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G = D4×C23  order 64 = 26

Direct product of C23 and D4

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

Aliases: D4×C23, C4⋊C24, C25⋊2C2, C22⋊C24, C2.1C25, C24⋊7C22, C23⋊3C23, (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, C2, C4, C22, C22, C2×C4, D4, C23, C23, C22×C4, C2×D4, C24, C24, C24, C23×C4, C22×D4, C25, D4×C23
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 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)]])
 

D4×C23 is a maximal subgroup of
 C24.50D4  C25⋊C4  C25.C4  C23.35D8  C24.90D4  C23.191C24  C24⋊7D4  C24.94D4  C23.308C24  C24⋊8D4  C23.333C24  C23.335C24  C24⋊9D4  C24.177D4  C22.73C25
D4×C23 is a maximal quotient of
 C22.38C25  C22.73C25  C22.74C25  C22.75C25  C22.76C25  C22.77C25  C22.78C25  C4⋊2+ 1+4  C4⋊2- 1+4  C22.87C25  C22.88C25  C22.89C25  C8.C24  D8⋊C23  C4.C25

40 conjugacy classes

class 1 2A···2O2P···2AE4A···4H
order12···22···24···4
size11···12···22···2

40 irreducible representations

dim11112
type+++++
imageC1C2C2C2D4
kernelD4×C23C23×C4C22×D4C25C23
# reps112828

Matrix representation of D4×C23 ►in GL5(ℤ)

-10000
01000
00100
000-10
0000-1
,
-10000
01000
00-100
00010
00001
,
-10000
0-1000
00100
000-10
0000-1
,
-10000
01000
00-100
0000-1
00010
,
-10000
01000
00100
000-10
00001

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
 

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