C2xC4^2:C2^2 - GroupNames

G = C₂×C₄²⋊C₂² order 128 = 2⁷

Direct product of C₂ and C₄²⋊C₂²

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

Aliases: C₂×C₄²⋊C₂², C₄²⋊₅C₂³, C₂⁴.₉₉D₄, M₄(2)⋊₁₂C₂³, C₄≀C₂⋊₁₅C₂², (C₂²×D₄)⋊₂₉C₄, C₄.₁₁(C₂³×C₄), (C₂²×Q₈)⋊₂₃C₄, (C₂×C₄).₁₈₁C₂⁴, (C₂×C₄²)⋊₃₄C₂², C₄○D₄.₂₀C₂³, D₄.₂₂(C₂²×C₄), C₂³.₆₄₁(C₂×D₄), C₄.₁₈₁(C₂²×D₄), (C₂²×C₄).₇₈₂D₄, Q₈.₂₂(C₂²×C₄), C₄ ○(C₄²⋊C₂²), C₂².₂₈(C₂²×D₄), C₄²⋊C₂⋊₇₆C₂², C₂³.₈₈(C₂²⋊C₄), (C₂²×M₄(2))⋊₂₂C₂, (C₂×M₄(2))⋊₇₃C₂², (C₂³×C₄).₅₁₆C₂², (C₂²×C₄).₁₄₉₉C₂³, (C₂×C₄≀C₂)⋊₂₉C₂, C₄○D₄⋊₁₆(C₂×C₄), (C₂×C₄○D₄)⋊₂₂C₄, (C₂×D₄)⋊₅₁(C₂×C₄), (C₂×Q₈)⋊₄₂(C₂×C₄), C₄.₇₇(C₂×C₂²⋊C₄), (C₂×C₄).₁₄₀₇(C₂×D₄), (C₂×C₄²⋊C₂)⋊₄₃C₂, (C₂×C₄).₂₄₆(C₂²×C₄), (C₂²×C₄).₃₂₆(C₂×C₄), (C₂²×C₄○D₄).₂₂C₂, C₂².₂₄(C₂×C₂²⋊C₄), C₂.₄₃(C₂²×C₂²⋊C₄), (C₂×C₄).₂₈₆(C₂²⋊C₄), (C₂×C₄○D₄).₂₇₆C₂², SmallGroup(128,1632)

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

Derived series

C₁ — C₄ — C₂×C₄²⋊C₂²

Chief series

C₁ — C₂ — C₄ — C₂×C₄ — C₂²×C₄ — C₂³×C₄ — C₂²×C₄○D₄ — C₂×C₄²⋊C₂²

Lower central

C₁ — C₂ — C₄ — C₂×C₄²⋊C₂²

Upper central

C₁ — C₂×C₄ — C₂³×C₄ — C₂×C₄²⋊C₂²

Jennings

C₁ — C₂ — C₂ — C₂×C₄ — C₂×C₄²⋊C₂²

Generators and relations for C₂×C₄²⋊C₂²
G = < a,b,c,d,e | a²=b⁴=c⁴=d²=e²=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd=b^-1c, ebe=bc², cd=dc, ce=ec, de=ed >

Subgroups: 668 in 386 conjugacy classes, 172 normal (34 characteristic)
C₁, C₂, C₂, C₂, C₄, C₄, C₄, C₂², C₂², C₂², C₈, C₂×C₄, C₂×C₄, C₂×C₄, D₄, D₄, Q₈, Q₈, C₂³, C₂³, C₂³, C₄², C₄², C₂²⋊C₄, C₄⋊C₄, C₂×C₈, M₄(2), M₄(2), C₂²×C₄, C₂²×C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂×Q₈, C₂×Q₈, C₄○D₄, C₄○D₄, C₂⁴, C₂⁴, C₄≀C₂, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊C₂, C₄²⋊C₂, C₂²×C₈, C₂×M₄(2), C₂×M₄(2), C₂³×C₄, C₂³×C₄, C₂²×D₄, C₂²×D₄, C₂²×Q₈, C₂×C₄○D₄, C₂×C₄○D₄, C₂×C₄≀C₂, C₄²⋊C₂², C₂×C₄²⋊C₂, C₂²×M₄(2), C₂²×C₄○D₄, C₂×C₄²⋊C₂²
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, C₂³, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂⁴, C₂×C₂²⋊C₄, C₂³×C₄, C₂²×D₄, C₄²⋊C₂², C₂²×C₂²⋊C₄, C₂×C₄²⋊C₂²

Smallest permutation representation of C₂×C₄²⋊C₂²
►On 32 points

Generators in S₃₂

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

(2 17)(3 25)(4 31)(5 22)(6 11)(7 13)(10 15)(12 24)(14 21)(19 26)(20 32)(28 29)

(1 27)(3 25)(6 11)(8 9)(14 21)(16 23)(18 30)(20 32)

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

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

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

44 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	4A	4B	4C	4D	4E	···	4J	4K	···	4V	8A	···	8H
order	1	2	2	2	2	···	2	2	2	2	2	4	4	4	4	4	···	4	4	···	4	8	···	8
size	1	1	1	1	2	···	2	4	4	4	4	1	1	1	1	2	···	2	4	···	4	4	···	4

44 irreducible representations

dim

type

image

C₁

C₂

C₄

D₄

C₄²⋊C₂²

kernel

C₂×C₄²⋊C₂²

C₂×C₄≀C₂

C₄²⋊C₂²

C₂×C₄²⋊C₂

C₂²×M₄(2)

C₂²×C₄○D₄

C₂²×D₄

C₂²×Q₈

C₂×C₄○D₄

C₂²×C₄

C₂⁴

C₂

# reps

Matrix representation of C₂×C₄²⋊C₂² ►in GL₆(𝔽₁₇)

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	1	0
0	0	0	0	0	1

0	13	0	0	0	0
13	0	0	0	0	0
0	0	0	0	0	8
0	0	0	0	13	4
0	0	8	0	0	0
0	0	9	16	0	0

1	0	0	0	0	0
0	1	0	0	0	0
0	0	4	0	0	0
0	0	0	4	0	0
0	0	0	0	4	0
0	0	0	0	0	4

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

G:=sub<GL(6,GF(17))| [16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,13,0,0,0,0,13,0,0,0,0,0,0,0,0,0,8,9,0,0,0,0,0,16,0,0,0,13,0,0,0,0,8,4,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[16,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,16,0],[16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C₂×C₄²⋊C₂² in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes C_2^2

% in TeX

G:=Group("C2xC4^2:C2^2");

// GroupNames label

G:=SmallGroup(128,1632);

// by ID

G=gap.SmallGroup(128,1632);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,2804,1411,172,124]);

// Polycyclic

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

// generators/relations

G = C2×C42⋊C22 order 128 = 27

Direct product of C2 and C42⋊C22

G = C₂×C₄²⋊C₂² order 128 = 2⁷

Direct product of C₂ and C₄²⋊C₂²