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G = C8oD12  order 96 = 25·3

Central product of C8 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8oD12, C8oDic6, C8.18D6, D12.2C4, Dic6.2C4, C24.27C22, C12.37C23, (C2xC8):7S3, (S3xC8):6C2, C8o(C3:D4), C3:1(C8oD4), C8o(C8:S3), C8:S3:7C2, (C2xC24):12C2, C8o(C4oD12), C4.10(C4xS3), D6.1(C2xC4), (C2xC4).78D6, C3:D4.2C4, C12.20(C2xC4), C4oD12.6C2, C3:C8.11C22, C8o(C4.Dic3), C22.2(C4xS3), C4.Dic3:11C2, C4.37(C22xS3), C6.14(C22xC4), Dic3.3(C2xC4), (C4xS3).15C22, (C2xC12).98C22, C2.15(S3xC2xC4), (C2xC6).16(C2xC4), SmallGroup(96,108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C8oD12
Chief series
C1 — C3 — C6 — C12 — C4xS3 — C4oD12 — C8oD12
Lower central
C3 — C6 — C8oD12
Upper central
C1 — C8 — C2xC8

Generators and relations for C8oD12
 G = < a,b,c | a8=c2=1, b6=a4, ab=ba, ac=ca, cbc=a4b5 >

Subgroups: 114 in 62 conjugacy classes, 37 normal (23 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2xC4, C2xC4, D4, Q8, Dic3, C12, D6, C2xC6, C2xC8, C2xC8, M4(2), C4oD4, C3:C8, C24, Dic6, C4xS3, D12, C3:D4, C2xC12, C8oD4, S3xC8, C8:S3, C4.Dic3, C2xC24, C4oD12, C8oD12
Quotients: C1, C2, C4, C22, S3, C2xC4, C23, D6, C22xC4, C4xS3, C22xS3, C8oD4, S3xC2xC4, C8oD12

Smallest permutation representation of C8oD12
►On 48 points
Generators in S48
(1 21 45 36 7 15 39 30)(2 22 46 25 8 16 40 31)(3 23 47 26 9 17 41 32)(4 24 48 27 10 18 42 33)(5 13 37 28 11 19 43 34)(6 14 38 29 12 20 44 35)

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

C8oD12 is a maximal subgroup of
 D12.C8  Dic6.C8  D24:11C4  D24:4C4  C24.18D4  C24.19D4  C24.42D4  D12.4C8  C16.12D6  C24.100D4  C24.54D4  C24.23D4  C24.44D4  C24.29D4  M4(2):26D6  S3xC8oD4  M4(2):28D6  D8:13D6  SD16:13D6  D12.30D4  D8:15D6  D8:11D6  D8.10D6  D36.2C4  C24.63D6  C24.64D6  D12.2Dic3  C3:C8.22D6  C24.95D6  C40.54D6  C40.34D6  D12.2Dic5  D60.5C4  D60.6C4  D12.2F5  D60.C4  C5:C8.D6
C8oD12 is a maximal quotient of
 C8xDic6  C24:12Q8  C8xD12  C8:6D12  D6.C42  C42.243D6  C24:C4:C2  D6:C8:C2  D6:2M4(2)  Dic3:M4(2)  C42.27D6  D6:3M4(2)  C42.30D6  C42.31D6  C12.12C42  Dic3:C8:C2  C8xC3:D4  (C22xC8):7S3  C24:33D4  D36.2C4  C24.63D6  C24.64D6  D12.2Dic3  C3:C8.22D6  C24.95D6  C40.54D6  C40.34D6  D12.2Dic5  D60.5C4  D60.6C4  D12.2F5  D60.C4  C5:C8.D6

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B6C8A8B8C8D8E8F8G8H8I8J12A12B12C12D24A···24H
order1222234444466688888888881212121224···24
size11266211266222111122666622222···2

36 irreducible representations

dim1111111112222222
type+++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6C4xS3C4xS3C8oD4C8oD12
kernelC8oD12S3xC8C8:S3C4.Dic3C2xC24C4oD12Dic6D12C3:D4C2xC8C8C2xC4C4C22C3C1
# reps1221112241212248

Matrix representation of C8oD12 ►in GL2(F73) generated by

630
063
,
667
6659
,
11
072
G:=sub<GL(2,GF(73))| [63,0,0,63],[66,66,7,59],[1,0,1,72] >;
 

C8oD12 in GAP, Magma, Sage, TeX 

C_8\circ D_{12}
 
% in TeX
 
G:=Group("C8oD12");
 
// GroupNames label
 
G:=SmallGroup(96,108);
 
// by ID
 
G=gap.SmallGroup(96,108);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-3,217,50,69,2309]);
 
// Polycyclic
 
G:=Group<a,b,c|a^8=c^2=1,b^6=a^4,a*b=b*a,a*c=c*a,c*b*c=a^4*b^5>;
 
// generators/relations
 

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