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G = C24⋊8D6  order 192 = 26·3

3rd semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24⋊8D6, C6.262+ 1+4, C3⋊2D42, D6⋊6(C2×D4), C3⋊D4⋊4D4, (C2×D4)⋊18D6, C22⋊4(S3×D4), C22≀C2⋊4S3, D6⋊D4⋊9C2, C23⋊2D6⋊4C2, C22⋊C4⋊24D6, Dic3⋊3(C2×D4), (C6×D4)⋊7C22, Dic3⋊D4⋊13C2, C12⋊3D4⋊11C2, D6⋊C4⋊11C22, C6.56(C22×D4), C23.14D6⋊2C2, Dic3⋊4D4⋊2C2, (C2×D12)⋊19C22, (C2×C6).134C24, (C2×C12).28C23, Dic3⋊C4⋊9C22, (S3×C23)⋊7C22, (C23×C6)⋊10C22, C2.28(D4⋊6D6), (C4×Dic3)⋊14C22, C6.D4⋊15C22, (C22×S3).53C23, C23.118(C22×S3), (C22×C6).181C23, C22.155(S3×C23), (C2×Dic3).221C23, (C22×Dic3)⋊13C22, (C2×S3×D4)⋊7C2, (C2×C6)⋊6(C2×D4), C2.29(C2×S3×D4), (S3×C2×C4)⋊7C22, (C3×C22≀C2)⋊5C2, (C22×C3⋊D4)⋊8C2, (C2×C3⋊D4)⋊39C22, (C3×C22⋊C4)⋊5C22, (C2×C4).28(C22×S3), SmallGroup(192,1149)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24⋊8D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C23 — C2×S3×D4 — C24⋊8D6
Lower central
C3 — C2×C6 — C24⋊8D6
Upper central
C1 — C22 — C22≀C2

Generators and relations for C24⋊8D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, fbf=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1344 in 428 conjugacy classes, 115 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C22×C6, C4×D4, C22≀C2, C22≀C2, C4⋊D4, C4⋊1D4, C22×D4, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C2×D12, S3×D4, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C6×D4, C6×D4, S3×C23, S3×C23, C23×C6, D42, Dic3⋊4D4, D6⋊D4, Dic3⋊D4, C23⋊2D6, C23.14D6, C12⋊3D4, C3×C22≀C2, C2×S3×D4, C22×C3⋊D4, C24⋊8D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, C22×S3, C22×D4, 2+ 1+4, S3×D4, S3×C23, D42, C2×S3×D4, D4⋊6D6, C24⋊8D6

Smallest permutation representation of C24⋊8D6
►On 48 points
Generators in S48
(1 25)(2 29)(3 27)(4 38)(5 42)(6 40)(7 28)(8 26)(9 30)(10 41)(11 39)(12 37)(13 32)(14 44)(15 34)(16 46)(17 36)(18 48)(19 43)(20 33)(21 45)(22 35)(23 47)(24 31)

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

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

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

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

39 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O 3 4A4B4C4D4E4F4G4H4I6A6B6C6D···6I6J12A12B12C
order122222222222222234444444446666···66121212
size1111222244666612122444666612122224···48888

39 irreducible representations

dim111111111122222444
type+++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D4D6D6D62+ 1+4S3×D4D4⋊6D6
kernelC24⋊8D6Dic3⋊4D4D6⋊D4Dic3⋊D4C23⋊2D6C23.14D6C12⋊3D4C3×C22≀C2C2×S3×D4C22×C3⋊D4C22≀C2C3⋊D4C22⋊C4C2×D4C24C6C22C2
# reps122212112218331142

Matrix representation of C24⋊8D6 ►in GL6(ℤ)

-100000
0-10000
00-1000
000-100
00001-2
00000-1
,
1-20000
0-10000
001000
000100
00001-2
00000-1
,
100000
010000
001000
000100
0000-10
00000-1
,
-100000
0-10000
001000
000100
0000-10
00000-1
,
100000
1-10000
00-1-100
001000
000010
00001-1
,
-100000
0-10000
00-1-100
000100
0000-10
0000-11

G:=sub<GL(6,Integers())| [-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,-2,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,-2,-1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,1,0,0,0,0,0,-1,0,0,0,0,0,0,-1,1,0,0,0,0,-1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,-1,1,0,0,0,0,0,0,-1,-1,0,0,0,0,0,1] >;
 

C24⋊8D6 in GAP, Magma, Sage, TeX 

C_2^4\rtimes_8D_6
 
% in TeX
 
G:=Group("C2^4:8D6");
 
// GroupNames label
 
G:=SmallGroup(192,1149);
 
// by ID
 
G=gap.SmallGroup(192,1149);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,1571,297,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^6=f^2=1,a*b=b*a,e*a*e^-1=f*a*f=a*c=c*a,a*d=d*a,f*b*f=b*c=c*b,e*b*e^-1=b*d=d*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
 
// generators/relations
 

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