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G = M4(2)⋊26D6  order 192 = 26·3

2nd semidirect product of M4(2) and D6 acting through Inn(M4(2))

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊26D6, C24.52C23, C12.70C24, (C2×C8)⋊22D6, C8○D12⋊15C2, C4○D12.5C4, C3⋊C8.32C23, D12.C4⋊12C2, (S3×C8)⋊11C22, (C2×C24)⋊30C22, D12.29(C2×C4), (C2×D12).17C4, C3⋊1(Q8○M4(2)), C23.27(C4×S3), C4.69(S3×C23), C6.33(C23×C4), C8.44(C22×S3), C8⋊S3⋊19C22, (C2×M4(2))⋊16S3, (C6×M4(2))⋊15C2, (S3×M4(2))⋊10C2, (C4×S3).36C23, C12.93(C22×C4), (C2×Dic6).17C4, Dic6.30(C2×C4), D6.14(C22×C4), (C22×C4).280D6, (C2×C12).510C23, C4○D12.59C22, C4.Dic3⋊40C22, (C3×M4(2))⋊31C22, Dic3.14(C22×C4), (C22×C12).265C22, C4.95(S3×C2×C4), (C2×C3⋊C8)⋊12C22, (C2×C4).58(C4×S3), C3⋊D4.4(C2×C4), C2.34(S3×C22×C4), C22.28(S3×C2×C4), (C4×S3).10(C2×C4), (C2×C3⋊D4).15C4, (C2×C12).132(C2×C4), (C2×C4○D12).22C2, (S3×C2×C4).153C22, (C2×C4.Dic3)⋊25C2, (C22×C6).79(C2×C4), (C2×C6).26(C22×C4), (C22×S3).27(C2×C4), (C2×C4).605(C22×S3), (C2×Dic3).37(C2×C4), SmallGroup(192,1304)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — M4(2)⋊26D6
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — C2×C4○D12 — M4(2)⋊26D6
Lower central
C3 — C6 — M4(2)⋊26D6

Subgroups: 504 in 258 conjugacy classes, 147 normal (41 characteristic)
C1, C2, C2 [×7], C3, C4 [×4], C4 [×4], C22 [×3], C22 [×7], S3 [×4], C6, C6 [×3], C8 [×4], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], Dic3 [×4], C12 [×4], D6 [×4], D6 [×2], C2×C6 [×3], C2×C6, C2×C8 [×2], C2×C8 [×10], M4(2) [×4], M4(2) [×12], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊C8 [×4], C24 [×4], Dic6 [×4], C4×S3 [×8], D12 [×4], C2×Dic3 [×2], C3⋊D4 [×8], C2×C12 [×6], C22×S3 [×2], C22×C6, C2×M4(2), C2×M4(2) [×5], C8○D4 [×8], C2×C4○D4, S3×C8 [×8], C8⋊S3 [×8], C2×C3⋊C8 [×2], C4.Dic3 [×4], C2×C24 [×2], C3×M4(2) [×4], C2×Dic6, S3×C2×C4 [×2], C2×D12, C4○D12 [×8], C2×C3⋊D4 [×2], C22×C12, Q8○M4(2), C8○D12 [×4], S3×M4(2) [×4], D12.C4 [×4], C2×C4.Dic3, C6×M4(2), C2×C4○D12, M4(2)⋊26D6

Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], S3, C2×C4 [×28], C23 [×15], D6 [×7], C22×C4 [×14], C24, C4×S3 [×4], C22×S3 [×7], C23×C4, S3×C2×C4 [×6], S3×C23, Q8○M4(2), S3×C22×C4, M4(2)⋊26D6

Generators and relations
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=dad=a5, bc=cb, dbd=a4b, dcd=c-1 >

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

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

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

Matrix representation ►G ⊆ GL4(𝔽73) generated by

47020
14602
500260
63607227
,
306000
134300
20574313
16366030
,
727200
1000
111011
2737720
,
727200
0100
111011
064072
G:=sub<GL(4,GF(73))| [47,1,50,63,0,46,0,60,2,0,26,72,0,2,0,27],[30,13,20,16,60,43,57,36,0,0,43,60,0,0,13,30],[72,1,11,27,72,0,10,37,0,0,1,72,0,0,1,0],[72,0,11,0,72,1,10,64,0,0,1,0,0,0,1,72] >;
 

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I6A6B6C6D6E8A···8H8I···8P12A12B12C12D12E12F24A···24H
order1222222223444444444666668···88···812121212121224···24
size1122266662112226666222442···26···62222444···4

54 irreducible representations

dim1111111111122222244
type+++++++++++
imageC1C2C2C2C2C2C2C4C4C4C4S3D6D6D6C4×S3C4×S3Q8○M4(2)M4(2)⋊26D6
kernelM4(2)⋊26D6C8○D12S3×M4(2)D12.C4C2×C4.Dic3C6×M4(2)C2×C4○D12C2×Dic6C2×D12C4○D12C2×C3⋊D4C2×M4(2)C2×C8M4(2)C22×C4C2×C4C23C3C1
# reps1444111228412416224

In GAP, Magma, Sage, TeX 

M_{4(2)}\rtimes_{26}D_6
 
% in TeX
 
G:=Group("M4(2):26D6");
 
// GroupNames label
 
G:=SmallGroup(192,1304);
 
// by ID
 
G=gap.SmallGroup(192,1304);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,570,80,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=d*a*d=a^5,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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