C2xC2^2:SD16 - GroupNames

G = C₂×C₂²⋊SD₁₆ order 128 = 2⁷

Direct product of C₂ and C₂²⋊SD₁₆

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

Aliases: C₂×C₂²⋊SD₁₆, C₂³⋊₆SD₁₆, C₂⁴.₁₇₄D₄, C₄⋊C₄⋊₂C₂³, (C₂×C₈)⋊₈C₂³, D₄.₃₇(C₂×D₄), (C₂×Q₈)⋊₁C₂³, C₄.₃₄C₂²≀C₂, (C₂×D₄).₂₉₂D₄, C₂²⋊₃(C₂×SD₁₆), C₄.₃₄(C₂²×D₄), C₂²⋊C₈⋊₆₅C₂², (C₂×C₄).₂₁₆C₂⁴, (C₂²×C₈)⋊₃₄C₂², (D₄×C₂³).₁₇C₂, C₂³.₈₄₇(C₂×D₄), (C₂²×C₄).₄₁₆D₄, C₂²⋊Q₈⋊₅₉C₂², D₄⋊C₄⋊₆₉C₂², C₂.₅(C₂²×SD₁₆), (C₂×SD₁₆)⋊₆₅C₂², (C₂²×SD₁₆)⋊₁₅C₂, (C₂×D₄).₃₇₈C₂³, (C₂²×Q₈)⋊₁₂C₂², C₂².₁₁₃C₂²≀C₂, (C₂²×C₄).₉₅₄C₂³, (C₂³×C₄).₅₃₆C₂², C₂².₄₇₆(C₂²×D₄), C₂².₁₁₀(C₈⋊C₂²), (C₂²×D₄).₅₆₀C₂², C₂.₈(C₂×C₈⋊C₂²), (C₂×C₄⋊C₄)⋊₄₅C₂², (C₂×C₂²⋊C₈)⋊₃₃C₂, (C₂×C₂²⋊Q₈)⋊₅₁C₂, (C₂×D₄⋊C₄)⋊₃₅C₂, C₂.₃₄(C₂×C₂²≀C₂), (C₂×C₄).₁₀₉₀(C₂×D₄), SmallGroup(128,1729)

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

Derived series

C₁ — C₂×C₄ — C₂×C₂²⋊SD₁₆

Chief series

C₁ — C₂ — C₂² — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — D₄×C₂³ — C₂×C₂²⋊SD₁₆

Lower central

C₁ — C₂ — C₂×C₄ — C₂×C₂²⋊SD₁₆

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂²⋊SD₁₆

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₂²⋊SD₁₆

Generators and relations for C₂×C₂²⋊SD₁₆
G = < a,b,c,d,e | a²=b²=c²=d⁸=e²=1, ab=ba, ac=ca, ad=da, ae=ea, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, ede=d³ >

Subgroups: 1148 in 498 conjugacy classes, 124 normal (18 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂×C₈, SD₁₆, C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂⁴, C₂²⋊C₈, D₄⋊C₄, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂×SD₁₆, C₂×SD₁₆, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂⁵, C₂×C₂²⋊C₈, C₂×D₄⋊C₄, C₂²⋊SD₁₆, C₂×C₂²⋊Q₈, C₂²×SD₁₆, D₄×C₂³, C₂×C₂²⋊SD₁₆
Quotients: C₁, C₂, C₂², D₄, C₂³, SD₁₆, C₂×D₄, C₂⁴, C₂²≀C₂, C₂×SD₁₆, C₈⋊C₂², C₂²×D₄, C₂²⋊SD₁₆, C₂×C₂²≀C₂, C₂²×SD₁₆, C₂×C₈⋊C₂², C₂×C₂²⋊SD₁₆

Smallest permutation representation of C₂×C₂²⋊SD₁₆
►On 32 points

Generators in S₃₂

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

(2 30)(4 32)(6 26)(8 28)(9 22)(11 24)(13 18)(15 20)

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

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

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

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

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

38 conjugacy classes

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

38 irreducible representations

dim

type

image

C₁

C₂

D₄

SD₁₆

C₈⋊C₂²

kernel

C₂×C₂²⋊SD₁₆

C₂×C₂²⋊C₈

C₂×D₄⋊C₄

C₂²⋊SD₁₆

C₂×C₂²⋊Q₈

C₂²×SD₁₆

D₄×C₂³

C₂²×C₄

C₂×D₄

C₂⁴

C₂³

C₂²

# reps

Matrix representation of C₂×C₂²⋊SD₁₆ ►in GL₅(𝔽₁₇)

16	0	0	0	0
0	16	0	0	0
0	0	16	0	0
0	0	0	16	0
0	0	0	0	16

16	0	0	0	0
0	1	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	7	16

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

1	0	0	0	0
0	5	12	0	0
0	5	5	0	0
0	0	0	7	15
0	0	0	7	10

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

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16,0,0,0,0,0,16],[16,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,7,0,0,0,0,16],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,16,0,0,0,0,0,16],[1,0,0,0,0,0,5,5,0,0,0,12,5,0,0,0,0,0,7,7,0,0,0,15,10],[16,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,16] >;

C₂×C₂²⋊SD₁₆ in GAP, Magma, Sage, TeX

C_2\times C_2^2\rtimes {\rm SD}_{16}

% in TeX

G:=Group("C2xC2^2:SD16");

// GroupNames label

G:=SmallGroup(128,1729);

// by ID

G=gap.SmallGroup(128,1729);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,758,2804,1411,172]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^8=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,d*b*d^-1=b*c=c*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^3>;

// generators/relations

G = C2×C22⋊SD16 order 128 = 27

Direct product of C2 and C22⋊SD16

G = C₂×C₂²⋊SD₁₆ order 128 = 2⁷

Direct product of C₂ and C₂²⋊SD₁₆