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G = C23×C4  order 32 = 25

Abelian group of type [2,2,2,4]

direct product, p-group, abelian, monomial

Aliases: C23×C4, SmallGroup(32,45)

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

Derived series
C1 — C23×C4
Chief series
C1 — C2 — C22 — C23 — C24 — C23×C4
Lower central
C1 — C23×C4
Upper central
C1 — C23×C4
Jennings
C1 — C2 — C23×C4

Generators and relations for C23×C4
 G = < a,b,c,d | a2=b2=c2=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, cd=dc >

Subgroups: 118, all normal (4 characteristic)
C1, C2, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C24, C23×C4

Smallest permutation representation of C23×C4
►Regular action on 32 points
Generators in S32
(1 6)(2 7)(3 8)(4 5)(9 23)(10 24)(11 21)(12 22)(13 19)(14 20)(15 17)(16 18)(25 29)(26 30)(27 31)(28 32)

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

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

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

C23×C4 is a maximal subgroup of   C23.7Q8  C23.34D4  C23.8Q8  C23.23D4  C24.4C4  C22.19C24
C23×C4 is a maximal quotient of   C22.11C24  C23.32C23  C23.33C23  Q8○M4(2)

32 conjugacy classes

class 1 2A···2O4A···4P
order12···24···4
size11···11···1

32 irreducible representations

dim1111
type+++
imageC1C2C2C4
kernelC23×C4C22×C4C24C23
# reps114116

Matrix representation of C23×C4 ►in GL4(𝔽5) generated by

4000
0400
0010
0001
,
4000
0100
0040
0004
,
4000
0100
0040
0001
,
2000
0200
0020
0001
G:=sub<GL(4,GF(5))| [4,0,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,0,0,0,0,1],[2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,1] >;
 

C23×C4 in GAP, Magma, Sage, TeX 

C_2^3\times C_4
 
% in TeX
 
G:=Group("C2^3xC4");
 
// GroupNames label
 
G:=SmallGroup(32,45);
 
// by ID
 
G=gap.SmallGroup(32,45);
 
# by ID
 
G:=PCGroup([5,-2,2,2,2,-2,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,c*d=d*c>;
 
// generators/relations
 

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