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G = C3×C23.12D6order 288 = 25·32

Direct product of C3 and C23.12D6

direct product, metabelian, supersoluble, monomial

Aliases: C3×C23.12D6, C62.201C23, (C6×D4).5C6, C6.48(C6×D4), (C4×Dic3)⋊5C6, (C6×D4).28S3, (C3×C12).87D4, C12.17(C3×D4), (C6×Dic6)⋊15C2, (C2×Dic6)⋊10C6, C6.D49C6, (C2×C12).326D6, C23.12(S3×C6), (C22×C6).31D6, (Dic3×C12)⋊15C2, C12.89(C3⋊D4), (C6×C12).121C22, C3215(C4.4D4), (C2×C62).56C22, C6.124(D42S3), (C6×Dic3).100C22, (D4×C3×C6).7C2, C4.7(C3×C3⋊D4), (C2×C4).50(S3×C6), (C2×D4).6(C3×S3), C6.30(C3×C4○D4), C33(C3×C4.4D4), C2.12(C6×C3⋊D4), C22.58(S3×C2×C6), (C2×C12).32(C2×C6), (C3×C6).258(C2×D4), C6.149(C2×C3⋊D4), C2.16(C3×D42S3), (C2×C6).56(C22×C6), (C22×C6).30(C2×C6), (C3×C6).138(C4○D4), (C3×C6.D4)⋊25C2, (C2×C6).334(C22×S3), (C2×Dic3).37(C2×C6), SmallGroup(288,707)

Series: Derived Chief Lower central Upper central

C1C2×C6 — C3×C23.12D6
C1C3C6C2×C6C62C6×Dic3Dic3×C12 — C3×C23.12D6
C3C2×C6 — C3×C23.12D6
C1C2×C6C6×D4

Generators and relations for C3×C23.12D6
 G = < a,b,c,d,e,f | a3=b2=c2=d2=1, e6=c, f2=dc=cd, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ebe-1=bc=cb, fbf-1=bd=db, ce=ec, cf=fc, de=ed, df=fd, fef-1=e5 >

Subgroups: 394 in 179 conjugacy classes, 66 normal (26 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], C6 [×2], C6 [×4], C6 [×11], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], C2×C6 [×2], C2×C6 [×21], C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×5], C3×D4 [×8], C3×Q8 [×2], C22×C6 [×4], C22×C6 [×2], C4.4D4, C3×Dic3 [×4], C3×C12 [×2], C62, C62 [×6], C4×Dic3, C6.D4 [×4], C4×C12, C3×C22⋊C4 [×4], C2×Dic6, C6×D4 [×2], C6×D4, C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C6×C12, D4×C32 [×2], C2×C62 [×2], C23.12D6, C3×C4.4D4, Dic3×C12, C3×C6.D4 [×4], C6×Dic6, D4×C3×C6, C3×C23.12D6
Quotients: C1, C2 [×7], C3, C22 [×7], S3, C6 [×7], D4 [×2], C23, D6 [×3], C2×C6 [×7], C2×D4, C4○D4 [×2], C3×S3, C3⋊D4 [×2], C3×D4 [×2], C22×S3, C22×C6, C4.4D4, S3×C6 [×3], D42S3 [×2], C2×C3⋊D4, C6×D4, C3×C4○D4 [×2], C3×C3⋊D4 [×2], S3×C2×C6, C23.12D6, C3×C4.4D4, C3×D42S3 [×2], C6×C3⋊D4, C3×C23.12D6

Smallest permutation representation of C3×C23.12D6
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(2 8)(4 10)(6 12)(13 37)(14 44)(15 39)(16 46)(17 41)(18 48)(19 43)(20 38)(21 45)(22 40)(23 47)(24 42)(26 32)(28 34)(30 36)
(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 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 25)(10 26)(11 27)(12 28)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 37)(20 38)(21 39)(22 40)(23 41)(24 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 40 35 16)(2 45 36 21)(3 38 25 14)(4 43 26 19)(5 48 27 24)(6 41 28 17)(7 46 29 22)(8 39 30 15)(9 44 31 20)(10 37 32 13)(11 42 33 18)(12 47 34 23)

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

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

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

72 conjugacy classes

class 1 2A2B2C2D2E3A3B3C3D3E4A4B4C4D4E4F4G4H6A···6F6G···6O6P···6AE12A12B12C12D12E···12J12K···12R12S12T12U12V
order12222233333444444446···66···66···61212121212···1212···1212121212
size1111441122222666612121···12···24···422224···46···612121212

72 irreducible representations

dim111111111122222222222244
type+++++++++-
imageC1C2C2C2C2C3C6C6C6C6S3D4D6D6C4○D4C3×S3C3⋊D4C3×D4S3×C6S3×C6C3×C4○D4C3×C3⋊D4D42S3C3×D42S3
kernelC3×C23.12D6Dic3×C12C3×C6.D4C6×Dic6D4×C3×C6C23.12D6C4×Dic3C6.D4C2×Dic6C6×D4C6×D4C3×C12C2×C12C22×C6C3×C6C2×D4C12C12C2×C4C23C6C4C6C2
# reps114112282212124244248824

Matrix representation of C3×C23.12D6 in GL6(𝔽13)

100000
010000
003000
000300
000010
000001
,
100000
3120000
0012000
0001200
000010
0000112
,
100000
010000
001000
000100
0000120
0000012
,
1200000
0120000
001000
000100
0000120
0000012
,
100000
010000
0010000
0010400
0000122
0000121
,
8120000
050000
004800
003900
000053
000058

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,3,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,1,0,0,0,0,0,12],[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],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,10,0,0,0,0,0,4,0,0,0,0,0,0,12,12,0,0,0,0,2,1],[8,0,0,0,0,0,12,5,0,0,0,0,0,0,4,3,0,0,0,0,8,9,0,0,0,0,0,0,5,5,0,0,0,0,3,8] >;

C3×C23.12D6 in GAP, Magma, Sage, TeX

C_3\times C_2^3._{12}D_6
% in TeX

G:=Group("C3xC2^3.12D6");
// GroupNames label

G:=SmallGroup(288,707);
// by ID

G=gap.SmallGroup(288,707);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,176,1598,303,9414]);
// Polycyclic

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

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