C2xC4^2:2C2 - GroupNames

G = C₂×C₄²⋊₂C₂ order 64 = 2⁶

Direct product of C₂ and C₄²⋊₂C₂

direct product, p-group, metabelian, nilpotent (class 2), monomial

Aliases: C₂×C₄²⋊₂C₂, C₄²⋊₁₄C₂², C₂².₂₂C₂⁴, C₂³.₇₂C₂³, C₂⁴.₁₄C₂², (C₂×C₄²)⋊₄C₂, C₄⋊C₄⋊₁₃C₂², (C₂×C₄).₅₃C₂³, C₂².₃₄(C₄○D₄), C₂²⋊C₄.₁₂C₂², (C₂²×C₄).₆₀C₂², (C₂×C₄⋊C₄)⋊₁₇C₂, C₂.₁₁(C₂×C₄○D₄), (C₂×C₂²⋊C₄).₁₂C₂, SmallGroup(64,209)

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

Derived series

C₁ — C₂² — C₂×C₄²⋊₂C₂

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂²×C₄ — C₂×C₄² — C₂×C₄²⋊₂C₂

Lower central

C₁ — C₂² — C₂×C₄²⋊₂C₂

Upper central

C₁ — C₂³ — C₂×C₄²⋊₂C₂

Jennings

C₁ — C₂² — C₂×C₄²⋊₂C₂

Generators and relations for C₂×C₄²⋊₂C₂
G = < a,b,c,d | a²=b⁴=c⁴=d²=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc², dcd=b²c^-1 >

Subgroups: 177 in 123 conjugacy classes, 81 normal (6 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₂×C₄, C₂×C₄, C₂³, C₂³, C₂³, C₄², C₂²⋊C₄, C₄⋊C₄, C₂²×C₄, C₂⁴, C₂×C₄², C₂×C₂²⋊C₄, C₂×C₄⋊C₄, C₄²⋊₂C₂, C₂×C₄²⋊₂C₂
Quotients: C₁, C₂, C₂², C₂³, C₄○D₄, C₂⁴, C₄²⋊₂C₂, C₂×C₄○D₄, C₂×C₄²⋊₂C₂

Character table of C₂×C₄²⋊₂C₂

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	4G	4H	4I	4J	4K	4L	4M	4N	4O	4P	4Q	4R
size	1	1	1	1	1	1	1	1	4	4	2	2	2	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4
ρ₁	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	trivial
ρ₂	1	-1	1	1	1	-1	-1	-1	-1	1	-1	1	1	-1	-1	1	-1	1	-1	1	1	-1	1	-1	1	-1	-1	1	linear of order 2
ρ₃	1	1	1	1	1	1	1	1	1	1	1	-1	1	1	-1	-1	-1	-1	-1	-1	1	-1	-1	-1	-1	-1	1	1	linear of order 2
ρ₄	1	-1	1	1	1	-1	-1	-1	-1	1	-1	-1	1	-1	1	-1	1	-1	1	-1	1	1	-1	1	-1	1	-1	1	linear of order 2
ρ₅	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	1	-1	-1	-1	-1	-1	-1	1	1	1	1	linear of order 2
ρ₆	1	-1	1	1	1	-1	-1	-1	1	-1	1	-1	-1	1	-1	1	-1	1	1	-1	-1	1	-1	1	1	-1	-1	1	linear of order 2
ρ₇	1	1	1	1	1	1	1	1	-1	-1	-1	1	-1	-1	-1	-1	-1	-1	1	1	-1	1	1	1	-1	-1	1	1	linear of order 2
ρ₈	1	-1	1	1	1	-1	-1	-1	1	-1	1	1	-1	1	1	-1	1	-1	-1	1	-1	-1	1	-1	-1	1	-1	1	linear of order 2
ρ₉	1	-1	1	1	1	-1	-1	-1	-1	1	1	1	-1	1	1	-1	1	-1	-1	1	-1	-1	-1	1	1	-1	1	-1	linear of order 2
ρ₁₀	1	1	1	1	1	1	1	1	1	1	-1	1	-1	-1	-1	-1	-1	-1	1	1	-1	1	-1	-1	1	1	-1	-1	linear of order 2
ρ₁₁	1	-1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	1	-1	1	-1	1	1	-1	-1	1	1	-1	-1	1	1	-1	linear of order 2
ρ₁₂	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	1	1	1	1	-1	-1	-1	-1	1	1	-1	-1	-1	-1	linear of order 2
ρ₁₃	1	-1	1	1	1	-1	-1	-1	1	-1	-1	-1	1	-1	1	-1	1	-1	1	-1	1	1	1	-1	1	-1	1	-1	linear of order 2
ρ₁₄	1	1	1	1	1	1	1	1	-1	-1	1	-1	1	1	-1	-1	-1	-1	-1	-1	1	-1	1	1	1	1	-1	-1	linear of order 2
ρ₁₅	1	-1	1	1	1	-1	-1	-1	1	-1	-1	1	1	-1	-1	1	-1	1	-1	1	1	-1	-1	1	-1	1	1	-1	linear of order 2
ρ₁₆	1	1	1	1	1	1	1	1	-1	-1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	linear of order 2
ρ₁₇	2	-2	-2	-2	2	2	2	-2	0	0	2i	0	2i	-2i	0	0	0	0	0	0	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₈	2	2	-2	-2	2	-2	-2	2	0	0	2i	0	-2i	-2i	0	0	0	0	0	0	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₁₉	2	2	-2	2	-2	2	-2	-2	0	0	0	-2i	0	0	0	0	0	0	-2i	2i	0	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₀	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	-2i	-2i	2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₁	2	-2	2	-2	-2	2	-2	2	0	0	0	0	0	0	2i	2i	-2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₂	2	-2	-2	2	-2	-2	2	2	0	0	0	2i	0	0	0	0	0	0	-2i	-2i	0	2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₃	2	2	-2	-2	2	-2	-2	2	0	0	-2i	0	2i	2i	0	0	0	0	0	0	-2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₄	2	-2	-2	-2	2	2	2	-2	0	0	-2i	0	-2i	2i	0	0	0	0	0	0	2i	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₅	2	2	-2	2	-2	2	-2	-2	0	0	0	2i	0	0	0	0	0	0	2i	-2i	0	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₆	2	2	2	-2	-2	-2	2	-2	0	0	0	0	0	0	-2i	2i	2i	-2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₇	2	-2	-2	2	-2	-2	2	2	0	0	0	-2i	0	0	0	0	0	0	2i	2i	0	-2i	0	0	0	0	0	0	complex lifted from C₄○D₄
ρ₂₈	2	2	2	-2	-2	-2	2	-2	0	0	0	0	0	0	2i	-2i	-2i	2i	0	0	0	0	0	0	0	0	0	0	complex lifted from C₄○D₄

