₁₂₃D₄

C₂×C₂²⋊C₈

Q₈⋊D₄

C₂²⋊Q₁₆

D₄.₇D₄

C₈⋊₈D₄

C₈.₁₈D₄

C₂².D₈

C₂³.₄₇D₄

C₂³.₂₀D₄

C₂×C₂²⋊Q₈

C₂².₁₉C₂⁴

C₂²×C₄

C₂⁴

C₂²

C₄

C₂²

# reps

Matrix representation of C₂⁴.₁₂₃D₄ ►in 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	14	8	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

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

15	0	0	0	0	0
0	9	0	0	0	0
0	0	15	11	7	15
0	0	0	0	0	12
0	0	0	0	2	0
0	0	10	7	0	0

0	9	0	0	0	0
2	0	0	0	0	0
0	0	0	0	0	1
0	0	7	4	1	7
0	0	16	2	13	16
0	0	1	0	0	0

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

C₂⁴.₁₂₃D₄ in GAP, Magma, Sage, TeX

C_2^4._{123}D_4

% in TeX

G:=Group("C2^4.123D4");

// GroupNames label

G:=SmallGroup(128,1922);

// by ID

G=gap.SmallGroup(128,1922);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,253,456,758,219,352,675,4037,1027,124]);

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

// generators/relations

G = C24.123D4 order 128 = 27

78th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₁₂₃D₄ order 128 = 2⁷

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