C2^4.59D4 - GroupNames

G = C₂⁴.₅₉D₄ order 128 = 2⁷

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

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

Aliases: C₂⁴.₅₉D₄, C₄⋊D₄⋊₇C₄, C₂²⋊Q₈⋊₇C₄, C₄²⋊C₂⋊₄C₄, (C₂²×C₄).₁₂₇D₄, C₂³.₅₀₆(C₂×D₄), C₂².SD₁₆⋊₁C₂, C₂².₁₈(C₄○D₈), C₂³.₃₄D₄⋊₅C₂, C₂³.₃₁D₄⋊₁C₂, C₂².₇(C₂³⋊C₄), C₄⋊D₄.₁₄₀C₂², C₂²⋊C₈.₁₆₅C₂², C₂³.₅₅(C₂²⋊C₄), (C₂³×C₄).₂₁₂C₂², C₂².₁₉C₂⁴.₄C₂, (C₂²×C₄).₆₃₈C₂³, C₂²⋊Q₈.₁₄₅C₂², C₂.C₄².₆C₂², C₂.₂₁(C₄²⋊C₂²), C₂.₁₀(C₂³.₂₄D₄), (C₂×C₄○D₄)⋊₄C₄, C₄⋊C₄.₁₆(C₂×C₄), (C₂×C₂²⋊C₈)⋊₇C₂, (C₂×D₄).₁₃(C₂×C₄), C₂.₂₁(C₂×C₂³⋊C₄), (C₂×Q₈).₁₃(C₂×C₄), (C₂×C₄).₁₁₆₂(C₂×D₄), (C₂²×C₄).₂₀₃(C₂×C₄), (C₂×C₄).₁₂₈(C₂²×C₄), (C₂×C₄).₁₇₄(C₂²⋊C₄), C₂².₁₉₂(C₂×C₂²⋊C₄), SmallGroup(128,248)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₅₉D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂².₁₉C₂⁴ — C₂⁴.₅₉D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₂⁴.₅₉D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₅₉D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂⁴.₅₉D₄

Generators and relations for C₂⁴.₅₉D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, faf^-1=ad=da, eae^-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: 340 in 142 conjugacy classes, 46 normal (34 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₄○D₄, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₄²⋊C₂, C₄×D₄, C₂²≀C₂, C₄⋊D₄, C₂²⋊Q₈, C₂².D₄, C₂²×C₈, C₂³×C₄, C₂×C₄○D₄, C₂².SD₁₆, C₂³.₃₁D₄, C₂³.₃₄D₄, C₂×C₂²⋊C₈, C₂².₁₉C₂⁴, C₂⁴.₅₉D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂³⋊C₄, C₂×C₂²⋊C₄, C₄○D₈, C₂×C₂³⋊C₄, C₂³.₂₄D₄, C₄²⋊C₂², C₂⁴.₅₉D₄

Smallest permutation representation of C₂⁴.₅₉D₄
►On 32 points

Generators in S₃₂

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

(2 32)(4 26)(6 28)(8 30)(9 23)(11 17)(13 19)(15 21)

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

(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)

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

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

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

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

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	1	1	1	1	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+					+	+		+
image	C₁	C₂	C₂	C₂	C₂	C₂	C₄	C₄	C₄	C₄	D₄	D₄	C₄○D₈	C₂³⋊C₄	C₄²⋊C₂²
kernel	C₂⁴.₅₉D₄	C₂².SD₁₆	C₂³.₃₁D₄	C₂³.₃₄D₄	C₂×C₂²⋊C₈	C₂².₁₉C₂⁴	C₄²⋊C₂	C₄⋊D₄	C₂²⋊Q₈	C₂×C₄○D₄	C₂²×C₄	C₂⁴	C₂²	C₂²	C₂
# reps	1	2	2	1	1	1	2	2	2	2	3	1	8	2	2

Matrix representation of C₂⁴.₅₉D₄ ►in GL₆(𝔽₁₇)

1	8	0	0	0	0
0	16	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	15	0	16	0
0	0	15	16	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	15	0	16	0
0	0	15	16	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

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

15	7	0	0	0	0
0	9	0	0	0	0
0	0	6	2	2	4
0	0	6	13	14	9
0	0	3	15	15	13
0	0	13	9	8	0

13	0	0	0	0	0
1	4	0	0	0	0
0	0	1	0	0	0
0	0	13	16	0	0
0	0	2	1	1	2
0	0	16	0	16	16

G:=sub<GL(6,GF(17))| [1,0,0,0,0,0,8,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,15,15,0,0,0,1,0,16,0,0,0,0,16,0,0,0,0,0,0,16],[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],[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],[15,0,0,0,0,0,7,9,0,0,0,0,0,0,6,6,3,13,0,0,2,13,15,9,0,0,2,14,15,8,0,0,4,9,13,0],[13,1,0,0,0,0,0,4,0,0,0,0,0,0,1,13,2,16,0,0,0,16,1,0,0,0,0,0,1,16,0,0,0,0,2,16] >;

C₂⁴.₅₉D₄ in GAP, Magma, Sage, TeX

C_2^4._{59}D_4

% in TeX

G:=Group("C2^4.59D4");

// GroupNames label

G:=SmallGroup(128,248);

// by ID

G=gap.SmallGroup(128,248);

# by ID

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

// 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,f*a*f^-1=a*d=d*a,e*a*e^-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 = C24.59D4 order 128 = 27

14th non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₅₉D₄ order 128 = 2⁷

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