C2^4.98D4 - GroupNames

G = C₂⁴.₉₈D₄ order 128 = 2⁷

53^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₉₈D₄, C₄.₇(C₂³×C₄), C₄⋊C₄.₃₄₃C₂³, (C₂×C₈).₃₉₂C₂³, (C₂×C₄).₁₇₇C₂⁴, (C₂²×C₈)⋊₄₈C₂², D₄.₁₉(C₂²×C₄), (C₂²×C₄).₇₈₁D₄, C₂³.₃₇₇(C₂×D₄), C₄.₁₄₂(C₂²×D₄), Q₈.₁₉(C₂²×C₄), D₄⋊C₄⋊₈₅C₂², Q₈⋊C₄⋊₈₈C₂², (C₂×D₄).₃₆₁C₂³, C₄ ○(C₂³.₃₇D₄), C₄ ○(C₂³.₃₆D₄), C₄ ○(C₂³.₃₈D₄), (C₂×Q₈).₃₃₄C₂³, C₄²⋊C₂⋊₇₅C₂², C₂³.₃₆D₄⋊₄₇C₂, C₂³.₂₄D₄⋊₃₄C₂, C₂.₁(D₈⋊C₂²), C₂³.₃₈D₄⋊₃₈C₂, C₂³.₃₇D₄⋊₃₈C₂, C₂³.₈₇(C₂²⋊C₄), (C₂²×M₄(2))⋊₂₀C₂, (C₂×M₄(2))⋊₇₀C₂², (C₂³×C₄).₅₁₅C₂², (C₂²×C₄).₉₀₁C₂³, C₂².₁₂₇(C₂²×D₄), (C₂²×D₄).₅₅₄C₂², (C₂²×Q₈).₄₅₈C₂², (C₂×C₄○D₄)⋊₂₀C₄, C₄○D₄⋊₁₄(C₂×C₄), (C₂×D₄)⋊₄₉(C₂×C₄), (C₂×Q₈)⋊₄₀(C₂×C₄), (C₂×C₄).₄₄₅(C₂×D₄), C₄.₇₅(C₂×C₂²⋊C₄), (C₂×C₄⋊C₄)⋊₁₁₅C₂², (C₂×C₄²⋊C₂)⋊₄₂C₂, (C₂×C₄).₂₄₅(C₂²×C₄), (C₂²×C₄).₃₂₅(C₂×C₄), (C₂²×C₄○D₄).₂₀C₂, C₂².₂₂(C₂×C₂²⋊C₄), C₂.₃₉(C₂²×C₂²⋊C₄), (C₂×C₄).₁₅₉(C₂²⋊C₄), (C₂×C₄○D₄).₂₇₄C₂², (C₂×C₄)○(C₂³.₃₆D₄), (C₂×C₄)○(C₂³.₃₈D₄), SmallGroup(128,1628)

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

Derived series

C₁ — C₄ — C₂⁴.₉₈D₄

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂⁴.₉₈D₄

Lower central

C₁ — C₂ — C₄ — C₂⁴.₉₈D₄

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂⁴.₉₈D₄

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂⁴.₉₈D₄

Generators and relations for C₂⁴.₉₈D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=c, ab=ba, ac=ca, eae^-1=faf^-1=ad=da, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=cde³ >

Subgroups: 668 in 386 conjugacy classes, 172 normal (26 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, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂⁴, D₄⋊C₄, Q₈⋊C₄, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂³.₂₄D₄, C₂³.₃₆D₄, C₂³.₃₇D₄, C₂³.₃₈D₄, C₂×C₄²⋊C₂, C₂²×M₄(2), C₂²×C₄○D₄, C₂⁴.₉₈D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₂²×C₂²⋊C₄, D₈⋊C₂², C₂⁴.₉₈D₄

Smallest permutation representation of C₂⁴.₉₈D₄
►On 32 points

Generators in S₃₂

(2 6)(4 8)(9 13)(11 15)(18 22)(20 24)(25 29)(27 31)

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

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4J	4K	···	4V	8A	···	8H
order	1	2	2	2	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	4	4	4	4	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

D₈⋊C₂²

kernel

C₂⁴.₉₈D₄

C₂³.₂₄D₄

C₂³.₃₆D₄

C₂³.₃₇D₄

C₂³.₃₈D₄

C₂×C₄²⋊C₂

C₂²×M₄(2)

C₂²×C₄○D₄

C₂×C₄○D₄

C₂²×C₄

C₂⁴

C₂

# reps

Matrix representation of C₂⁴.₉₈D₄ ►in GL₆(𝔽₁₇)

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	1	0	16	0
0	0	16	0	0	16

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

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

13	7	0	0	0	0
10	4	0	0	0	0
0	0	16	0	0	15
0	0	0	0	1	1
0	0	6	13	0	16
0	0	7	0	0	1

10	4	0	0	0	0
13	7	0	0	0	0
0	0	1	0	15	0
0	0	0	0	1	1
0	0	1	0	16	0
0	0	16	16	1	0

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

C₂⁴.₉₈D₄ in GAP, Magma, Sage, TeX

C_2^4._{98}D_4

% in TeX

G:=Group("C2^4.98D4");

// GroupNames label

G:=SmallGroup(128,1628);

// by ID

G=gap.SmallGroup(128,1628);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,248,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^4=d,f^2=c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^3>;

// generators/relations

G = C24.98D4 order 128 = 27

53rd non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₉₈D₄ order 128 = 2⁷

53^rd non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²