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G = C2×D6.4D6order 288 = 25·32

Direct product of C2 and D6.4D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×D6.4D6, C62.143C23, C3⋊D48D6, C23.37S32, C64(D42S3), (C3×C6).30C24, C6.30(S3×C23), (S3×C6).17C23, (C2×Dic3).89D6, (S3×Dic3)⋊9C22, D6.17(C22×S3), (C22×S3).57D6, (C22×C6).100D6, D6⋊S317C22, C322Q816C22, C3⋊Dic3.43C23, (C2×C62).78C22, (C3×Dic3).18C23, Dic3.16(C22×S3), (C6×Dic3).49C22, (C6×C3⋊D4)⋊5C2, (C2×S3×Dic3)⋊5C2, (C3×C6)⋊6(C4○D4), C35(C2×D42S3), (C2×C3⋊D4)⋊10S3, C22.14(C2×S32), C2.31(C22×S32), C3212(C2×C4○D4), (S3×C2×C6).67C22, (C2×D6⋊S3)⋊16C2, (C2×C322Q8)⋊18C2, (C3×C3⋊D4)⋊11C22, (C2×C6).158(C22×S3), (C22×C3⋊Dic3)⋊13C2, (C2×C3⋊Dic3)⋊24C22, SmallGroup(288,971)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×D6.4D6
C1C3C32C3×C6S3×C6S3×Dic3C2×S3×Dic3 — C2×D6.4D6
C32C3×C6 — C2×D6.4D6
C1C22C23

Generators and relations for C2×D6.4D6
 G = < a,b,c,d,e | a2=b6=c2=1, d6=e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=dbd-1=ebe-1=b-1, dcd-1=bc, ece-1=b4c, ede-1=d5 >

Subgroups: 1106 in 355 conjugacy classes, 116 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×2], C3, C4 [×8], C22, C22 [×2], C22 [×10], S3 [×4], C6 [×6], C6 [×15], C2×C4 [×16], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×4], Dic3 [×12], C12 [×4], D6 [×4], D6 [×4], C2×C6 [×6], C2×C6 [×19], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3×S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×8], C4×S3 [×8], C2×Dic3 [×2], C2×Dic3 [×26], C3⋊D4 [×8], C3⋊D4 [×8], C2×C12 [×2], C3×D4 [×8], C22×S3 [×2], C22×C6 [×2], C22×C6 [×3], C2×C4○D4, C3×Dic3 [×4], C3⋊Dic3 [×4], S3×C6 [×4], S3×C6 [×4], C62, C62 [×2], C62 [×2], C2×Dic6 [×2], S3×C2×C4 [×2], D42S3 [×16], C22×Dic3 [×5], C2×C3⋊D4 [×2], C2×C3⋊D4 [×2], C6×D4 [×2], S3×Dic3 [×8], D6⋊S3 [×4], C322Q8 [×4], C6×Dic3 [×2], C3×C3⋊D4 [×8], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×4], S3×C2×C6 [×2], C2×C62, C2×D42S3 [×2], C2×S3×Dic3 [×2], D6.4D6 [×8], C2×D6⋊S3, C2×C322Q8, C6×C3⋊D4 [×2], C22×C3⋊Dic3, C2×D6.4D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C4○D4 [×2], C24, C22×S3 [×14], C2×C4○D4, S32, D42S3 [×4], S3×C23 [×2], C2×S32 [×3], C2×D42S3 [×2], D6.4D6 [×2], C22×S32, C2×D6.4D6

Smallest permutation representation of C2×D6.4D6
On 48 points
Generators in S48
(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 44)(26 45)(27 46)(28 47)(29 48)(30 37)(31 38)(32 39)(33 40)(34 41)(35 42)(36 43)
(1 11 9 7 5 3)(2 4 6 8 10 12)(13 15 17 19 21 23)(14 24 22 20 18 16)(25 27 29 31 33 35)(26 36 34 32 30 28)(37 47 45 43 41 39)(38 40 42 44 46 48)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 24)(14 15)(16 17)(18 19)(20 21)(22 23)(25 36)(26 27)(28 29)(30 31)(32 33)(34 35)(37 38)(39 40)(41 42)(43 44)(45 46)(47 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 30 7 36)(2 35 8 29)(3 28 9 34)(4 33 10 27)(5 26 11 32)(6 31 12 25)(13 46 19 40)(14 39 20 45)(15 44 21 38)(16 37 22 43)(17 42 23 48)(18 47 24 41)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I3A3B3C4A4B4C4D4E4F4G4H4I4J6A···6F6G···6Q6R6S6T6U12A12B12C12D
order122222222233344444444446···66···6666612121212
size11112266662246666999918182···24···41212121212121212

48 irreducible representations

dim11111112222224444
type+++++++++++++-+-
imageC1C2C2C2C2C2C2S3D6D6D6D6C4○D4S32D42S3C2×S32D6.4D6
kernelC2×D6.4D6C2×S3×Dic3D6.4D6C2×D6⋊S3C2×C322Q8C6×C3⋊D4C22×C3⋊Dic3C2×C3⋊D4C2×Dic3C3⋊D4C22×S3C22×C6C3×C6C23C6C22C2
# reps12811212282241434

Matrix representation of C2×D6.4D6 in GL6(𝔽13)

100000
010000
001000
000100
0000120
0000012
,
1200000
0120000
001000
000100
0000012
0000112
,
0120000
1200000
001000
000100
0000121
000001
,
010000
1200000
0001200
0011200
000001
000010
,
800000
080000
0011200
0001200
000001
000010

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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[0,12,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[8,0,0,0,0,0,0,8,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×D6.4D6 in GAP, Magma, Sage, TeX

C_2\times D_6._4D_6
% in TeX

G:=Group("C2xD6.4D6");
// GroupNames label

G:=SmallGroup(288,971);
// by ID

G=gap.SmallGroup(288,971);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,346,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^6=c^2=1,d^6=e^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=b*c,e*c*e^-1=b^4*c,e*d*e^-1=d^5>;
// generators/relations

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