C2^3:3Q16 - GroupNames

G = C₂³⋊₃Q₁₆ order 128 = 2⁷

2^nd semidirect product of C₂³ and Q₁₆ acting via Q₁₆/C₄=C₂²

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

Aliases: C₂³⋊₃Q₁₆, C₂⁴.₁₂₂D₄, C₄.₄2₊¹⁺⁴, C₂.D₈⋊₅C₂², C₈.₁₈D₄⋊₃C₂, C₂²⋊Q₁₆⋊₄C₂, (C₂×Q₁₆)⋊₂C₂², C₄⋊C₄.₁₂₈C₂³, (C₂×C₈).₁₅₁C₂³, (C₂×C₄).₃₈₇C₂⁴, Q₈⋊C₄⋊₂C₂², (C₂²×C₄).₄₈₅D₄, C₂³.₄₀₁(C₂×D₄), C₂².₁₇(C₂×Q₁₆), C₂.₁₄(C₂²×Q₁₆), C₂³.₄₈D₄⋊₅C₂, (C₂×Q₈).₁₂₇C₂³, C₂.₆₈(C₂³⋊₃D₄), C₂²⋊C₈.₁₇₆C₂², (C₂³×C₄).₅₆₇C₂², (C₂²×C₈).₁₄₉C₂², C₂².₆₄₇(C₂²×D₄), C₂²⋊Q₈.₁₈₅C₂², C₂.₄₉(D₈⋊C₂²), (C₂²×C₄).₁₀₆₅C₂³, (C₂²×Q₈).₃₁₃C₂², (C₂×C₄).₅₂₇(C₂×D₄), (C₂×C₂²⋊C₈).₃₂C₂, (C₂×C₂²⋊Q₈).₅₈C₂, (C₂×C₄⋊C₄).₆₃₇C₂², SmallGroup(128,1921)

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

Derived series

C₁ — C₂×C₄ — C₂³⋊₃Q₁₆

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂²×Q₈ — C₂×C₂²⋊Q₈ — C₂³⋊₃Q₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂³⋊₃Q₁₆

Upper central

C₁ — C₂² — C₂³×C₄ — C₂³⋊₃Q₁₆

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂³⋊₃Q₁₆

Generators and relations for C₂³⋊₃Q₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=1, e²=d⁴, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede^-1=d^-1 >

Subgroups: 404 in 208 conjugacy classes, 94 normal (14 characteristic)
C₁, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, Q₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂²⋊C₈, Q₈⋊C₄, C₂.D₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂×Q₁₆, C₂³×C₄, C₂²×Q₈, C₂×C₂²⋊C₈, C₂²⋊Q₁₆, C₈.₁₈D₄, C₂³.₄₈D₄, C₂×C₂²⋊Q₈, C₂³⋊₃Q₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, Q₁₆, C₂×D₄, C₂⁴, C₂×Q₁₆, C₂²×D₄, 2₊¹⁺⁴, C₂³⋊₃D₄, C₂²×Q₁₆, D₈⋊C₂², C₂³⋊₃Q₁₆

Smallest permutation representation of C₂³⋊₃Q₁₆
►On 32 points

Generators in S₃₂

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

(2 31)(4 25)(6 27)(8 29)(10 24)(12 18)(14 20)(16 22)

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

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

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

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

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

32 conjugacy classes

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

32 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	4
type	+	+	+	+	+	+	+	+	-	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₄	D₄	Q₁₆	2₊¹⁺⁴	D₈⋊C₂²
kernel	C₂³⋊₃Q₁₆	C₂×C₂²⋊C₈	C₂²⋊Q₁₆	C₈.₁₈D₄	C₂³.₄₈D₄	C₂×C₂²⋊Q₈	C₂²×C₄	C₂⁴	C₂³	C₄	C₂
# reps	1	1	4	4	4	2	3	1	8	2	2

Matrix representation of C₂³⋊₃Q₁₆ ►in GL₆(𝔽₁₇)

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

3	14	0	0	0	0
3	3	0	0	0	0
0	0	0	0	16	15
0	0	0	0	0	1
0	0	16	15	0	0
0	0	0	1	0	0

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

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

C₂³⋊₃Q₁₆ in GAP, Magma, Sage, TeX

C_2^3\rtimes_3Q_{16}

% in TeX

G:=Group("C2^3:3Q16");

// GroupNames label

G:=SmallGroup(128,1921);

// by ID

G=gap.SmallGroup(128,1921);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,219,352,675,4037,1027,124]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=1,e^2=d^4,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;

// generators/relations

G = C23⋊3Q16 order 128 = 27

2nd semidirect product of C23 and Q16 acting via Q16/C4=C22

G = C₂³⋊₃Q₁₆ order 128 = 2⁷

2^nd semidirect product of C₂³ and Q₁₆ acting via Q₁₆/C₄=C₂²