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G = C8○D12  order 96 = 25·3

Central product of C8 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8○D12, C8○Dic6, C8.18D6, D12.2C4, Dic6.2C4, C24.27C22, C12.37C23, (C2×C8)⋊7S3, (S3×C8)⋊6C2, C8○(C3⋊D4), C3⋊1(C8○D4), C8○(C8⋊S3), C8⋊S3⋊7C2, (C2×C24)⋊12C2, C8○(C4○D12), C4.10(C4×S3), D6.1(C2×C4), (C2×C4).78D6, C3⋊D4.2C4, C12.20(C2×C4), C4○D12.6C2, C3⋊C8.11C22, C8○(C4.Dic3), C22.2(C4×S3), C4.Dic3⋊11C2, C4.37(C22×S3), C6.14(C22×C4), Dic3.3(C2×C4), (C4×S3).15C22, (C2×C12).98C22, C2.15(S3×C2×C4), (C2×C6).16(C2×C4), SmallGroup(96,108)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — C8○D12
Chief series
C1 — C3 — C6 — C12 — C4×S3 — C4○D12 — C8○D12
Lower central
C3 — C6 — C8○D12
Upper central
C1 — C8 — C2×C8

Generators and relations for C8○D12
 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, C2×C4, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C2×C8, C2×C8, M4(2), C4○D4, C3⋊C8, C24, Dic6, C4×S3, D12, C3⋊D4, C2×C12, C8○D4, S3×C8, C8⋊S3, C4.Dic3, C2×C24, C4○D12, C8○D12
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C4×S3, C22×S3, C8○D4, S3×C2×C4, C8○D12

Smallest permutation representation of C8○D12
►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)]])
 

C8○D12 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  S3×C8○D4  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
C8○D12 is a maximal quotient of
 C8×Dic6  C24⋊12Q8  C8×D12  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  C8×C3⋊D4  (C22×C8)⋊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+++++++++
imageC1C2C2C2C2C2C4C4C4S3D6D6C4×S3C4×S3C8○D4C8○D12
kernelC8○D12S3×C8C8⋊S3C4.Dic3C2×C24C4○D12Dic6D12C3⋊D4C2×C8C8C2×C4C4C22C3C1
# reps1221112241212248

Matrix representation of C8○D12 ►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] >;
 

C8○D12 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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