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G = C24.47D6  order 192 = 26·3

36th non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.47D6, C6.312+ 1+4, C22≀C2⋊9S3, (C2×D4).88D6, C22⋊C4.3D6, Dic3⋊D4⋊15C2, D6⋊3D4⋊15C2, C12⋊3D4⋊13C2, D6⋊C4⋊16C22, C24⋊4S3⋊10C2, C23.14D6⋊6C2, C23.9D6⋊15C2, (C2×D12)⋊21C22, (C2×C6).139C24, (C2×C12).33C23, C4⋊Dic3⋊28C22, C2.33(D4⋊6D6), Dic3⋊C4⋊13C22, (C4×Dic3)⋊19C22, (C6×D4).113C22, C23.8D6⋊13C2, C23.23D6⋊6C2, (C23×C6).71C22, C3⋊1(C22.54C24), C6.D4⋊19C22, (C22×S3).58C23, C22.160(S3×C23), (C22×C6).184C23, C23.121(C22×S3), (C2×Dic3).64C23, (C22×Dic3)⋊17C22, (S3×C2×C4)⋊11C22, (C3×C22≀C2)⋊10C2, (C2×C3⋊D4)⋊11C22, (C2×C4).33(C22×S3), (C3×C22⋊C4).4C22, SmallGroup(192,1154)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.47D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×S3 — S3×C2×C4 — C23.9D6 — C24.47D6
Lower central
C3 — C2×C6 — C24.47D6
Upper central
C1 — C22 — C22≀C2

Generators and relations for C24.47D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 736 in 252 conjugacy classes, 91 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C23, C23, Dic3, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C2×D4, C24, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C22×C6, C22≀C2, C22≀C2, C4⋊D4, C22.D4, C42⋊2C2, C4⋊1D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C3×C22⋊C4, S3×C2×C4, C2×D12, C22×Dic3, C2×C3⋊D4, C6×D4, C6×D4, C23×C6, C22.54C24, C23.8D6, C23.9D6, Dic3⋊D4, C23.23D6, D6⋊3D4, C23.14D6, C12⋊3D4, C24⋊4S3, C3×C22≀C2, C24.47D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S3×C23, C22.54C24, D4⋊6D6, C24.47D6

Smallest permutation representation of C24.47D6
►On 48 points
Generators in S48
(2 17)(4 19)(6 21)(8 23)(10 13)(12 15)(25 31)(26 40)(27 33)(28 42)(29 35)(30 44)(32 46)(34 48)(36 38)(37 43)(39 45)(41 47)

(1 7)(3 9)(5 11)(14 20)(16 22)(18 24)(25 45)(26 40)(27 47)(28 42)(29 37)(30 44)(31 39)(32 46)(33 41)(34 48)(35 43)(36 38)

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

33 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I 3 4A4B4C4D···4I6A6B6C6D···6I6J12A12B12C
order122222222234444···46666···66121212
size111144441212244412···122224···48888

33 irreducible representations

dim1111111111222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2S3D6D6D62+ 1+4D4⋊6D6
kernelC24.47D6C23.8D6C23.9D6Dic3⋊D4C23.23D6D6⋊3D4C23.14D6C12⋊3D4C24⋊4S3C3×C22≀C2C22≀C2C22⋊C4C2×D4C24C6C2
# reps1222122121133136

Matrix representation of C24.47D6 ►in GL8(𝔽13)

10000000
01000000
001200000
000120000
00001000
000031200
00000010
000000312
,
120000000
01000000
00100000
000120000
00001000
00000100
000000120
000000012
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
120000000
012000000
001200000
000120000
00001000
00000100
00000010
00000001
,
09000000
40000000
00030000
001000000
000010200
00000300
00000046
00000009
,
00030000
001000000
09000000
40000000
00000046
00000009
000010200
00000300

G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,6,9],[0,0,0,4,0,0,0,0,0,0,9,0,0,0,0,0,0,10,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,0,0,0,10,0,0,0,0,0,0,0,2,3,0,0,0,0,4,0,0,0,0,0,0,0,6,9,0,0] >;
 

C24.47D6 in GAP, Magma, Sage, TeX 

C_2^4._{47}D_6
 
% in TeX
 
G:=Group("C2^4.47D6");
 
// GroupNames label
 
G:=SmallGroup(192,1154);
 
// by ID
 
G=gap.SmallGroup(192,1154);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,758,219,1571,570,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=f^2=d,a*b=b*a,e*a*e^-1=a*c=c*a,a*d=d*a,f*a*f^-1=a*c*d,f*b*f^-1=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^-1=e^5>;
 
// generators/relations
 

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