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G = C24.12D6  order 192 = 26·3

1st non-split extension by C24 of D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.12D6, C23.13D12, C23.3Dic6, (C22×C12)⋊1C4, C6.D4⋊5C4, (C22×C6).6Q8, (C2×C6).17C42, C23.13(C4×S3), (C22×C4)⋊3Dic3, (C22×C6).41D4, C6.17(C23⋊C4), C3⋊1(C23.9D4), C22.8(C4×Dic3), C22.16(D6⋊C4), C23.53(C3⋊D4), (C23×C6).22C22, C22.8(C4⋊Dic3), C23.26(C2×Dic3), C6.3(C2.C42), C2.4(C6.C42), C22.1(Dic3⋊C4), C2.2(C23.7D6), C22.24(C6.D4), (C2×C6).1(C4⋊C4), (C2×C22⋊C4).2S3, (C6×C22⋊C4).1C2, (C2×C6).9(C22⋊C4), (C22×C6).28(C2×C4), (C2×C6.D4).1C2, SmallGroup(192,85)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C6 — C24.12D6
Chief series
C1 — C3 — C6 — C2×C6 — C22×C6 — C23×C6 — C2×C6.D4 — C24.12D6
Lower central
C3 — C6 — C2×C6 — C24.12D6
Upper central
C1 — C22 — C24 — C2×C22⋊C4

Generators and relations for C24.12D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e6=b, f2=abc, ab=ba, ac=ca, eae-1=faf-1=ad=da, bc=cb, bd=db, be=eb, bf=fb, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=cde5 >

Subgroups: 408 in 142 conjugacy classes, 51 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C22, C6, C6, C6, C2×C4, C23, C23, C23, Dic3, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C22×C4, C22×C4, C24, C2×Dic3, C2×C12, C22×C6, C22×C6, C22×C6, C2×C22⋊C4, C2×C22⋊C4, C6.D4, C6.D4, C3×C22⋊C4, C22×Dic3, C22×C12, C23×C6, C23.9D4, C2×C6.D4, C6×C22⋊C4, C24.12D6
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C23⋊C4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C23.9D4, C6.C42, C23.7D6, C24.12D6

Smallest permutation representation of C24.12D6
►On 48 points
Generators in S48
(2 20)(4 22)(6 24)(8 14)(10 16)(12 18)(25 43)(27 45)(29 47)(31 37)(33 39)(35 41)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D···2I 3 4A4B4C4D4E···4L6A···6G6H6I6J6K12A···12H
order12222···2344444···46···6666612···12
size11112···22444412···122···244444···4

42 irreducible representations

dim1111122222222244
type+++++--+-++
imageC1C2C2C4C4S3D4Q8Dic3D6Dic6C4×S3D12C3⋊D4C23⋊C4C23.7D6
kernelC24.12D6C2×C6.D4C6×C22⋊C4C6.D4C22×C12C2×C22⋊C4C22×C6C22×C6C22×C4C24C23C23C23C23C6C2
# reps1218413121242424

Matrix representation of C24.12D6 ►in GL8(𝔽13)

120000000
012000000
001200000
000120000
00001000
00000100
000000120
000055012
,
120000000
012000000
00100000
00010000
000012000
000001200
000000120
000000012
,
120000000
012000000
001200000
000120000
00000100
00001000
0000551211
00000001
,
10000000
01000000
00100000
00010000
000012000
000001200
000000120
000000012
,
01000000
120000000
001120000
001190000
00000010
0000551211
000012000
00000008
,
08000000
80000000
003100000
007100000
0000551211
000000120
00001000
000012088

G:=sub<GL(8,GF(13))| [12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,5,0,0,0,0,0,1,0,5,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,5,0,0,0,0,0,1,0,5,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,11,1],[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],[0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,11,11,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,5,0,0,0,0,0,0,1,12,0,0,0,0,0,0,0,11,0,8],[0,8,0,0,0,0,0,0,8,0,0,0,0,0,0,0,0,0,3,7,0,0,0,0,0,0,10,10,0,0,0,0,0,0,0,0,5,0,1,12,0,0,0,0,5,0,0,0,0,0,0,0,12,12,0,8,0,0,0,0,11,0,0,8] >;
 

C24.12D6 in GAP, Magma, Sage, TeX 

C_2^4._{12}D_6
 
% in TeX
 
G:=Group("C2^4.12D6");
 
// GroupNames label
 
G:=SmallGroup(192,85);
 
// by ID
 
G=gap.SmallGroup(192,85);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,387,1684,6278]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^6=b,f^2=a*b*c,a*b=b*a,a*c=c*a,e*a*e^-1=f*a*f^-1=a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*d*e^5>;
 
// generators/relations
 

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