C2^4.60D4 - GroupNames

G = C₂⁴.₆₀D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₆₀D₄, C₂³.₁₀D₈, C₂³.₁₅SD₁₆, C₄⋊D₄⋊₈C₄, (C₂²×D₄)⋊₅C₄, C₂².₁₂(C₂×D₈), C₄.₁₃(C₂³⋊C₄), C₂²⋊C₈⋊₅₀C₂², C₂².SD₁₆⋊₃C₂, (C₂²×C₄).₂₂₀D₄, C₂³.₅₀₉(C₂×D₄), C₂³.₇Q₈⋊₂C₂, C₂².₃₀(C₂×SD₁₆), C₄⋊D₄.₁₄₂C₂², C₂.C₄²⋊₂C₂², (C₂²×C₄).₆₄₁C₂³, (C₂³×C₄).₂₁₃C₂², C₂².₂₅(D₄⋊C₄), C₂³.₁₇₅(C₂²⋊C₄), C₂.₂₄(C₄²⋊C₂²), (C₂×C₄⋊C₄)⋊₁₁C₄, C₄⋊C₄.₁₉(C₂×C₄), (C₂×C₂²⋊C₈)⋊₈C₂, (C₂×D₄).₁₅(C₂×C₄), (C₂×C₄⋊D₄).₅C₂, C₂.₂₂(C₂×C₂³⋊C₄), C₂.₁₀(C₂×D₄⋊C₄), (C₂×C₄).₁₁₆₅(C₂×D₄), (C₂×C₄).₁₃₁(C₂²×C₄), (C₂²×C₄).₂₀₄(C₂×C₄), (C₂×C₄).₂₄₁(C₂²⋊C₄), C₂².₁₉₅(C₂×C₂²⋊C₄), SmallGroup(128,251)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₆₀D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₄⋊D₄ — C₂⁴.₆₀D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₂⁴.₆₀D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₆₀D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₆₀D₄

Generators and relations for C₂⁴.₆₀D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, ad=da, ae=ea, faf^-1=acd, bc=cb, bd=db, ebe^-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bde³ >

Subgroups: 452 in 170 conjugacy classes, 54 normal (28 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₄⋊D₄, C₄⋊D₄, C₂²×C₈, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂².SD₁₆, C₂³.₇Q₈, C₂×C₂²⋊C₈, C₂×C₄⋊D₄, C₂⁴.₆₀D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, D₈, SD₁₆, C₂²×C₄, C₂×D₄, C₂³⋊C₄, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×D₈, C₂×SD₁₆, C₂×C₂³⋊C₄, C₂×D₄⋊C₄, C₄²⋊C₂², C₂⁴.₆₀D₄

Smallest permutation representation of C₂⁴.₆₀D₄
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	4	4
type	+	+	+	+	+				+	+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	D₈	SD₁₆	C₂³⋊C₄	C₄²⋊C₂²
kernel	C₂⁴.₆₀D₄	C₂².SD₁₆	C₂³.₇Q₈	C₂×C₂²⋊C₈	C₂×C₄⋊D₄	C₂×C₄⋊C₄	C₄⋊D₄	C₂²×D₄	C₂²×C₄	C₂⁴	C₂³	C₂³	C₄	C₂
# reps	1	4	1	1	1	2	4	2	3	1	4	4	2	2

Matrix representation of C₂⁴.₆₀D₄ ►in GL₆(𝔽₁₇)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	15	0	0
0	0	0	1	0	0
0	0	4	4	0	16
0	0	13	13	16	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	16	15	0	0
0	0	0	1	0	0
0	0	0	4	0	1
0	0	0	13	1	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	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

3	14	0	0	0	0
3	3	0	0	0	0
0	0	13	0	15	0
0	0	4	0	1	16
0	0	16	16	4	0
0	0	0	0	13	0

3	14	0	0	0	0
14	14	0	0	0	0
0	0	13	0	15	0
0	0	0	0	1	1
0	0	0	1	4	0
0	0	0	0	13	0

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,4,13,0,0,15,1,4,13,0,0,0,0,0,16,0,0,0,0,16,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,15,1,4,13,0,0,0,0,0,1,0,0,0,0,1,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[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],[3,3,0,0,0,0,14,3,0,0,0,0,0,0,13,4,16,0,0,0,0,0,16,0,0,0,15,1,4,13,0,0,0,16,0,0],[3,14,0,0,0,0,14,14,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,0,0,0,15,1,4,13,0,0,0,1,0,0] >;

C₂⁴.₆₀D₄ in GAP, Magma, Sage, TeX

C_2^4._{60}D_4

% in TeX

G:=Group("C2^4.60D4");

// GroupNames label

G:=SmallGroup(128,251);

// by ID

G=gap.SmallGroup(128,251);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,1430,387,352,1123,1018,248,1971]);

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

// generators/relations

G = C24.60D4 order 128 = 27

15th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₆₀D₄ order 128 = 2⁷

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