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G = C423D6order 192 = 26·3

1st semidirect product of C42 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C423D6, M4(2)⋊16D6, C4≀C25S3, (S3×D4)⋊4C4, (S3×Q8)⋊4C4, D42S34C4, Q83S34C4, C4○D4.35D6, D12.5(C2×C4), D4.11(C4×S3), (C4×S3).48D4, C4.201(S3×D4), Q8.16(C4×S3), C424S34C2, (C4×C12)⋊10C22, C422S39C2, C12.360(C2×D4), (S3×M4(2))⋊9C2, Dic6.5(C2×C4), D12⋊C410C2, C22.28(S3×D4), Q83Dic32C2, C12.18(C22×C4), D6.8(C22⋊C4), (C2×Dic3).37D4, (C4×Dic3)⋊3C22, (C22×S3).23D4, C4.Dic33C22, (C2×C12).261C23, C31(C42⋊C22), C4○D12.10C22, (C3×M4(2))⋊18C22, Dic3.14(C22⋊C4), C4.18(S3×C2×C4), (C3×C4≀C2)⋊10C2, (C4×S3).5(C2×C4), (S3×C4○D4).2C2, (C3×D4).5(C2×C4), (C2×C6).25(C2×D4), (C3×Q8).5(C2×C4), C2.26(S3×C22⋊C4), C6.25(C2×C22⋊C4), (S3×C2×C4).29C22, (C3×C4○D4).2C22, (C2×C4).368(C22×S3), SmallGroup(192,380)

Series: Derived Chief Lower central Upper central

C1C12 — C423D6
C1C3C6C12C2×C12S3×C2×C4S3×C4○D4 — C423D6
C3C6C12 — C423D6
C1C4C2×C4C4≀C2

Generators and relations for C423D6
 G = < a,b,c,d | a4=b4=c6=d2=1, cac-1=ab=ba, dad=ab-1, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 448 in 154 conjugacy classes, 51 normal (all characteristic)
C1, C2, C2 [×5], C3, C4 [×2], C4 [×6], C22, C22 [×7], S3 [×3], C6, C6 [×2], C8 [×2], C2×C4, C2×C4 [×12], D4, D4 [×6], Q8, Q8 [×2], C23 [×2], Dic3 [×2], Dic3 [×2], C12 [×2], C12 [×2], D6 [×2], D6 [×4], C2×C6, C2×C6, C42, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2) [×2], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4, C4○D4 [×5], C3⋊C8, C24, Dic6, Dic6, C4×S3 [×4], C4×S3 [×3], D12, D12, C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×3], C2×C12, C2×C12 [×2], C3×D4, C3×D4, C3×Q8, C22×S3, C22×S3, C4≀C2, C4≀C2 [×3], C42⋊C2, C2×M4(2), C2×C4○D4, S3×C8, C8⋊S3, C4.Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C4×C12, C3×M4(2), S3×C2×C4, S3×C2×C4, C4○D12, C4○D12, S3×D4, S3×D4, D42S3, D42S3, S3×Q8, Q83S3, C3×C4○D4, C42⋊C22, C424S3, D12⋊C4, Q83Dic3, C3×C4≀C2, C422S3, S3×M4(2), S3×C4○D4, C423D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, S3×C2×C4, S3×D4 [×2], C42⋊C22, S3×C22⋊C4, C423D6

Smallest permutation representation of C423D6
On 48 points
Generators in S48
(1 15 22 25)(2 13 20 29)(3 17 24 27)(4 31 38 18)(5 35 42 16)(6 33 40 14)(7 30 37 46)(8 28 41 44)(9 26 39 48)(10 45 23 32)(11 43 21 36)(12 47 19 34)
(1 38 12 41)(2 42 10 39)(3 40 11 37)(4 19 8 22)(5 23 9 20)(6 21 7 24)(13 16 45 48)(14 43 46 17)(15 18 47 44)(25 31 34 28)(26 29 35 32)(27 33 36 30)
(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 6)(7 8)(11 12)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 24)(20 23)(21 22)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)(37 38)(39 42)(40 41)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F4G4H4I4J4K6A6B6C8A8B8C8D12A12B12C···12G12H24A24B
order12222223444444444446668888121212···12122424
size11246612211244466121212248441212224···4888

36 irreducible representations

dim1111111111112222222224444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D4D4D4D6D6D6C4×S3C4×S3S3×D4S3×D4C42⋊C22C423D6
kernelC423D6C424S3D12⋊C4Q83Dic3C3×C4≀C2C422S3S3×M4(2)S3×C4○D4S3×D4D42S3S3×Q8Q83S3C4≀C2C4×S3C2×Dic3C22×S3C42M4(2)C4○D4D4Q8C4C22C3C1
# reps1111111122221211111221124

Matrix representation of C423D6 in GL4(𝔽73) generated by

55375537
36183618
18365537
37553618
,
00720
00072
1000
0100
,
007272
0010
727200
1000
,
0011
00072
1100
07200
G:=sub<GL(4,GF(73))| [55,36,18,37,37,18,36,55,55,36,55,36,37,18,37,18],[0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[0,0,72,1,0,0,72,0,72,1,0,0,72,0,0,0],[0,0,1,0,0,0,1,72,1,0,0,0,1,72,0,0] >;

C423D6 in GAP, Magma, Sage, TeX

C_4^2\rtimes_3D_6
% in TeX

G:=Group("C4^2:3D6");
// GroupNames label

G:=SmallGroup(192,380);
// by ID

G=gap.SmallGroup(192,380);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,219,58,136,851,438,102,6278]);
// Polycyclic

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

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