C2xC2^3.C8 - GroupNames

G = C₂×C₂³.C₈ order 128 = 2⁷

Direct product of C₂ and C₂³.C₈

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

Aliases: C₂×C₂³.C₈, C₂⁴.₃C₈, M₅(2)⋊₁₀C₂², C₈.₁₂₃(C₂×D₄), (C₂×C₈).₃₈₉D₄, C₄ ○(C₂³.C₈), (C₂²×C₄).₆C₈, (C₂×M₅(2))⋊₈C₂, C₂³.₁₇(C₂×C₈), (C₂³×C₄).₂₂C₄, (C₂²×C₈).₁₉C₄, C₈.₃₀(C₂²⋊C₄), C₄.₂₅(C₂²⋊C₈), (C₂×C₈).₃₈₄C₂³, C₄.₄₉(C₂×M₄(2)), (C₂×C₄).₈₁M₄(2), (C₂×M₄(2)).₂₉C₄, C₂².₁₂(C₂²×C₈), C₂².₄₄(C₂²⋊C₈), (C₂²×C₈).₄₁₅C₂², (C₂²×M₄(2)).₂₃C₂, (C₂×M₄(2)).₃₂₈C₂², (C₂×C₄).₃₈(C₂×C₈), (C₂×C₈).₁₄₉(C₂×C₄), C₂.₂₄(C₂×C₂²⋊C₈), C₄.₁₁₅(C₂×C₂²⋊C₄), (C₂²×C₄).₄₈₆(C₂×C₄), (C₂×C₄).₅₅₈(C₂²×C₄), (C₂×C₄).₃₆₂(C₂²⋊C₄), SmallGroup(128,846)

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

Derived series

C₁ — C₂² — C₂×C₂³.C₈

Chief series

C₁ — C₂ — C₄ — C₈ — C₂×C₈ — C₂²×C₈ — C₂²×M₄(2) — C₂×C₂³.C₈

Lower central

C₁ — C₂ — C₂² — C₂×C₂³.C₈

Upper central

C₁ — C₂×C₄ — C₂²×C₈ — C₂×C₂³.C₈

Jennings

C₁ — C₂ — C₂ — C₂ — C₂ — C₄ — C₄ — C₂×C₈ — C₂×C₂³.C₈

Generators and relations for C₂×C₂³.C₈
G = < a,b,c,d,e | a²=b²=c²=d²=1, e⁸=d, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe^-1=bcd, ece^-1=cd=dc, de=ed >

Subgroups: 196 in 118 conjugacy classes, 60 normal (26 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₂², C₂², C₈, C₈, C₂×C₄, C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₁₆, C₂×C₈, C₂×C₈, C₂×C₈, M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂⁴, C₂×C₁₆, M₅(2), M₅(2), C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂³.C₈, C₂×M₅(2), C₂²×M₄(2), C₂×C₂³.C₈
Quotients: C₁, C₂, C₄, C₂², C₈, C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂×C₈, M₄(2), C₂²×C₄, C₂×D₄, C₂²⋊C₈, C₂×C₂²⋊C₄, C₂²×C₈, C₂×M₄(2), C₂³.C₈, C₂×C₂²⋊C₈, C₂×C₂³.C₈

Smallest permutation representation of C₂×C₂³.C₈
►On 32 points

Generators in S₃₂

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

(1 25)(2 10)(3 19)(5 29)(6 14)(7 23)(9 17)(11 27)(13 21)(15 31)(18 26)(22 30)

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

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

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	4B	4C	4D	4E	4F	4G	4H	8A	···	8H	8I	8J	8K	8L	16A	···	16P
order	1	2	2	2	2	2	2	2	4	4	4	4	4	4	4	4	8	···	8	8	8	8	8	16	···	16
size	1	1	1	1	2	2	4	4	1	1	1	1	2	2	4	4	2	···	2	4	4	4	4	4	···	4

44 irreducible representations

dim	1	1	1	1	1	1	1	1	1	2	2	4
type	+	+	+	+						+
image	C₁	C₂	C₂	C₂	C₄	C₄	C₄	C₈	C₈	D₄	M₄(2)	C₂³.C₈
kernel	C₂×C₂³.C₈	C₂³.C₈	C₂×M₅(2)	C₂²×M₄(2)	C₂²×C₈	C₂×M₄(2)	C₂³×C₄	C₂²×C₄	C₂⁴	C₂×C₈	C₂×C₄	C₂
# reps	1	4	2	1	2	4	2	12	4	4	4	4

Matrix representation of C₂×C₂³.C₈ ►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	16	0
0	0	0	0	0	16

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	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

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

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

C₂×C₂³.C₈ in GAP, Magma, Sage, TeX

C_2\times C_2^3.C_8

% in TeX

G:=Group("C2xC2^3.C8");

// GroupNames label

G:=SmallGroup(128,846);

// by ID

G=gap.SmallGroup(128,846);

# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,-2,112,141,2019,1411,102,124]);

// Polycyclic

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

// generators/relations

G = C2×C23.C8 order 128 = 27

Direct product of C2 and C23.C8

G = C₂×C₂³.C₈ order 128 = 2⁷

Direct product of C₂ and C₂³.C₈