C2^4.55D4 - GroupNames

G = C₂⁴.₅₅D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₅D₄, C₂²⋊Q₈⋊₄C₄, (C₂²×Q₈)⋊₃C₄, C₂².₂₄C₄≀C₂, C₄.₁₂(C₂³⋊C₄), C₂³.₄₉₈(C₂×D₄), (C₂²×C₄).₇₃₇D₄, C₂³.₃₁D₄⋊₁₈C₂, C₂²⋊C₈.₁₃₀C₂², C₂³.₅₃(C₂²⋊C₄), C₂⁴.₄C₄.₁₃C₂, (C₂²×C₄).₆₃₀C₂³, (C₂³×C₄).₂₀₈C₂², C₂²⋊Q₈.₁₃₉C₂², C₂.₈(C₂³.₃₈D₄), C₂².₂₄(C₈.C₂²), C₂.C₄².₅₀₆C₂², (C₂×C₄⋊C₄)⋊₈C₄, C₄⋊C₄.₈(C₂×C₄), C₂.₂₅(C₂×C₄≀C₂), (C₂×Q₈).₇(C₂×C₄), C₂.₁₇(C₂×C₂³⋊C₄), (C₂×C₂²⋊Q₈).₃C₂, (C₂×C₄).₁₁₅₄(C₂×D₄), (C₄×C₂²⋊C₄).₁₀C₂, (C₂×C₄).₁₂₀(C₂²×C₄), (C₂²×C₄).₁₉₉(C₂×C₄), (C₂×C₄).₁₇₂(C₂²⋊C₄), C₂².₁₈₄(C₂×C₂²⋊C₄), SmallGroup(128,240)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₅₅D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊Q₈ — C₂⁴.₅₅D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₂⁴.₅₅D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₅₅D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₅₅D₄

Generators and relations for C₂⁴.₅₅D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, faf^-1=ac=ca, eae^-1=ad=da, bc=cb, bd=db, ebe^-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bcde³ >

Subgroups: 324 in 147 conjugacy classes, 48 normal (26 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂×M₄(2), C₂³×C₄, C₂²×Q₈, C₂³.₃₁D₄, C₄×C₂²⋊C₄, C₂⁴.₄C₄, C₂×C₂²⋊Q₈, C₂⁴.₅₅D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₄≀C₂, C₂×C₂²⋊C₄, C₈.C₂², C₂×C₂³⋊C₄, C₂³.₃₈D₄, C₂×C₄≀C₂, C₂⁴.₅₅D₄

Smallest permutation representation of C₂⁴.₅₅D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

32 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	4A	···	4F	4G	···	4O	4P	4Q	4R	4S	8A	8B	8C	8D
order	1	2	2	2	2	2	2	2	2	4	···	4	4	···	4	4	4	4	4	8	8	8	8
size	1	1	1	1	2	2	2	2	4	2	···	2	4	···	4	8	8	8	8	8	8	8	8

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+				+	+		+	-
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	C₄≀C₂	C₂³⋊C₄	C₈.C₂²
kernel	C₂⁴.₅₅D₄	C₂³.₃₁D₄	C₄×C₂²⋊C₄	C₂⁴.₄C₄	C₂×C₂²⋊Q₈	C₂×C₄⋊C₄	C₂²⋊Q₈	C₂²×Q₈	C₂²×C₄	C₂⁴	C₂²	C₄	C₂²
# reps	1	4	1	1	1	2	4	2	3	1	8	2	2

Matrix representation of C₂⁴.₅₅D₄ ►in GL₆(𝔽₁₇)

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

0	16	0	0	0	0
4	0	0	0	0	0
0	0	1	0	0	2
0	0	1	0	1	1
0	0	0	0	0	16
0	0	16	1	0	16

4	0	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	1	16	0	0
0	0	16	0	0	16
0	0	0	0	1	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,16,16,0,0,15,16,1,1,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,16,16,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[0,4,0,0,0,0,16,0,0,0,0,0,0,0,1,1,0,16,0,0,0,0,0,1,0,0,0,1,0,0,0,0,2,1,16,16],[4,0,0,0,0,0,0,16,0,0,0,0,0,0,1,1,16,0,0,0,0,16,0,0,0,0,0,0,0,1,0,0,0,0,16,0] >;

C₂⁴.₅₅D₄ in GAP, Magma, Sage, TeX

C_2^4._{55}D_4

% in TeX

G:=Group("C2^4.55D4");

// GroupNames label

G:=SmallGroup(128,240);

// by ID

G=gap.SmallGroup(128,240);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,387,352,1123,1018,248,1971]);

// Polycyclic

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

// generators/relations

G = C24.55D4 order 128 = 27

10th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₅₅D₄ order 128 = 2⁷

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