C2xC2^4:3C4

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

Aliases: C₂×C₂⁴⋊₃C₄, C₂⁵⋊₄C₄, C₂⁶.₁C₂, C₂⁴.₁₆₃D₄, C₂⁵.₈₂C₂², C₂⁴.₅₂₆C₂³, C₂³.₁₅₉C₂⁴, C₂⁴⋊₁₇(C₂×C₄), (C₂³×C₄)⋊₂C₂², (C₂²×C₄)⋊₇C₂³, C₂³⋊₇(C₂²⋊C₄), C₂³.₅₉₆(C₂×D₄), C₂².₅₀(C₂³×C₄), C₂².₅₉(C₂²×D₄), C₂².₁₀₅C₂²≀C₂, C₂³.₂₀₅(C₂²×C₄), C₂.₁(C₂×C₂²≀C₂), C₂²⋊₃(C₂×C₂²⋊C₄), (C₂²×C₂²⋊C₄)⋊₃C₂, C₂.₄(C₂²×C₂²⋊C₄), (C₂×C₂²⋊C₄)⋊₆₈C₂², SmallGroup(128,1009)

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

Derived series

C₁ — C₂² — C₂×C₂⁴⋊₃C₄

Chief series

C₁ — C₂ — C₂² — C₂³ — C₂⁴ — C₂⁵ — C₂⁶ — C₂×C₂⁴⋊₃C₄

Lower central

C₁ — C₂² — C₂×C₂⁴⋊₃C₄

Upper central

C₁ — C₂⁴ — C₂×C₂⁴⋊₃C₄

Jennings

C₁ — C₂³ — C₂×C₂⁴⋊₃C₄

Subgroups: 3212 in 1732 conjugacy classes, 300 normal (6 characteristic)
C₁, C₂^[×15], C₂^[×24], C₄^[×8], C₂², C₂²^[×58], C₂²^[×296], C₂×C₄^[×40], C₂³, C₂³^[×98], C₂³^[×648], C₂²⋊C₄^[×48], C₂²×C₄^[×8], C₂²×C₄^[×24], C₂⁴, C₂⁴^[×58], C₂⁴^[×296], C₂×C₂²⋊C₄^[×24], C₂×C₂²⋊C₄^[×24], C₂³×C₄^[×4], C₂⁵^[×15], C₂⁵^[×24], C₂⁴⋊₃C₄^[×8], C₂²×C₂²⋊C₄^[×6], C₂⁶, C₂×C₂⁴⋊₃C₄

Quotients:
C₁, C₂^[×15], C₄^[×8], C₂²^[×35], C₂×C₄^[×28], D₄^[×24], C₂³^[×15], C₂²⋊C₄^[×48], C₂²×C₄^[×14], C₂×D₄^[×36], C₂⁴, C₂×C₂²⋊C₄^[×36], C₂²≀C₂^[×16], C₂³×C₄, C₂²×D₄^[×6], C₂⁴⋊₃C₄^[×8], C₂²×C₂²⋊C₄^[×3], C₂×C₂²≀C₂^[×4], C₂×C₂⁴⋊₃C₄

Generators and relations
G = < a,b,c,d,e,f | a²=b²=c²=d²=e²=f⁴=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf^-1=bd=db, be=eb, cd=dc, fcf^-1=ce=ec, de=ed, df=fd, ef=fe >

Smallest permutation representation
►On 32 points

Generators in S₃₂

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

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

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

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₅)

4	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	0
0	0	0	0	0	4

4	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	1	0
0	0	0	0	2	4

4	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	4	0
0	0	0	0	0	4

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	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	1	0
0	0	0	0	0	1

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

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

56 conjugacy classes

class	1	2A	···	2O	2P	···	2AM	4A	···	4P
order	1	2	···	2	2	···	2	4	···	4
size	1	1	···	1	2	···	2	4	···	4

56 irreducible representations

dim	1	1	1	1	1	2
type	+	+	+	+		+
image	C₁	C₂	C₂	C₂	C₄	D₄
kernel	C₂×C₂⁴⋊₃C₄	C₂⁴⋊₃C₄	C₂²×C₂²⋊C₄	C₂⁶	C₂⁵	C₂⁴
# reps	1	8	6	1	16	24

In GAP, Magma, Sage, TeX

C_2\times C_2^4\rtimes_3C_4

% in TeX

G:=Group("C2xC2^4:3C4");

// GroupNames label

G:=SmallGroup(128,1009);

// by ID

G=gap.SmallGroup(128,1009);

# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758]);

// Polycyclic

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

// generators/relations

G = C₂×C₂⁴⋊₃C₄ order 128 = 2⁷

Direct product of C₂ and C₂⁴⋊₃C₄

G = C2×C24⋊3C4 order 128 = 27

Direct product of C2 and C24⋊3C4

G = C₂×C₂⁴⋊₃C₄ order 128 = 2⁷

Direct product of C₂ and C₂⁴⋊₃C₄