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G = C8○D12order 96 = 25·3

Central product of C8 and D12

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C8D12, C8Dic6, C8.18D6, D12.2C4, Dic6.2C4, C24.27C22, C12.37C23, (C2×C8)⋊7S3, (S3×C8)⋊6C2, C8(C3⋊D4), C31(C8○D4), C8(C8⋊S3), C8⋊S37C2, (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.Dic311C2, 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

C1C6 — C8○D12
C1C3C6C12C4×S3C4○D12 — C8○D12
C3C6 — C8○D12
C1C8C2×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 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×2], S3 [×2], C6, C6, C8 [×2], C8 [×2], C2×C4, C2×C4 [×2], D4 [×3], Q8, Dic3 [×2], C12 [×2], D6 [×2], C2×C6, C2×C8, C2×C8 [×2], M4(2) [×3], C4○D4, C3⋊C8 [×2], C24 [×2], Dic6, C4×S3 [×2], D12, C3⋊D4 [×2], C2×C12, C8○D4, S3×C8 [×2], C8⋊S3 [×2], C4.Dic3, C2×C24, C4○D12, C8○D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, D6 [×3], C22×C4, C4×S3 [×2], C22×S3, C8○D4, S3×C2×C4, C8○D12

Smallest permutation representation of C8○D12
On 48 points
Generators in S48
(1 34 17 47 7 28 23 41)(2 35 18 48 8 29 24 42)(3 36 19 37 9 30 13 43)(4 25 20 38 10 31 14 44)(5 26 21 39 11 32 15 45)(6 27 22 40 12 33 16 46)
(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 23)(14 22)(15 21)(16 20)(17 19)(25 33)(26 32)(27 31)(28 30)(34 36)(37 47)(38 46)(39 45)(40 44)(41 43)

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

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

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

C8○D12 is a maximal subgroup of
D12.C8  Dic6.C8  D2411C4  D244C4  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  D813D6  SD1613D6  D12.30D4  D815D6  D811D6  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  C2412Q8  C8×D12  C86D12  D6.C42  C42.243D6  C24⋊C4⋊C2  D6⋊C8⋊C2  D62M4(2)  Dic3⋊M4(2)  C42.27D6  D63M4(2)  C42.30D6  C42.31D6  C12.12C42  Dic3⋊C8⋊C2  C8×C3⋊D4  (C22×C8)⋊7S3  C2433D4  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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