Smallest permutation representation of C₂×C₄²⋊₂C₂
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

C₂×C₄²⋊₂C₂ is a maximal subgroup of
C₄².₃₇₂D₄ C₂⁴.₂₀₃C₂³ C₂³.₂₅₅C₂⁴ C₂⁴.₂₃₀C₂³ C₂⁴.₂₈₆C₂³ C₂³.₃₆₇C₂⁴ C₂³.₃₆₈C₂⁴ C₂³.₃₆₉C₂⁴ C₂⁴.₂₉₅C₂³ C₂³.₃₇₉C₂⁴ C₂³.₃₈₀C₂⁴ C₂⁴.₅₇₃C₂³ C₂³.₃₉₆C₂⁴ C₂³.₄₁₂C₂⁴ C₂³.₄₁₉C₂⁴ C₂⁴.₃₁₁C₂³ C₂⁴.₃₂₆C₂³ C₂⁴.₃₂₇C₂³ C₂⁴.₃₃₁C₂³ C₂⁴.₃₃₂C₂³ C₄²⋊₂₃D₄ C₄²⋊₂₄D₄ C₄².₁₈₄D₄ C₄²⋊₃₀D₄ C₄².₁₉₂D₄ C₄²⋊₃₂D₄ C₄².₁₉₈D₄ C₂³.₅₈₅C₂⁴ C₂³.₅₈₉C₂⁴ C₂³.₅₉₁C₂⁴ C₂³.₅₉₅C₂⁴ C₂⁴.₄₀₅C₂³ C₂³.₆₀₂C₂⁴ C₂³.₆₀₅C₂⁴ C₂⁴.₄₁₃C₂³ C₂³.₆₁₅C₂⁴ C₂³.₆₁₆C₂⁴ C₂³.₆₁₈C₂⁴ C₂³.₆₂₁C₂⁴ C₂³.₆₂₂C₂⁴ C₂⁴.₄₁₈C₂³ C₂³.₆₂₅C₂⁴ C₂³.₆₂₇C₂⁴ C₄²⋊₃₃D₄ C₄².₂₀₀D₄ C₄²⋊₃₅D₄ C₄²⋊₄₃D₄ C₄³⋊₁₃C₂ C₂².₁₁₀C₂⁵ C₂².₁₄₉C₂⁵ C₂².₁₅₃C₂⁵
C₂×C₄²⋊₂C₂ is a maximal quotient of
C₂³.₃₀₁C₂⁴ C₂³.₃₈₀C₂⁴ C₂⁴.₅₇₃C₂³ C₂⁴.₅₇₇C₂³ C₂⁴.₃₀₄C₂³ C₂³.₃₉₅C₂⁴ C₂³.₃₉₆C₂⁴ C₂³.₃₉₇C₂⁴ C₂³.₄₁₀C₂⁴ C₂³.₄₁₁C₂⁴ C₂³.₄₁₂C₂⁴ C₂³.₄₁₃C₂⁴ C₂³.₄₁₄C₂⁴ C₂³.₅₄₃C₂⁴ C₂³.₅₄₄C₂⁴ C₂³.₅₄₅C₂⁴ C₄²⋊₄₃D₄ C₄³⋊₁₃C₂ C₄²⋊₁₅Q₈

Matrix representation of C₂×C₄²⋊₂C₂ ►in GL₅(𝔽₅)

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

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

1	0	0	0	0
0	2	0	0	0
0	0	2	0	0
0	0	0	3	3
0	0	0	0	2

1	0	0	0	0
0	4	0	0	0
0	0	1	0	0
0	0	0	1	0
0	0	0	3	4

G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,4,1],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,3,2],[1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,3,0,0,0,0,4] >;

C₂×C₄²⋊₂C₂ in GAP, Magma, Sage, TeX

C_2\times C_4^2\rtimes_2C_2

% in TeX

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

// GroupNames label

G:=SmallGroup(64,209);

// by ID

G=gap.SmallGroup(64,209);

# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,295,650,86]);

// Polycyclic

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

// generators/relations

Export

Character table of C₂×C₄²⋊₂C₂ in TeX

G = C2×C42⋊2C2 order 64 = 26

Direct product of C2 and C42⋊2C2

G = C₂×C₄²⋊₂C₂ order 64 = 2⁶

Direct product of C₂ and C₄²⋊₂C₂