C2^4.2Q8 - GroupNames

G = C₂⁴.₂Q₈ order 128 = 2⁷

1^st non-split extension by C₂⁴ of Q₈ acting via Q₈/C₂=C₂²

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

Aliases: C₂⁴.₂Q₈, C₂²⋊C₈.₃C₄, (C₂²×C₈).₄C₄, (C₂×C₄).₃₇C₄², C₂³.₄₁(C₄⋊C₄), C₄.₂₃(C₂³⋊C₄), (C₂²×C₄).₁₈₁D₄, C₂⁴.₄C₄.₅C₂, C₂².₂(C₈.C₄), C₂.₅(C₄.C₄²), (C₂³×C₄).₁₉₁C₂², C₂.₅(C₂³.₉D₄), C₂.₄(C₄.₁₀C₄²), C₂².₄₃(C₂.C₄²), (C₂×C₂²⋊C₈).₄C₂, (C₂²×C₄).₉₃(C₂×C₄), (C₂×C₄).₂₉₈(C₂²⋊C₄), SmallGroup(128,25)

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

Derived series

C₁ — C₂×C₄ — C₂⁴.₂Q₈

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊C₈ — C₂⁴.₂Q₈

Lower central

C₁ — C₂ — C₂×C₄ — C₂⁴.₂Q₈

Upper central

C₁ — C₂² — C₂³×C₄ — C₂⁴.₂Q₈

Jennings

C₁ — C₂ — C₂² — C₂³×C₄ — C₂⁴.₂Q₈

Generators and relations for C₂⁴.₂Q₈
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=dc=cd, f²=bcde², ab=ba, ac=ca, faf^-1=ad=da, ae=ea, bc=cb, ebe^-1=bd=db, fbf^-1=bcd, ce=ec, cf=fc, de=ed, df=fd, fef^-1=ace³ >

Subgroups: 200 in 92 conjugacy classes, 34 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂²⋊C₈, C₂²⋊C₈, C₂²×C₈, C₂×M₄(2), C₂³×C₄, C₂×C₂²⋊C₈, C₂⁴.₄C₄, C₂⁴.₂Q₈
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, Q₈, C₄², C₂²⋊C₄, C₄⋊C₄, C₂.C₄², C₂³⋊C₄, C₈.C₄, C₄.₁₀C₄², C₄.C₄², C₂³.₉D₄, C₂⁴.₂Q₈

Smallest permutation representation of C₂⁴.₂Q₈
►On 32 points

Generators in S₃₂

(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)

(2 18)(4 20)(6 22)(8 24)(9 13)(10 31)(11 15)(12 25)(14 27)(16 29)(26 30)(28 32)

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

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	2	2	2	4	4
type	+	+	+			+	-		+
image	C₁	C₂	C₂	C₄	C₄	D₄	Q₈	C₈.C₄	C₂³⋊C₄	C₄.₁₀C₄²
kernel	C₂⁴.₂Q₈	C₂×C₂²⋊C₈	C₂⁴.₄C₄	C₂²⋊C₈	C₂²×C₈	C₂²×C₄	C₂⁴	C₂²	C₄	C₂
# reps	1	1	2	8	4	3	1	8	2	2

Matrix representation of C₂⁴.₂Q₈ ►in GL₆(𝔽₁₇)

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	1	0
0	0	0	0	0	1

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

9	0	0	0	0	0
0	15	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	16
0	0	0	0	16	0

0	1	0	0	0	0
13	0	0	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	16	0	0

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[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],[9,0,0,0,0,0,0,15,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[0,13,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,1,0,0,0,0,0,0,1,0,0] >;

C₂⁴.₂Q₈ in GAP, Magma, Sage, TeX

C_2^4._2Q_8

% in TeX

G:=Group("C2^4.2Q8");

// GroupNames label

G:=SmallGroup(128,25);

// by ID

G=gap.SmallGroup(128,25);

# by ID

G:=PCGroup([7,-2,2,-2,2,2,-2,2,56,85,120,758,520,1018,360,3924,242]);

// 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=b*c*d*e^2,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,b*c=c*b,e*b*e^-1=b*d=d*b,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*c*e^3>;

// generators/relations

G = C24.2Q8 order 128 = 27

1st non-split extension by C24 of Q8 acting via Q8/C2=C22

G = C₂⁴.₂Q₈ order 128 = 2⁷

1^st non-split extension by C₂⁴ of Q₈ acting via Q₈/C₂=C₂²