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G = C22×D12order 96 = 25·3

Direct product of C22 and D12

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

Aliases: C22×D12, C122C23, D61C23, C6.3C24, C23.40D6, C61(C2×D4), (C2×C6)⋊6D4, (C2×C4)⋊9D6, C31(C22×D4), (C22×C4)⋊7S3, C42(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

C1C6 — C22×D12
C1C3C6D6C22×S3S3×C23 — C22×D12
C3C6 — C22×D12
C1C23C22×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 [×6], C2 [×8], C3, C4 [×4], C22 [×7], C22 [×32], S3 [×8], C6, C6 [×6], C2×C4 [×6], D4 [×16], C23, C23 [×20], C12 [×4], D6 [×8], D6 [×24], C2×C6 [×7], C22×C4, C2×D4 [×12], C24 [×2], D12 [×16], C2×C12 [×6], C22×S3 [×12], C22×S3 [×8], C22×C6, C22×D4, C2×D12 [×12], C22×C12, S3×C23 [×2], C22×D12
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, C22×D12

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

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

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

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

C22×D12 is a maximal subgroup of
(C2×C4)⋊9D12  (C2×C12)⋊5D4  C6.C22≀C2  D12.31D4  D1213D4  (C2×C4)⋊6D12  C233D12  (C2×D12)⋊10C4  (C2×C4)⋊3D12  C4⋊C436D6  D1216D4  D12.36D4  C23.53D12  C429D6  C4211D6  D1223D4  D1219D4  D1221D4  C6.1202+ 1+4  C6.1462+ 1+4  C22×S3×D4
C22×D12 is a maximal quotient of
C42.276D6  C234D12  C6.2+ 1+4  C4210D6  C4211D6  C42.92D6  D45D12  D46D12  Q86D12  Q87D12  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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