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G = D6.C24order 192 = 26·3

9th non-split extension by D6 of C24 acting via C24/C23=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C6.16C25, D6.9C24, C12.51C24, D12.37C23, 2+ 1+410S3, Dic6.38C23, Dic3.11C24, C4○D414D6, (C2×D4)⋊32D6, Q8○D1211C2, (C2×C6).7C24, D46D610C2, (S3×D4)⋊14C22, (C6×D4)⋊26C22, C4.48(S3×C23), C2.17(S3×C24), (S3×Q8)⋊16C22, C3⋊D4.3C23, C4○D1213C22, (C4×S3).20C23, C32(C2.C25), D4.31(C22×S3), (C3×D4).31C23, C22.4(S3×C23), (C3×Q8).32C23, Q8.42(C22×S3), D42S316C22, (C2×C12).122C23, Q83S319C22, (C2×Dic6)⋊44C22, (C22×C6).79C23, C23.81(C22×S3), (C3×2+ 1+4)⋊5C2, (C22×S3).143C23, (C2×Dic3).169C23, (C22×Dic3)⋊39C22, (S3×C4○D4)⋊8C2, (S3×C2×C4)⋊37C22, (C2×D42S3)⋊31C2, (C3×C4○D4)⋊11C22, (C2×C3⋊D4)⋊33C22, (C2×C4).106(C22×S3), SmallGroup(192,1525)

Series: Derived Chief Lower central Upper central

C1C6 — D6.C24
C1C3C6D6C22×S3S3×C2×C4S3×C4○D4 — D6.C24
C3C6 — D6.C24
C1C22+ 1+4

Generators and relations for D6.C24
 G = < a,b,c,d,e,f | a6=b2=c2=e2=f2=1, d2=a3, bab=eae=a-1, ac=ca, ad=da, af=fa, cbc=fbf=a3b, bd=db, ebe=a4b, cd=dc, ce=ec, cf=fc, de=ed, fdf=a3d, fef=a3e >

Subgroups: 1640 in 810 conjugacy classes, 443 normal (8 characteristic)
C1, C2, C2 [×15], C3, C4 [×6], C4 [×10], C22 [×9], C22 [×21], S3 [×6], C6, C6 [×9], C2×C4 [×9], C2×C4 [×51], D4 [×18], D4 [×42], Q8 [×2], Q8 [×18], C23 [×6], C23 [×9], Dic3, Dic3 [×9], C12 [×6], D6 [×6], D6 [×9], C2×C6 [×9], C2×C6 [×6], C22×C4 [×15], C2×D4 [×9], C2×D4 [×36], C2×Q8 [×15], C4○D4 [×6], C4○D4 [×74], Dic6 [×18], C4×S3 [×24], D12 [×6], C2×Dic3 [×27], C3⋊D4 [×36], C2×C12 [×9], C3×D4 [×18], C3×Q8 [×2], C22×S3 [×9], C22×C6 [×6], C2×C4○D4 [×15], 2+ 1+4, 2+ 1+4 [×9], 2- 1+4 [×6], C2×Dic6 [×9], S3×C2×C4 [×9], C4○D12 [×18], S3×D4 [×18], D42S3 [×54], S3×Q8 [×6], Q83S3 [×2], C22×Dic3 [×6], C2×C3⋊D4 [×18], C6×D4 [×9], C3×C4○D4 [×6], C2.C25, C2×D42S3 [×9], D46D6 [×9], S3×C4○D4 [×6], Q8○D12 [×6], C3×2+ 1+4, D6.C24
Quotients: C1, C2 [×31], C22 [×155], S3, C23 [×155], D6 [×15], C24 [×31], C22×S3 [×35], C25, S3×C23 [×15], C2.C25, S3×C24, D6.C24

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

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

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

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

51 conjugacy classes

class 1 2A2B···2J2K···2P 3 4A···4F4G4H4I···4Q6A6B···6J12A···12F
order122···22···234···4444···466···612···12
size112···26···622···2336···624···44···4

51 irreducible representations

dim11111122248
type+++++++++-
imageC1C2C2C2C2C2S3D6D6C2.C25D6.C24
kernelD6.C24C2×D42S3D46D6S3×C4○D4Q8○D12C3×2+ 1+42+ 1+4C2×D4C4○D4C3C1
# reps19966119621

Matrix representation of D6.C24 in GL6(𝔽13)

0120000
1120000
0012000
0001200
0000120
0000012
,
1120000
0120000
000008
000080
000500
005000
,
100000
010000
000050
000008
008000
000500
,
100000
010000
000010
000001
0012000
0001200
,
0120000
1200000
000050
000005
008000
000800
,
100000
010000
000100
001000
0000012
0000120

G:=sub<GL(6,GF(13))| [0,1,0,0,0,0,12,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0,0,0,0,8,0,0,0,0,8,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,5,0,0,5,0,0,0,0,0,0,8,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,1,0,0,0,0,0,0,1,0,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,0,0,8,0,0,0,0,0,0,8,0,0,5,0,0,0,0,0,0,5,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0] >;

D6.C24 in GAP, Magma, Sage, TeX

D_6.C_2^4
% in TeX

G:=Group("D6.C2^4");
// GroupNames label

G:=SmallGroup(192,1525);
// by ID

G=gap.SmallGroup(192,1525);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,184,570,1684,438,6278]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^6=b^2=c^2=e^2=f^2=1,d^2=a^3,b*a*b=e*a*e=a^-1,a*c=c*a,a*d=d*a,a*f=f*a,c*b*c=f*b*f=a^3*b,b*d=d*b,e*b*e=a^4*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f=a^3*d,f*e*f=a^3*e>;
// generators/relations

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