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G = C247D6order 192 = 26·3

2nd semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C247D6, C6.252+ 1+4, (C2×D4)⋊5D6, C22⋊C45D6, D6.2(C2×D4), C22≀C23S3, C232D63C2, (C6×D4)⋊6C22, (C22×S3)⋊6D4, Dic3⋊D412C2, D63D412C2, C244S36C2, D6⋊C410C22, C32(C233D4), (C23×C6)⋊9C22, C22.40(S3×D4), C6.55(C22×D4), C23.9D612C2, (C2×D12)⋊18C22, (C2×C6).133C24, (C2×C12).27C23, Dic3⋊C48C22, (S3×C23)⋊6C22, C4⋊Dic325C22, (C22×C6).8C23, C2.27(D46D6), C23.21D69C2, C23.23D64C2, C6.D414C22, C23.117(C22×S3), C22.154(S3×C23), (C2×Dic3).60C23, (C22×S3).182C23, (C22×Dic3)⋊12C22, (C2×S3×D4)⋊6C2, C2.28(C2×S3×D4), (S3×C2×C4)⋊6C22, (S3×C22⋊C4)⋊2C2, (C2×C6).53(C2×D4), (C3×C22≀C2)⋊4C2, (C22×C3⋊D4)⋊7C2, (C2×C3⋊D4)⋊38C22, (C3×C22⋊C4)⋊4C22, (C2×C4).27(C22×S3), SmallGroup(192,1148)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C247D6
C1C3C6C2×C6C22×S3S3×C23C2×S3×D4 — C247D6
C3C2×C6 — C247D6
C1C22C22≀C2

Generators and relations for C247D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, eae-1=faf=ac=ca, ad=da, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 1072 in 346 conjugacy classes, 103 normal (39 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×2], C22 [×34], S3 [×5], C6, C6 [×2], C6 [×5], C2×C4, C2×C4 [×2], C2×C4 [×11], D4 [×20], C23 [×2], C23 [×2], C23 [×17], Dic3 [×5], C12 [×3], D6 [×4], D6 [×15], C2×C6, C2×C6 [×2], C2×C6 [×15], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×4], C22×C4 [×4], C2×D4, C2×D4 [×2], C2×D4 [×17], C24, C24 [×2], C4×S3 [×4], D12 [×2], C2×Dic3, C2×Dic3 [×4], C2×Dic3 [×2], C3⋊D4 [×14], C2×C12, C2×C12 [×2], C3×D4 [×4], C22×S3 [×3], C22×S3 [×4], C22×S3 [×6], C22×C6 [×2], C22×C6 [×2], C22×C6 [×4], C2×C22⋊C4, C22≀C2, C22≀C2 [×3], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], Dic3⋊C4 [×2], C4⋊Dic3 [×2], D6⋊C4 [×4], C6.D4, C6.D4 [×4], C3×C22⋊C4, C3×C22⋊C4 [×2], S3×C2×C4, S3×C2×C4 [×2], C2×D12, S3×D4 [×4], C22×Dic3, C2×C3⋊D4 [×2], C2×C3⋊D4 [×6], C2×C3⋊D4 [×4], C6×D4, C6×D4 [×2], S3×C23 [×2], C23×C6, C233D4, S3×C22⋊C4, C23.9D6 [×2], Dic3⋊D4 [×2], C23.21D6, C23.23D6, C232D6 [×2], D63D4 [×2], C244S3, C3×C22≀C2, C2×S3×D4, C22×C3⋊D4, C247D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C22×S3 [×7], C22×D4, 2+ 1+4 [×2], S3×D4 [×2], S3×C23, C233D4, C2×S3×D4, D46D6 [×2], C247D6

Smallest permutation representation of C247D6
On 48 points
Generators in S48
(1 37)(2 24)(3 39)(4 20)(5 41)(6 22)(7 27)(8 44)(9 29)(10 46)(11 25)(12 48)(13 45)(14 30)(15 47)(16 26)(17 43)(18 28)(19 33)(21 35)(23 31)(32 38)(34 40)(36 42)
(1 17)(2 44)(3 13)(4 46)(5 15)(6 48)(7 31)(8 24)(9 33)(10 20)(11 35)(12 22)(14 40)(16 42)(18 38)(19 29)(21 25)(23 27)(26 36)(28 32)(30 34)(37 43)(39 45)(41 47)
(1 31)(2 32)(3 33)(4 34)(5 35)(6 36)(7 17)(8 18)(9 13)(10 14)(11 15)(12 16)(19 39)(20 40)(21 41)(22 42)(23 37)(24 38)(25 47)(26 48)(27 43)(28 44)(29 45)(30 46)
(1 37)(2 38)(3 39)(4 40)(5 41)(6 42)(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 45)(14 46)(15 47)(16 48)(17 43)(18 44)(19 33)(20 34)(21 35)(22 36)(23 31)(24 32)
(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 16)(8 15)(9 14)(10 13)(11 18)(12 17)(19 20)(21 24)(22 23)(25 44)(26 43)(27 48)(28 47)(29 46)(30 45)(31 36)(32 35)(33 34)(37 42)(38 41)(39 40)

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

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

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

36 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M 3 4A4B4C4D···4H6A6B6C6D···6I6J12A12B12C
order1222222222222234444···46666···66121212
size111122444666612244412···122224···48888

36 irreducible representations

dim11111111111122222444
type+++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2S3D4D6D6D62+ 1+4S3×D4D46D6
kernelC247D6S3×C22⋊C4C23.9D6Dic3⋊D4C23.21D6C23.23D6C232D6D63D4C244S3C3×C22≀C2C2×S3×D4C22×C3⋊D4C22≀C2C22×S3C22⋊C4C2×D4C24C6C22C2
# reps11221122111114331224

Matrix representation of C247D6 in GL6(𝔽13)

100000
010000
001000
000100
0000120
0000012
,
860000
950000
002400
0091100
000024
0000911
,
100000
010000
0012000
0001200
0000120
0000012
,
1200000
0120000
001000
000100
000010
000001
,
1200000
710000
00001212
000010
00121200
001000
,
1200000
710000
00001212
000001
00121200
000100

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[8,9,0,0,0,0,6,5,0,0,0,0,0,0,2,9,0,0,0,0,4,11,0,0,0,0,0,0,2,9,0,0,0,0,4,11],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,1,0,0,0,0,12,0,0,0,12,1,0,0,0,0,12,0,0,0],[12,7,0,0,0,0,0,1,0,0,0,0,0,0,0,0,12,0,0,0,0,0,12,1,0,0,12,0,0,0,0,0,12,1,0,0] >;

C247D6 in GAP, Magma, Sage, TeX

C_2^4\rtimes_7D_6
% in TeX

G:=Group("C2^4:7D6");
// GroupNames label

G:=SmallGroup(192,1148);
// by ID

G=gap.SmallGroup(192,1148);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,219,675,297,6278]);
// 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,b*c=c*b,e*b*e^-1=b*d=d*b,f*b*f=b*c*d,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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