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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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, C23, C23, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C24, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C22⋊C8, C22⋊C8, C2×M4(2), C23×C4, C8⋊S3, C2×C3⋊C8, C2×C24, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, C24.4C4, D6⋊C8, C12.55D4, C3×C22⋊C8, C2×C8⋊S3, S3×C22×C4, D6⋊M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, M4(2), C22×C4, C2×D4, C4×S3, C22×S3, C2×C22⋊C4, C2×M4(2), C8⋊S3, S3×C2×C4, S3×D4, 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 24 35 12 46 29)(2 30 47 13 36 17)(3 18 37 14 48 31)(4 32 41 15 38 19)(5 20 39 16 42 25)(6 26 43 9 40 21)(7 22 33 10 44 27)(8 28 45 11 34 23)
(1 35)(2 17)(3 37)(4 19)(5 39)(6 21)(7 33)(8 23)(9 43)(10 27)(11 45)(12 29)(13 47)(14 31)(15 41)(16 25)(26 40)(28 34)(30 36)(32 38)
(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 16)(2 13)(3 10)(4 15)(5 12)(6 9)(7 14)(8 11)(17 36)(18 33)(19 38)(20 35)(21 40)(22 37)(23 34)(24 39)(25 46)(26 43)(27 48)(28 45)(29 42)(30 47)(31 44)(32 41)

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

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

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

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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