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G = M4(2)⋊28D6  order 192 = 26·3

4th semidirect product of M4(2) and D6 acting through Inn(M4(2))

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊28D6, C12.72C24, C24.54C23, (S3×D4).C4, (S3×Q8).C4, (C2×C8)⋊23D6, C8○D4⋊12S3, D4⋊2S3.C4, Q8⋊3S3.C4, C8○D12⋊17C2, C4○D4.59D6, D4.13(C4×S3), C3⋊C8.33C23, Q8.19(C4×S3), D12.C4⋊13C2, (S3×C8)⋊12C22, (C2×C24)⋊32C22, D12.21(C2×C4), D4.Dic3⋊8C2, C3⋊2(Q8○M4(2)), C8.56(C22×S3), C4.71(S3×C23), C6.35(C23×C4), C8⋊S3⋊22C22, (S3×M4(2))⋊11C2, (C4×S3).37C23, C12.39(C22×C4), Dic6.22(C2×C4), D6.16(C22×C4), (C2×C12).514C23, C4○D12.52C22, C4.Dic3⋊27C22, (C3×M4(2))⋊33C22, Dic3.16(C22×C4), C4.39(S3×C2×C4), C22.5(S3×C2×C4), (C3×C8○D4)⋊14C2, (C2×C3⋊C8)⋊13C22, (S3×C4○D4).3C2, C3⋊D4.2(C2×C4), (C2×C8⋊S3)⋊28C2, C2.36(S3×C22×C4), (C4×S3).11(C2×C4), (C3×D4).17(C2×C4), (C2×C6).5(C22×C4), (C3×Q8).18(C2×C4), (S3×C2×C4).154C22, (C22×S3).28(C2×C4), (C2×C4).607(C22×S3), (C2×Dic3).38(C2×C4), (C3×C4○D4).44C22, SmallGroup(192,1309)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C6 — M4(2)⋊28D6
Chief series
C1 — C3 — C6 — C12 — C4×S3 — S3×C2×C4 — S3×C4○D4 — M4(2)⋊28D6
Lower central
C3 — C6 — M4(2)⋊28D6
Upper central
C1 — C4 — C8○D4

Generators and relations for M4(2)⋊28D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a5, ad=da, cbc-1=dbd=a4b, dcd=c-1 >

Subgroups: 512 in 258 conjugacy classes, 147 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, Dic3, C12, C12, D6, D6, D6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, C3⋊C8, C3⋊C8, C24, C24, Dic6, C4×S3, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C2×M4(2), C8○D4, C8○D4, C2×C4○D4, S3×C8, C8⋊S3, C8⋊S3, C2×C3⋊C8, C4.Dic3, C2×C24, C3×M4(2), S3×C2×C4, C4○D12, S3×D4, D4⋊2S3, S3×Q8, Q8⋊3S3, C3×C4○D4, Q8○M4(2), C2×C8⋊S3, C8○D12, S3×M4(2), D12.C4, D4.Dic3, C3×C8○D4, S3×C4○D4, M4(2)⋊28D6
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C22×C4, C24, C4×S3, C22×S3, C23×C4, S3×C2×C4, S3×C23, Q8○M4(2), S3×C22×C4, M4(2)⋊28D6

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

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

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

54 conjugacy classes

class 1 2A2B2C2D2E2F2G2H 3 4A4B4C4D4E4F4G4H4I6A6B6C6D8A···8H8I···8P12A12B12C12D12E24A24B24C24D24E···24J
order122222222344444444466668···88···812121212122424242424···24
size112226666211222666624442···26···62244422224···4

54 irreducible representations

dim11111111111122222244
type++++++++++++
imageC1C2C2C2C2C2C2C2C4C4C4C4S3D6D6D6C4×S3C4×S3Q8○M4(2)M4(2)⋊28D6
kernelM4(2)⋊28D6C2×C8⋊S3C8○D12S3×M4(2)D12.C4D4.Dic3C3×C8○D4S3×C4○D4S3×D4D4⋊2S3S3×Q8Q8⋊3S3C8○D4C2×C8M4(2)C4○D4D4Q8C3C1
# reps13333111662213316224

Matrix representation of M4(2)⋊28D6 ►in GL4(𝔽5) generated by

0001
0330
0120
2000
,
0400
4000
1002
0330
,
0120
1004
3004
0030
,
0430
4001
0002
0030
G:=sub<GL(4,GF(5))| [0,0,0,2,0,3,1,0,0,3,2,0,1,0,0,0],[0,4,1,0,4,0,0,3,0,0,0,3,0,0,2,0],[0,1,3,0,1,0,0,0,2,0,0,3,0,4,4,0],[0,4,0,0,4,0,0,0,3,0,0,3,0,1,2,0] >;
 

M4(2)⋊28D6 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_{28}D_6
 
% in TeX
 
G:=Group("M4(2):28D6");
 
// GroupNames label
 
G:=SmallGroup(192,1309);
 
// by ID
 
G=gap.SmallGroup(192,1309);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,1123,80,102,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^8=b^2=c^6=d^2=1,b*a*b=c*a*c^-1=a^5,a*d=d*a,c*b*c^-1=d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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