C2^4.46D4 - GroupNames

G = C₂⁴.₄₆D₄ order 128 = 2⁷

1^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²

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

Aliases: C₂⁴.₄₆D₄, C₂²⋊C₈⋊₁C₄, (C₂×C₄²)⋊₁C₄, C₂².₉C₄≀C₂, (C₂×C₄).₂₈C₄², (C₂²×C₄).₁₆Q₈, C₂³.₁₁(C₄⋊C₄), C₂.C₄²⋊₁C₄, (C₂²×C₄).₁₁₈D₄, C₂.₄(C₄²⋊₆C₄), C₂.₃(C₄.₉C₄²), C₂⁴.₄C₄.₄C₂, C₂².₁₁(C₂³⋊C₄), (C₂³×C₄).₁₈₉C₂², C₂³.₃₄D₄.₂C₂, C₂.₃(C₂³.₉D₄), C₂³.₁₃₆(C₂²⋊C₄), C₂².₃₄(C₂.C₄²), (C₂×C₄).₁₃(C₄⋊C₄), (C₄×C₂²⋊C₄).₈C₂, (C₂²×C₄).₁₄₅(C₂×C₄), (C₂×C₄).₂₉₂(C₂²⋊C₄), SmallGroup(128,16)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₄₆D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂³×C₄ — C₄×C₂²⋊C₄ — C₂⁴.₄₆D₄

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₄₆D₄

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₄₆D₄

Jennings

C₁ — C₂ — C₂² — C₂³×C₄ — C₂⁴.₄₆D₄

Generators and relations for C₂⁴.₄₆D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=dc=cd, f²=a, ab=ba, ac=ca, eae^-1=ad=da, af=fa, bc=cb, bd=db, ebe^-1=fbf^-1=bcd, ce=ec, cf=fc, de=ed, df=fd, fef^-1=abcde³ >

Subgroups: 272 in 113 conjugacy classes, 34 normal (24 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂³, C₂³, C₄², C₂²⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₄², C₂×C₂²⋊C₄, C₂×M₄(2), C₂³×C₄, C₄×C₂²⋊C₄, C₂³.₃₄D₄, C₂⁴.₄C₄, C₂⁴.₄₆D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₂³⋊C₄, C₄≀C₂, C₄.₉C₄², C₄²⋊₆C₄, C₂³.₉D₄, C₂⁴.₄₆D₄

Smallest permutation representation of C₂⁴.₄₆D₄
►On 32 points

Generators in S₃₂

(2 30)(4 32)(6 26)(8 28)(10 17)(12 19)(14 21)(16 23)

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

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

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

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

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

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

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

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	2	2	2	2	4	4
type	+	+	+	+				+	-	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	D₄	Q₈	D₄	C₄≀C₂	C₂³⋊C₄	C₄.₉C₄²
kernel	C₂⁴.₄₆D₄	C₄×C₂²⋊C₄	C₂³.₃₄D₄	C₂⁴.₄C₄	C₂.C₄²	C₂²⋊C₈	C₂×C₄²	C₂²×C₄	C₂²×C₄	C₂⁴	C₂²	C₂²	C₂
# reps	1	1	1	1	4	4	4	2	1	1	8	2	2

Matrix representation of C₂⁴.₄₆D₄ ►in GL₆(𝔽₁₇)

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

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

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

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

11	7	0	0	0	0
10	6	0	0	0	0
0	0	13	13	11	13
0	0	0	0	0	4
0	0	2	2	4	0
0	0	2	1	4	0

4	0	0	0	0	0
0	13	0	0	0	0
0	0	1	1	10	1
0	0	0	16	0	0
0	0	0	0	13	8
0	0	0	0	0	4

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

C₂⁴.₄₆D₄ in GAP, Magma, Sage, TeX

C_2^4._{46}D_4

% in TeX

G:=Group("C2^4.46D4");

// GroupNames label

G:=SmallGroup(128,16);

// by ID

G=gap.SmallGroup(128,16);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,3924]);

// Polycyclic

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

// generators/relations

G = C24.46D4 order 128 = 27

1st non-split extension by C24 of D4 acting via D4/C2=C22

G = C₂⁴.₄₆D₄ order 128 = 2⁷

1^st non-split extension by C₂⁴ of D₄ acting via D₄/C₂=C₂²