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G = D6⋊M4(2)  order 192 = 26·3

1st semidirect product of D6 and M4(2) acting via M4(2)/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D65M4(2), (C2×C8)⋊18D6, D6⋊C822C2, C22⋊C811S3, (C4×S3).46D4, C4.195(S3×D4), (C2×C6)⋊1M4(2), (C2×C24)⋊26C22, C12.354(C2×D4), (S3×C23).5C4, C23.52(C4×S3), C6.5(C2×M4(2)), C224(C8⋊S3), C31(C24.4C4), (C22×C4).320D6, C2.12(S3×M4(2)), C12.55D423C2, (C2×C12).821C23, D6.10(C22⋊C4), (C22×Dic3).10C4, Dic3.11(C22⋊C4), (C22×C12).338C22, (S3×C2×C4).18C4, C2.9(C2×C8⋊S3), (C2×C3⋊C8)⋊27C22, (C2×C8⋊S3)⋊11C2, C6.8(C2×C22⋊C4), C2.9(S3×C22⋊C4), (C3×C22⋊C8)⋊20C2, (C2×C4).132(C4×S3), (S3×C22×C4).17C2, C22.103(S3×C2×C4), (C2×C12).154(C2×C4), (S3×C2×C4).273C22, (C22×C6).39(C2×C4), (C2×C6).76(C22×C4), (C22×S3).55(C2×C4), (C2×C4).763(C22×S3), (C2×Dic3).85(C2×C4), SmallGroup(192,285)

Series: Derived Chief Lower central Upper central

C1C2×C6 — D6⋊M4(2)
C1C3C6C12C2×C12S3×C2×C4S3×C22×C4 — D6⋊M4(2)
C3C2×C6 — D6⋊M4(2)
C1C2×C4C22⋊C8

Generators and relations for D6⋊M4(2)
 G = < a,b,c,d | a6=b2=c8=d2=1, bab=cac-1=dad=a-1, cbc-1=ab, dbd=a4b, dcd=c5 >

Subgroups: 512 in 190 conjugacy classes, 61 normal (33 characteristic)
C1, C2 [×3], C2 [×6], C3, C4 [×2], C4 [×4], C22, C22 [×2], C22 [×18], S3 [×4], C6 [×3], C6 [×2], C8 [×4], C2×C4 [×2], C2×C4 [×16], C23, C23 [×8], Dic3 [×2], Dic3, C12 [×2], C12, D6 [×4], D6 [×12], C2×C6, C2×C6 [×2], C2×C6 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×9], C24, C3⋊C8 [×2], C24 [×2], C4×S3 [×4], C4×S3 [×6], C2×Dic3 [×2], C2×Dic3 [×2], C2×C12 [×2], C2×C12 [×2], C22×S3 [×2], C22×S3 [×6], C22×C6, C22⋊C8, C22⋊C8 [×3], C2×M4(2) [×2], C23×C4, C8⋊S3 [×4], C2×C3⋊C8 [×2], C2×C24 [×2], S3×C2×C4 [×4], S3×C2×C4 [×4], C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8 [×2], C12.55D4, C3×C22⋊C8, C2×C8⋊S3 [×2], S3×C22×C4, D6⋊M4(2)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×4], C23, D6 [×3], C22⋊C4 [×4], M4(2) [×4], C22×C4, C2×D4 [×2], C4×S3 [×2], C22×S3, C2×C22⋊C4, C2×M4(2) [×2], C8⋊S3 [×2], S3×C2×C4, S3×D4 [×2], C24.4C4, S3×C22⋊C4, C2×C8⋊S3, S3×M4(2), D6⋊M4(2)

Smallest permutation representation of D6⋊M4(2)
On 48 points
Generators in S48
(1 22 32 15 46 40)(2 33 47 16 25 23)(3 24 26 9 48 34)(4 35 41 10 27 17)(5 18 28 11 42 36)(6 37 43 12 29 19)(7 20 30 13 44 38)(8 39 45 14 31 21)
(1 32)(2 23)(3 26)(4 17)(5 28)(6 19)(7 30)(8 21)(9 34)(10 41)(11 36)(12 43)(13 38)(14 45)(15 40)(16 47)(25 33)(27 35)(29 37)(31 39)
(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 11)(2 16)(3 13)(4 10)(5 15)(6 12)(7 9)(8 14)(17 27)(18 32)(19 29)(20 26)(21 31)(22 28)(23 25)(24 30)(33 47)(34 44)(35 41)(36 46)(37 43)(38 48)(39 45)(40 42)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E8A8B8C8D8E8F8G8H12A12B12C12D12E12F24A···24H
order122222222234444444444666668888888812121212121224···24
size111122666621111226666222444444121212122222444···4

48 irreducible representations

dim11111111122222222244
type+++++++++++
imageC1C2C2C2C2C2C4C4C4S3D4D6D6M4(2)M4(2)C4×S3C4×S3C8⋊S3S3×D4S3×M4(2)
kernelD6⋊M4(2)D6⋊C8C12.55D4C3×C22⋊C8C2×C8⋊S3S3×C22×C4S3×C2×C4C22×Dic3S3×C23C22⋊C8C4×S3C2×C8C22×C4D6C2×C6C2×C4C23C22C4C2
# reps12112142214214422822

Matrix representation of D6⋊M4(2) in GL4(𝔽73) generated by

0100
72100
00720
00072
,
17200
07200
0010
007272
,
166500
85700
0012
00072
,
0100
1000
00720
00072
G:=sub<GL(4,GF(73))| [0,72,0,0,1,1,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,72,72,0,0,0,0,1,72,0,0,0,72],[16,8,0,0,65,57,0,0,0,0,1,0,0,0,2,72],[0,1,0,0,1,0,0,0,0,0,72,0,0,0,0,72] >;

D6⋊M4(2) in GAP, Magma, Sage, TeX

D_6\rtimes M_4(2)
% in TeX

G:=Group("D6:M4(2)");
// GroupNames label

G:=SmallGroup(192,285);
// by ID

G=gap.SmallGroup(192,285);
# by ID

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

G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a*b,d*b*d=a^4*b,d*c*d=c^5>;
// generators/relations

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