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G = C2×C4.Dic3order 96 = 25·3

Direct product of C2 and C4.Dic3

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

Aliases: C2×C4.Dic3, C62M4(2), C12.41C23, C23.4Dic3, C3⋊C812C22, (C2×C12).9C4, C33(C2×M4(2)), C12.36(C2×C4), (C2×C4).100D6, C4(C4.Dic3), (C22×C4).6S3, (C22×C6).7C4, (C2×C4).6Dic3, C4.9(C2×Dic3), (C22×C12).9C2, C6.21(C22×C4), C4.41(C22×S3), C2.3(C22×Dic3), (C2×C12).100C22, C22.12(C2×Dic3), (C2×C3⋊C8)⋊12C2, (C2×C6).32(C2×C4), SmallGroup(96,128)

Series: Derived Chief Lower central Upper central

C1C6 — C2×C4.Dic3
C1C3C6C12C3⋊C8C2×C3⋊C8 — C2×C4.Dic3
C3C6 — C2×C4.Dic3
C1C2×C4C22×C4

Generators and relations for C2×C4.Dic3
 G = < a,b,c,d | a2=b4=1, c6=b2, d2=b2c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 98 in 68 conjugacy classes, 49 normal (19 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×2], C22, C22 [×2], C22 [×2], C6, C6 [×2], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×4], C23, C12 [×2], C12 [×2], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C3⋊C8 [×4], C2×C12 [×2], C2×C12 [×4], C22×C6, C2×M4(2), C2×C3⋊C8 [×2], C4.Dic3 [×4], C22×C12, C2×C4.Dic3
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], C23, Dic3 [×4], D6 [×3], M4(2) [×2], C22×C4, C2×Dic3 [×6], C22×S3, C2×M4(2), C4.Dic3 [×2], C22×Dic3, C2×C4.Dic3

Smallest permutation representation of C2×C4.Dic3
On 48 points
Generators in S48
(1 22)(2 23)(3 24)(4 13)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(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 10 7 4)(2 11 8 5)(3 12 9 6)(13 22 19 16)(14 23 20 17)(15 24 21 18)(25 28 31 34)(26 29 32 35)(27 30 33 36)(37 40 43 46)(38 41 44 47)(39 42 45 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 25 10 34 7 31 4 28)(2 30 11 27 8 36 5 33)(3 35 12 32 9 29 6 26)(13 45 22 42 19 39 16 48)(14 38 23 47 20 44 17 41)(15 43 24 40 21 37 18 46)

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

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

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

C2×C4.Dic3 is a maximal subgroup of
C12.8C42  C12.(C4⋊C4)  C423Dic3  C12.2C42  (C2×C12).Q8  C12.10C42  M4(2)⋊Dic3  (C2×C24)⋊C4  C12.4C42  M4(2)⋊4Dic3  C12.21C42  D62M4(2)  Dic3⋊M4(2)  C127M4(2)  C4⋊C4.225D6  C4○D12⋊C4  C4⋊C4.232D6  C12.5C42  C42.43D6  C4⋊C436D6  C4⋊C4.237D6  C426D6  C42.47D6  C123M4(2)  C4⋊D4⋊S3  C3⋊C85D4  C3⋊C86D4  C3⋊C8.6D4  C12.12C42  Dic3⋊C8⋊C2  (C22×C8)⋊7S3  Dic3×M4(2)  Dic34M4(2)  C23.8Dic6  C23.9Dic6  D66M4(2)  M4(2).31D6  C24.6Dic3  (C6×D4)⋊6C4  (C6×Q8)⋊6C4  C4○D43Dic3  (C6×D4).11C4  (C6×D4)⋊9C4  (C6×D4).16C4  C2×S3×M4(2)  M4(2)⋊26D6  C12.76C24  C12.C24  C60.59(C2×C4)
C2×C4.Dic3 is a maximal quotient of
C127M4(2)  C42.285D6  C42.270D6  C42.47D6  C123M4(2)  C42.210D6  C24.6Dic3  C60.59(C2×C4)

36 conjugacy classes

class 1 2A2B2C2D2E 3 4A4B4C4D4E4F6A···6G8A···8H12A···12H
order12222234444446···68···812···12
size11112221111222···26···62···2

36 irreducible representations

dim111111222222
type+++++-+-
imageC1C2C2C2C4C4S3Dic3D6Dic3M4(2)C4.Dic3
kernelC2×C4.Dic3C2×C3⋊C8C4.Dic3C22×C12C2×C12C22×C6C22×C4C2×C4C2×C4C23C6C2
# reps124162133148

Matrix representation of C2×C4.Dic3 in GL3(𝔽73) generated by

7200
0720
0072
,
100
0460
0027
,
7200
030
0024
,
4600
001
0460
G:=sub<GL(3,GF(73))| [72,0,0,0,72,0,0,0,72],[1,0,0,0,46,0,0,0,27],[72,0,0,0,3,0,0,0,24],[46,0,0,0,0,46,0,1,0] >;

C2×C4.Dic3 in GAP, Magma, Sage, TeX

C_2\times C_4.{\rm Dic}_3
% in TeX

G:=Group("C2xC4.Dic3");
// GroupNames label

G:=SmallGroup(96,128);
// by ID

G=gap.SmallGroup(96,128);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-3,48,362,69,2309]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=1,c^6=b^2,d^2=b^2*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations

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