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G = M4(2).25D6order 192 = 26·3

8th non-split extension by M4(2) of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2).25D6, C8.10(C4×S3), (C2×C8).68D6, C8.C43S3, C8⋊S3.1C4, C24.28(C2×C4), D6.7(C4⋊C4), (C4×S3).50D4, C4.211(S3×D4), C24.C47C2, C12.370(C2×D4), C22.4(S3×Q8), (C2×Dic3).4Q8, (C22×S3).3Q8, Dic3.9(C4⋊C4), C12.53(C22×C4), (C2×C24).40C22, (S3×M4(2)).3C2, C12.53D412C2, C31(M4(2).C4), (C2×C12).309C23, C4.Dic3.13C22, (C3×M4(2)).27C22, C3⋊C8.3(C2×C4), C4.83(S3×C2×C4), C6.17(C2×C4⋊C4), C2.18(S3×C4⋊C4), (C2×C6).2(C2×Q8), (C4×S3).8(C2×C4), (C3×C8.C4)⋊3C2, (C2×C8⋊S3).1C2, (C2×C3⋊C8).77C22, (S3×C2×C4).43C22, (C2×C4).412(C22×S3), SmallGroup(192,452)

Series: Derived Chief Lower central Upper central

C1C12 — M4(2).25D6
C1C3C6C12C2×C12S3×C2×C4C2×C8⋊S3 — M4(2).25D6
C3C6C12 — M4(2).25D6
C1C4C2×C4C8.C4

Generators and relations for M4(2).25D6
 G = < a,b,c,d | a8=b2=1, c6=d2=a6b, bab=a5, cac-1=a-1b, dad-1=a3b, bc=cb, bd=db, dcd-1=c5 >

Subgroups: 224 in 102 conjugacy classes, 51 normal (27 characteristic)
C1, C2, C2 [×3], C3, C4 [×2], C4 [×2], C22, C22 [×3], S3 [×2], C6, C6, C8 [×2], C8 [×6], C2×C4, C2×C4 [×5], C23, Dic3 [×2], C12 [×2], D6 [×2], D6, C2×C6, C2×C8, C2×C8 [×3], M4(2) [×2], M4(2) [×8], C22×C4, C3⋊C8 [×2], C3⋊C8 [×2], C24 [×2], C24 [×2], C4×S3 [×4], C2×Dic3, C2×C12, C22×S3, C8.C4, C8.C4 [×3], C2×M4(2) [×3], S3×C8 [×2], C8⋊S3 [×4], C8⋊S3 [×2], C2×C3⋊C8, C4.Dic3 [×2], C2×C24, C3×M4(2) [×2], S3×C2×C4, M4(2).C4, C24.C4, C12.53D4 [×2], C3×C8.C4, C2×C8⋊S3, S3×M4(2) [×2], M4(2).25D6
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3, C2×C4 [×6], D4 [×2], Q8 [×2], C23, D6 [×3], C4⋊C4 [×4], C22×C4, C2×D4, C2×Q8, C4×S3 [×2], C22×S3, C2×C4⋊C4, S3×C2×C4, S3×D4, S3×Q8, M4(2).C4, S3×C4⋊C4, M4(2).25D6

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

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

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

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

36 conjugacy classes

class 1 2A2B2C2D 3 4A4B4C4D4E6A6B8A···8F8G···8L12A12B12C24A24B24C24D24E24F24G24H
order12222344444668···88···81212122424242424242424
size11266211266244···412···1222444448888

36 irreducible representations

dim111111122222224444
type++++++++--+++-
imageC1C2C2C2C2C2C4S3D4Q8Q8D6D6C4×S3S3×D4S3×Q8M4(2).C4M4(2).25D6
kernelM4(2).25D6C24.C4C12.53D4C3×C8.C4C2×C8⋊S3S3×M4(2)C8⋊S3C8.C4C4×S3C2×Dic3C22×S3C2×C8M4(2)C8C4C22C3C1
# reps112112812111241124

Matrix representation of M4(2).25D6 in GL4(𝔽5) generated by

0033
0012
2200
4300
,
1000
0100
0040
0004
,
2400
4400
0032
0024
,
1400
4400
0012
0014
G:=sub<GL(4,GF(5))| [0,0,2,4,0,0,2,3,3,1,0,0,3,2,0,0],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[2,4,0,0,4,4,0,0,0,0,3,2,0,0,2,4],[1,4,0,0,4,4,0,0,0,0,1,1,0,0,2,4] >;

M4(2).25D6 in GAP, Magma, Sage, TeX

M_4(2)._{25}D_6
% in TeX

G:=Group("M4(2).25D6");
// GroupNames label

G:=SmallGroup(192,452);
// by ID

G=gap.SmallGroup(192,452);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,219,58,136,438,102,6278]);
// Polycyclic

G:=Group<a,b,c,d|a^8=b^2=1,c^6=d^2=a^6*b,b*a*b=a^5,c*a*c^-1=a^-1*b,d*a*d^-1=a^3*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^5>;
// generators/relations

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