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G = C22×D12  order 96 = 25·3

Direct product of C22 and D12

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22×D12, C12⋊2C23, D6⋊1C23, C6.3C24, C23.40D6, C6⋊1(C2×D4), (C2×C6)⋊6D4, (C2×C4)⋊9D6, C3⋊1(C22×D4), (C22×C4)⋊7S3, C4⋊2(C22×S3), (S3×C23)⋊3C2, (C22×C12)⋊7C2, C2.4(S3×C23), (C2×C12)⋊12C22, (C2×C6).64C23, (C22×S3)⋊5C22, C22.30(C22×S3), (C22×C6).45C22, SmallGroup(96,207)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C22×D12
Chief series
C1 — C3 — C6 — D6 — C22×S3 — S3×C23 — C22×D12
Lower central
C3 — C6 — C22×D12
Upper central
C1 — C23 — C22×C4

Generators and relations for C22×D12
 G = < a,b,c,d | a2=b2=c12=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 578 in 236 conjugacy classes, 105 normal (9 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C2×C4, D4, C23, C23, C12, D6, D6, C2×C6, C22×C4, C2×D4, C24, D12, C2×C12, C22×S3, C22×S3, C22×C6, C22×D4, C2×D12, C22×C12, S3×C23, C22×D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, C22×D12

Smallest permutation representation of C22×D12
►On 48 points
Generators in S48
(1 28)(2 29)(3 30)(4 31)(5 32)(6 33)(7 34)(8 35)(9 36)(10 25)(11 26)(12 27)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 37)(21 38)(22 39)(23 40)(24 41)

(1 16)(2 17)(3 18)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 13)(11 14)(12 15)(25 42)(26 43)(27 44)(28 45)(29 46)(30 47)(31 48)(32 37)(33 38)(34 39)(35 40)(36 41)

(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 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)

(1 30)(2 29)(3 28)(4 27)(5 26)(6 25)(7 36)(8 35)(9 34)(10 33)(11 32)(12 31)(13 38)(14 37)(15 48)(16 47)(17 46)(18 45)(19 44)(20 43)(21 42)(22 41)(23 40)(24 39)
 
G:=sub<Sym(48)| (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)>;
 
G:=Group( (1,28)(2,29)(3,30)(4,31)(5,32)(6,33)(7,34)(8,35)(9,36)(10,25)(11,26)(12,27)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,37)(21,38)(22,39)(23,40)(24,41), (1,16)(2,17)(3,18)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,13)(11,14)(12,15)(25,42)(26,43)(27,44)(28,45)(29,46)(30,47)(31,48)(32,37)(33,38)(34,39)(35,40)(36,41), (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48), (1,30)(2,29)(3,28)(4,27)(5,26)(6,25)(7,36)(8,35)(9,34)(10,33)(11,32)(12,31)(13,38)(14,37)(15,48)(16,47)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39) );
 
G=PermutationGroup([[(1,28),(2,29),(3,30),(4,31),(5,32),(6,33),(7,34),(8,35),(9,36),(10,25),(11,26),(12,27),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,37),(21,38),(22,39),(23,40),(24,41)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,13),(11,14),(12,15),(25,42),(26,43),(27,44),(28,45),(29,46),(30,47),(31,48),(32,37),(33,38),(34,39),(35,40),(36,41)], [(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48)], [(1,30),(2,29),(3,28),(4,27),(5,26),(6,25),(7,36),(8,35),(9,34),(10,33),(11,32),(12,31),(13,38),(14,37),(15,48),(16,47),(17,46),(18,45),(19,44),(20,43),(21,42),(22,41),(23,40),(24,39)]])
 

C22×D12 is a maximal subgroup of
 (C2×C4)⋊9D12  (C2×C12)⋊5D4  C6.C22≀C2  D12.31D4  D12⋊13D4  (C2×C4)⋊6D12  C23⋊3D12  (C2×D12)⋊10C4  (C2×C4)⋊3D12  C4⋊C4⋊36D6  D12⋊16D4  D12.36D4  C23.53D12  C42⋊9D6  C42⋊11D6  D12⋊23D4  D12⋊19D4  D12⋊21D4  C6.1202+ 1+4  C6.1462+ 1+4  C22×S3×D4
C22×D12 is a maximal quotient of
 C42.276D6  C23⋊4D12  C6.2+ 1+4  C42⋊10D6  C42⋊11D6  C42.92D6  D4⋊5D12  D4⋊6D12  Q8⋊6D12  Q8⋊7D12  C24.9C23  D4.11D12  D4.12D12  D4.13D12

36 conjugacy classes

class 1 2A···2G2H···2O 3 4A4B4C4D6A···6G12A···12H
order12···22···2344446···612···12
size11···16···6222222···22···2

36 irreducible representations

dim111122222
type+++++++++
imageC1C2C2C2S3D4D6D6D12
kernelC22×D12C2×D12C22×C12S3×C23C22×C4C2×C6C2×C4C23C22
# reps1121214618

Matrix representation of C22×D12 ►in GL5(ℤ)

10000
0-1000
00-100
00010
00001
,
-10000
01000
00100
000-10
0000-1
,
-10000
0-1-100
01000
00001
000-10
,
-10000
01100
00-100
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,-1,1,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,1,0,0,0,0,1,-1,0,0,0,0,0,-1,0,0,0,0,0,1] >;
 

C22×D12 in GAP, Magma, Sage, TeX 

C_2^2\times D_{12}
 
% in TeX
 
G:=Group("C2^2xD12");
 
// GroupNames label
 
G:=SmallGroup(96,207);
 
// by ID
 
G=gap.SmallGroup(96,207);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,579,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^12=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
 
// generators/relations
 

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