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G = M4(2)⋊24D6  order 192 = 26·3

8th semidirect product of M4(2) and D6 acting via D6/C6=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: M4(2)⋊24D6, C23.25D12, C4○D12⋊8C4, (C2×D12)⋊16C4, (C2×C4).55D12, C4.30(D6⋊C4), (C2×Dic6)⋊16C4, D12.27(C2×C4), (C2×C12).178D4, C12.446(C2×D4), D12⋊C4⋊13C2, (C6×M4(2))⋊23C2, (C2×M4(2))⋊15S3, C12.70(C22×C4), Dic6.28(C2×C4), (C4×Dic3)⋊4C22, C22.17(C2×D12), (C22×C6).107D4, (C22×C4).162D6, C12.29(C22⋊C4), (C2×C12).420C23, C22.29(D6⋊C4), C3⋊3(C42⋊C22), C4○D12.43C22, C23.26D6⋊17C2, (C3×M4(2))⋊36C22, (C22×C12).193C22, C4.55(S3×C2×C4), (C2×C4).56(C4×S3), C2.35(C2×D6⋊C4), (C2×C6).33(C2×D4), C4.137(C2×C3⋊D4), C6.63(C2×C22⋊C4), (C2×C12).113(C2×C4), (C2×C4○D12).15C2, (C2×C4).258(C3⋊D4), (C2×C6).23(C22⋊C4), (C2×C4).513(C22×S3), SmallGroup(192,698)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C12 — M4(2)⋊24D6
Chief series
C1 — C3 — C6 — C12 — C2×C12 — C4○D12 — C2×C4○D12 — M4(2)⋊24D6
Lower central
C3 — C6 — C12 — M4(2)⋊24D6
Upper central
C1 — C4 — C22×C4 — C2×M4(2)

Generators and relations for M4(2)⋊24D6
 G = < a,b,c,d | a8=b2=c6=d2=1, bab=cac-1=a5, dad=a3b, bc=cb, dbd=a4b, dcd=c-1 >

Subgroups: 440 in 154 conjugacy classes, 59 normal (41 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C4×Dic3, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C4○D12, C2×C3⋊D4, C22×C12, C42⋊C22, D12⋊C4, C23.26D6, C6×M4(2), C2×C4○D12, M4(2)⋊24D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, D6⋊C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C42⋊C22, C2×D6⋊C4, M4(2)⋊24D6

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F 3 4A4B4C4D4E4F···4K6A6B6C6D6E8A8B8C8D12A12B12C12D12E12F24A···24H
order12222223444444···466666888812121212121224···24
size11222121221122212···122224444442222444···4

42 irreducible representations

dim1111111122222222244
type++++++++++++
imageC1C2C2C2C2C4C4C4S3D4D4D6D6C4×S3D12C3⋊D4D12C42⋊C22M4(2)⋊24D6
kernelM4(2)⋊24D6D12⋊C4C23.26D6C6×M4(2)C2×C4○D12C2×Dic6C2×D12C4○D12C2×M4(2)C2×C12C22×C6M4(2)C22×C4C2×C4C2×C4C2×C4C23C3C1
# reps1411122413121424224

Matrix representation of M4(2)⋊24D6 ►in GL6(𝔽73)

7200000
0720000
0000027
0000460
0007200
001000
,
100000
010000
0004600
0027000
0000027
0000460
,
110000
7200000
001000
000100
0000720
0000072
,
72720000
010000
001000
0007200
0000072
0000720

G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,1,0,0,0,0,72,0,0,0,0,46,0,0,0,0,27,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,27,0,0,0,0,46,0,0,0,0,0,0,0,0,46,0,0,0,0,27,0],[1,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;
 

M4(2)⋊24D6 in GAP, Magma, Sage, TeX 

M_4(2)\rtimes_{24}D_6
 
% in TeX
 
G:=Group("M4(2):24D6");
 
// GroupNames label
 
G:=SmallGroup(192,698);
 
// by ID
 
G=gap.SmallGroup(192,698);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,422,387,58,136,1684,438,102,6278]);
 
// 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,d*a*d=a^3*b,b*c=c*b,d*b*d=a^4*b,d*c*d=c^-1>;
 
// generators/relations
 

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