C2^3:C4^2

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

Aliases: C₂³⋊₃C₄², C₂⁵.₁₃C₂², C₂³.₁₅₅C₂⁴, C₂⁴.₅₂₃C₂³, C₂².₂₆2₊⁽¹⁺⁴⁾, C₂⁴.₆₄(C₂×C₄), (C₂×C₄²)⋊₁C₂², C₂.₈(C₂²×C₄²), C₂².₁₄(C₂×C₄²), C₂².₂₇(C₂³×C₄), (C₂³×C₄).₃₀C₂², C₂³.₂₀₄(C₂²×C₄), C₂.C₄²⋊₇₀C₂², (C₂²×C₄).₁₂₃₁C₂³, C₂.₁(C₂².₁₁C₂⁴), (C₄×C₂²⋊C₄)⋊₂C₂, (C₂×C₂²⋊C₄)⋊₁₆C₄, C₂²⋊C₄⋊₄₅(C₂×C₄), (C₂²×C₄)⋊₁₁(C₂×C₄), C₂²⋊C₄ ○₂(C₂²⋊C₄), (C₂×C₄).₂₈₈(C₂²×C₄), (C₂²×C₂²⋊C₄).₈C₂, (C₂×C₂²⋊C₄).₅₅₁C₂², C₂.C₄² ○(C₂.C₄²), C₂²⋊C₄ ○(C₂×C₂²⋊C₄), SmallGroup(128,1005)

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

Derived series

C₁ — C₂ — C₂³⋊C₄²

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂²×C₂²⋊C₄ — C₂³⋊C₄²

Lower central

C₁ — C₂ — C₂³⋊C₄²

Upper central

C₁ — C₂³ — C₂³⋊C₄²

Jennings

C₁ — C₂³ — C₂³⋊C₄²

Subgroups: 828 in 456 conjugacy classes, 260 normal (5 characteristic)
C₁, C₂, C₂^[×6], C₂^[×12], C₄^[×24], C₂²^[×19], C₂²^[×52], C₂×C₄^[×24], C₂×C₄^[×48], C₂³, C₂³^[×34], C₂³^[×36], C₄²^[×8], C₂²⋊C₄^[×48], C₂²×C₄^[×36], C₂²×C₄^[×12], C₂⁴^[×15], C₂⁴^[×4], C₂.C₄²^[×8], C₂×C₄²^[×8], C₂×C₂²⋊C₄^[×36], C₂³×C₄^[×6], C₂⁵, C₄×C₂²⋊C₄^[×12], C₂²×C₂²⋊C₄^[×3], C₂³⋊C₄²

Quotients:
C₁, C₂^[×15], C₄^[×24], C₂²^[×35], C₂×C₄^[×84], C₂³^[×15], C₄²^[×16], C₂²×C₄^[×42], C₂⁴, C₂×C₄²^[×12], C₂³×C₄^[×3], 2₊⁽¹⁺⁴⁾^[×4], C₂²×C₄², C₂².₁₁C₂⁴^[×6], C₂³⋊C₄²

Generators and relations
G = < a,b,c,d,e | a²=b²=c²=d⁴=e⁴=1, ab=ba, eae^-1=ac=ca, ad=da, dbd^-1=bc=cb, be=eb, cd=dc, ce=ec, de=ed >

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

(2 24)(4 22)(5 14)(7 16)(10 28)(12 26)(17 32)(19 30)

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

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

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

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

4	0	0	0	0	0
0	2	0	0	0	0
0	0	0	0	4	0
0	0	0	0	0	1
0	0	1	0	0	0
0	0	0	4	0	0

2	0	0	0	0	0
0	3	0	0	0	0
0	0	0	1	0	0
0	0	4	0	0	0
0	0	0	0	0	4
0	0	0	0	1	0

G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[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],[4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,0,1,0,0],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0] >;

68 conjugacy classes

class	1	2A	···	2G	2H	···	2S	4A	···	4AV
order	1	2	···	2	2	···	2	4	···	4
size	1	1	···	1	2	···	2	2	···	2

68 irreducible representations

dim	1	1	1	1	4
type	+	+	+		+
image	C₁	C₂	C₂	C₄	2₊⁽¹⁺⁴⁾
kernel	C₂³⋊C₄²	C₄×C₂²⋊C₄	C₂²×C₂²⋊C₄	C₂×C₂²⋊C₄	C₂²
# reps	1	12	3	48	4

In GAP, Magma, Sage, TeX

C_2^3\rtimes C_4^2

% in TeX

G:=Group("C2^3:C4^2");

// GroupNames label

G:=SmallGroup(128,1005);

// by ID

G=gap.SmallGroup(128,1005);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,219,675]);

// Polycyclic

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

// generators/relations

G = C₂³⋊C₄² order 128 = 2⁷

2^nd semidirect product of C₂³ and C₄² acting via C₄²/C₂²=C₂²

G = C23⋊C42 order 128 = 27

2nd semidirect product of C23 and C42 acting via C42/C22=C22

G = C₂³⋊C₄² order 128 = 2⁷

2^nd semidirect product of C₂³ and C₄² acting via C₄²/C₂²=C₂²