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## G = C23×C4order 32 = 25

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

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

32 irreducible representations

 dim 1 1 1 1 type + + + image C1 C2 C2 C4 kernel C23×C4 C22×C4 C24 C23 # reps 1 14 1 16

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

 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
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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