Copied to
clipboard

## G = D4×C23order 64 = 26

### Direct product of C23 and D4

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

Aliases: D4×C23, C4⋊C24, C252C2, C22⋊C24, C2.1C25, C247C22, C233C23, (C23×C4)⋊7C2, (C2×C4)⋊4C23, (C22×C4)⋊19C22, SmallGroup(64,261)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — D4×C23
 Chief series C1 — C2 — C22 — C23 — C24 — C25 — D4×C23
 Lower central C1 — C2 — D4×C23
 Upper central C1 — C24 — D4×C23
 Jennings C1 — C2 — D4×C23

Generators and relations for D4×C23
G = < a,b,c,d,e | a2=b2=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 937 in 681 conjugacy classes, 425 normal (5 characteristic)
C1, C2, C2 [×14], C2 [×16], C4 [×8], C22 [×51], C22 [×112], C2×C4 [×28], D4 [×64], C23 [×71], C23 [×112], C22×C4 [×14], C2×D4 [×112], C24, C24 [×28], C24 [×16], C23×C4, C22×D4 [×28], C25 [×2], D4×C23
Quotients: C1, C2 [×31], C22 [×155], D4 [×8], C23 [×155], C2×D4 [×28], C24 [×31], C22×D4 [×14], C25, D4×C23

Smallest permutation representation of D4×C23
On 32 points
Generators in S32
(1 9)(2 10)(3 11)(4 12)(5 16)(6 13)(7 14)(8 15)(17 26)(18 27)(19 28)(20 25)(21 32)(22 29)(23 30)(24 31)
(1 25)(2 26)(3 27)(4 28)(5 24)(6 21)(7 22)(8 23)(9 20)(10 17)(11 18)(12 19)(13 32)(14 29)(15 30)(16 31)
(1 30)(2 31)(3 32)(4 29)(5 17)(6 18)(7 19)(8 20)(9 23)(10 24)(11 21)(12 22)(13 27)(14 28)(15 25)(16 26)
(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 15)(2 14)(3 13)(4 16)(5 12)(6 11)(7 10)(8 9)(17 22)(18 21)(19 24)(20 23)(25 30)(26 29)(27 32)(28 31)

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

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

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

40 conjugacy classes

 class 1 2A ··· 2O 2P ··· 2AE 4A ··· 4H order 1 2 ··· 2 2 ··· 2 4 ··· 4 size 1 1 ··· 1 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 2 type + + + + + image C1 C2 C2 C2 D4 kernel D4×C23 C23×C4 C22×D4 C25 C23 # reps 1 1 28 2 8

Matrix representation of D4×C23 in GL5(ℤ)

 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 -1 0 0 0 1 0
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,-1,0],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

D4×C23 in GAP, Magma, Sage, TeX

D_4\times C_2^3
% in TeX

G:=Group("D4xC2^3");
// GroupNames label

G:=SmallGroup(64,261);
// by ID

G=gap.SmallGroup(64,261);
# by ID

G:=PCGroup([6,-2,2,2,2,2,-2,409]);
// Polycyclic

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

׿
×
𝔽