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G = C2412D6order 192 = 26·3

7th semidirect product of C24 and D6 acting via D6/C3=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2412D6, C6.892+ 1+4, (C2×D4)⋊40D6, (C22×C4)⋊31D6, (C22×C6)⋊13D4, C232D630C2, D6⋊C436C22, (C22×D4)⋊13S3, (C6×D4)⋊58C22, C235(C3⋊D4), C35(C233D4), C244S312C2, (C2×C6).299C24, (C23×C6)⋊14C22, C6.146(C22×D4), C23.14D641C2, C2.92(D46D6), (C2×C12).644C23, Dic3⋊C438C22, (S3×C23)⋊15C22, (C22×C12)⋊44C22, C6.D464C22, C23.23D629C2, C23.28D628C2, C23.216(C22×S3), (C22×C6).233C23, C22.312(S3×C23), (C22×S3).130C23, (C2×Dic3).154C23, (C22×Dic3)⋊34C22, (D4×C2×C6)⋊17C2, (C2×C6).582(C2×D4), (C22×C3⋊D4)⋊17C2, (C2×C3⋊D4)⋊48C22, C2.19(C22×C3⋊D4), C22.20(C2×C3⋊D4), (C2×C6.D4)⋊30C2, (C2×C4).238(C22×S3), SmallGroup(192,1363)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C2412D6
C1C3C6C2×C6C22×S3S3×C23C22×C3⋊D4 — C2412D6
C3C2×C6 — C2412D6
C1C22C22×D4

Generators and relations for C2412D6
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e6=f2=1, ab=ba, ac=ca, faf=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef=e-1 >

Subgroups: 968 in 346 conjugacy classes, 111 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×10], C3, C4 [×8], C22, C22 [×6], C22 [×30], S3 [×2], C6, C6 [×2], C6 [×8], C2×C4 [×2], C2×C4 [×12], D4 [×20], C23 [×3], C23 [×6], C23 [×12], Dic3 [×6], C12 [×2], D6 [×10], C2×C6, C2×C6 [×6], C2×C6 [×20], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4, C22×C4 [×3], C2×D4 [×4], C2×D4 [×16], C24 [×2], C24, C2×Dic3 [×6], C2×Dic3 [×4], C3⋊D4 [×12], C2×C12 [×2], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×S3 [×4], C22×C6 [×3], C22×C6 [×6], C22×C6 [×6], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4, C22×D4, Dic3⋊C4 [×4], D6⋊C4 [×4], C6.D4 [×8], C22×Dic3, C22×Dic3 [×2], C2×C3⋊D4 [×8], C2×C3⋊D4 [×4], C22×C12, C6×D4 [×4], C6×D4 [×4], S3×C23, C23×C6 [×2], C233D4, C23.28D6 [×2], C23.23D6 [×2], C232D6 [×2], C23.14D6 [×4], C2×C6.D4, C244S3 [×2], C22×C3⋊D4, D4×C2×C6, C2412D6
Quotients: C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, C3⋊D4 [×4], C22×S3 [×7], C22×D4, 2+ 1+4 [×2], C2×C3⋊D4 [×6], S3×C23, C233D4, D46D6 [×2], C22×C3⋊D4, C2412D6

Smallest permutation representation of C2412D6
On 48 points
Generators in S48
(1 9)(2 7)(3 8)(4 20)(5 21)(6 19)(10 18)(11 16)(12 17)(13 23)(14 24)(15 22)(25 43)(26 44)(27 45)(28 46)(29 47)(30 48)(31 38)(32 39)(33 40)(34 41)(35 42)(36 37)
(1 36)(2 34)(3 32)(4 33)(5 31)(6 35)(7 41)(8 39)(9 37)(10 28)(11 26)(12 30)(13 29)(14 27)(15 25)(16 44)(17 48)(18 46)(19 42)(20 40)(21 38)(22 43)(23 47)(24 45)
(1 15)(2 13)(3 14)(4 10)(5 11)(6 12)(7 23)(8 24)(9 22)(16 21)(17 19)(18 20)(25 36)(26 31)(27 32)(28 33)(29 34)(30 35)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)
(1 4)(2 5)(3 6)(7 21)(8 19)(9 20)(10 15)(11 13)(12 14)(16 23)(17 24)(18 22)(25 28)(26 29)(27 30)(31 34)(32 35)(33 36)(37 40)(38 41)(39 42)(43 46)(44 47)(45 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 3)(4 6)(7 21)(8 20)(9 19)(10 12)(14 15)(16 23)(17 22)(18 24)(25 35)(26 34)(27 33)(28 32)(29 31)(30 36)(37 45)(38 44)(39 43)(40 48)(41 47)(42 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M 3 4A4B4C···4H6A···6G6H···6O12A12B12C12D
order12222···222223444···46···66···612121212
size11112···244121224412···122···24···44444

42 irreducible representations

dim11111111122222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2S3D4D6D6D6C3⋊D42+ 1+4D46D6
kernelC2412D6C23.28D6C23.23D6C232D6C23.14D6C2×C6.D4C244S3C22×C3⋊D4D4×C2×C6C22×D4C22×C6C22×C4C2×D4C24C23C6C2
# reps12224121114142824

Matrix representation of C2412D6 in GL6(𝔽13)

100000
010000
0010110
0001011
0000120
0000012
,
240000
9110000
000100
001000
000001
000010
,
1200000
0120000
0012000
0001200
0000120
0000012
,
100000
010000
0012000
0001200
0000120
0000012
,
110000
1200000
001000
0001200
000010
0000012
,
12120000
010000
001000
000100
0010120
0001012

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

C2412D6 in GAP, Magma, Sage, TeX

C_2^4\rtimes_{12}D_6
% in TeX

G:=Group("C2^4:12D6");
// GroupNames label

G:=SmallGroup(192,1363);
// by ID

G=gap.SmallGroup(192,1363);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,758,675,570,6278]);
// 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,a*c=c*a,f*a*f=a*d=d*a,a*e=e*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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