C2xC2^3.31D4 - GroupNames

G = C₂×C₂³.₃₁D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₃₁D₄

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

Aliases: C₂×C₂³.₃₁D₄, C₂⁴.₁₄₉D₄, C₂³.₁₆Q₁₆, C₂³.₃₁SD₁₆, C₂²⋊Q₈⋊₁C₄, (C₂²×Q₈)⋊₂C₄, C₂².₃₇C₄≀C₂, C₂².₇(C₂×Q₁₆), C₂³.₄₈₉(C₂×D₄), (C₂²×C₄).₂₁₀D₄, C₂².₈(C₂×SD₁₆), C₂²⋊C₈.₁₆₀C₂², C₂².₄₆(C₂³⋊C₄), (C₂³×C₄).₂₀₄C₂², (C₂²×C₄).₆₂₁C₂³, C₂².₆(Q₈⋊C₄), C₂²⋊Q₈.₁₃₃C₂², C₂³.₁₀₁(C₂²⋊C₄), C₂.C₄².₄₉₈C₂², (C₂×C₄⋊C₄)⋊₆C₄, C₄⋊C₄⋊₂(C₂×C₄), (C₂×Q₈)⋊₁(C₂×C₄), C₂.₁₆(C₂×C₄≀C₂), C₂.₁₃(C₂×C₂³⋊C₄), C₂.₇(C₂×Q₈⋊C₄), (C₂×C₂²⋊Q₈).₂C₂, (C₂×C₄).₁₁₄₅(C₂×D₄), (C₂×C₂²⋊C₈).₁₀C₂, (C₂×C₄).₈₆(C₂²⋊C₄), (C₂×C₄).₁₁₁(C₂²×C₄), (C₂²×C₄).₁₉₅(C₂×C₄), C₂².₁₇₅(C₂×C₂²⋊C₄), (C₂×C₂.C₄²).₁₆C₂, SmallGroup(128,231)

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

Derived series

C₁ — C₂×C₄ — C₂×C₂³.₃₁D₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂³×C₄ — C₂×C₂²⋊Q₈ — C₂×C₂³.₃₁D₄

Lower central

C₁ — C₂² — C₂×C₄ — C₂×C₂³.₃₁D₄

Upper central

C₁ — C₂³ — C₂³×C₄ — C₂×C₂³.₃₁D₄

Jennings

C₁ — C₂ — C₂² — C₂²×C₄ — C₂×C₂³.₃₁D₄

Generators and relations for C₂×C₂³.₃₁D₄
G = < a,b,c,d,e,f | a²=b²=c²=d²=1, e⁴=d, f²=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe^-1=bcd, bf=fb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef^-1=bcde³ >

Subgroups: 364 in 168 conjugacy classes, 60 normal (28 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, Q₈, C₂³, C₂³, C₂³, C₂²⋊C₄, C₄⋊C₄, C₄⋊C₄, C₂×C₈, C₂²×C₄, C₂²×C₄, C₂×Q₈, C₂×Q₈, C₂⁴, C₂.C₄², C₂.C₄², C₂²⋊C₈, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₂×C₄⋊C₄, C₂²⋊Q₈, C₂²⋊Q₈, C₂²×C₈, C₂³×C₄, C₂³×C₄, C₂²×Q₈, C₂³.₃₁D₄, C₂×C₂.C₄², C₂×C₂²⋊C₈, C₂×C₂²⋊Q₈, C₂×C₂³.₃₁D₄
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, SD₁₆, Q₁₆, C₂²×C₄, C₂×D₄, C₂³⋊C₄, Q₈⋊C₄, C₄≀C₂, C₂×C₂²⋊C₄, C₂×SD₁₆, C₂×Q₁₆, C₂³.₃₁D₄, C₂×C₂³⋊C₄, C₂×Q₈⋊C₄, C₂×C₄≀C₂, C₂×C₂³.₃₁D₄

Smallest permutation representation of C₂×C₂³.₃₁D₄
►On 32 points

Generators in S₃₂

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

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

(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 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)

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

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

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

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

38 conjugacy classes

class	1	2A	···	2G	2H	2I	2J	2K	4A	4B	4C	4D	4E	···	4N	4O	4P	4Q	4R	8A	···	8H
order	1	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	2	2	2	2	4	···	4	8	8	8	8	4	···	4

38 irreducible representations

dim	1	1	1	1	1	1	1	1	2	2	2	2	2	4
type	+	+	+	+	+				+	+		-		+
image	C₁	C₂	C₂	C₂	C₂	C₄	C₄	C₄	D₄	D₄	SD₁₆	Q₁₆	C₄≀C₂	C₂³⋊C₄
kernel	C₂×C₂³.₃₁D₄	C₂³.₃₁D₄	C₂×C₂.C₄²	C₂×C₂²⋊C₈	C₂×C₂²⋊Q₈	C₂×C₄⋊C₄	C₂²⋊Q₈	C₂²×Q₈	C₂²×C₄	C₂⁴	C₂³	C₂³	C₂²	C₂²
# reps	1	4	1	1	1	2	4	2	3	1	4	4	8	2

Matrix representation of C₂×C₂³.₃₁D₄ ►in GL₆(𝔽₁₇)

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

1	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	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	1	0
0	0	0	0	0	1

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

0	10	0	0	0	0
12	10	0	0	0	0
0	0	12	5	0	0
0	0	12	12	0	0
0	0	0	0	0	4
0	0	0	0	16	0

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

G:=sub<GL(6,GF(17))| [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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,16],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[0,12,0,0,0,0,10,10,0,0,0,0,0,0,12,12,0,0,0,0,5,12,0,0,0,0,0,0,0,16,0,0,0,0,4,0],[16,16,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,0,0,0,0,0,4] >;

C₂×C₂³.₃₁D₄ in GAP, Magma, Sage, TeX

C_2\times C_2^3._{31}D_4

% in TeX

G:=Group("C2xC2^3.31D4");

// GroupNames label

G:=SmallGroup(128,231);

// by ID

G=gap.SmallGroup(128,231);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,232,1123,1018,248,1971]);

// 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,a*d=d*a,a*e=e*a,a*f=f*a,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*c*d*e^3>;

// generators/relations

G = C2×C23.31D4 order 128 = 27

Direct product of C2 and C23.31D4

G = C₂×C₂³.₃₁D₄ order 128 = 2⁷

Direct product of C₂ and C₂³.₃₁D₄