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G = C3×Q8○D12order 288 = 25·32

Direct product of C3 and Q8○D12

direct product, metabelian, supersoluble, monomial

Aliases: C3×Q8○D12, C3282- 1+4, C62.152C23, (S3×Q8)⋊8C6, C4○D129C6, D42S35C6, (C3×D4).50D6, D4.10(S3×C6), Q8.21(S3×C6), (C3×Q8).76D6, (C2×Dic6)⋊14C6, (C6×Dic6)⋊24C2, D12.14(C2×C6), (C2×C12).256D6, C6.81(S3×C23), (C3×C6).50C24, C6.13(C23×C6), D6.7(C22×C6), (S3×C6).35C23, C12.27(C22×C6), Dic6.14(C2×C6), C32(C3×2- 1+4), (S3×C12).37C22, (C6×C12).167C22, C12.178(C22×S3), (C3×C12).128C23, (C3×D12).48C22, Dic3.8(C22×C6), (C3×Dic3).36C23, (C3×Dic6).49C22, (D4×C32).30C22, (Q8×C32).32C22, (C6×Dic3).104C22, C4.34(S3×C2×C6), C3⋊D4.(C2×C6), (C3×S3×Q8)⋊12C2, C4○D48(C3×S3), (C3×C4○D4)⋊9C6, C22.4(S3×C2×C6), (C3×C4○D4)⋊13S3, (C4×S3).6(C2×C6), (C2×C4).23(S3×C6), C2.14(S3×C22×C6), (C3×C4○D12)⋊19C2, (C2×C12).49(C2×C6), (C3×D4).10(C2×C6), (C32×C4○D4)⋊7C2, (C2×C6).5(C22×C6), (C3×Q8).23(C2×C6), (C3×D42S3)⋊12C2, (C2×C6).24(C22×S3), (C3×C3⋊D4).4C22, (C2×Dic3).14(C2×C6), SmallGroup(288,1000)

Series: Derived Chief Lower central Upper central

C1C6 — C3×Q8○D12
C1C3C6C3×C6S3×C6S3×C12C3×S3×Q8 — C3×Q8○D12
C3C6 — C3×Q8○D12
C1C6C3×C4○D4

Generators and relations for C3×Q8○D12
 G = < a,b,c,d,e | a3=b4=e2=1, c2=d6=b2, ab=ba, ac=ca, ad=da, ae=ea, cbc-1=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=b2d5 >

Subgroups: 578 in 312 conjugacy classes, 170 normal (24 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4, C4 [×3], C4 [×6], C22 [×3], C22 [×2], S3 [×2], C6 [×2], C6 [×12], C2×C4 [×3], C2×C4 [×12], D4 [×3], D4 [×7], Q8, Q8 [×9], C32, Dic3 [×6], C12 [×2], C12 [×6], C12 [×10], D6 [×2], C2×C6 [×6], C2×C6 [×5], C2×Q8 [×5], C4○D4, C4○D4 [×9], C3×S3 [×2], C3×C6, C3×C6 [×3], Dic6 [×9], C4×S3 [×6], D12, C2×Dic3 [×6], C3⋊D4 [×6], C2×C12 [×6], C2×C12 [×15], C3×D4 [×6], C3×D4 [×10], C3×Q8 [×2], C3×Q8 [×10], 2- 1+4, C3×Dic3 [×6], C3×C12, C3×C12 [×3], S3×C6 [×2], C62 [×3], C2×Dic6 [×3], C4○D12 [×3], D42S3 [×6], S3×Q8 [×2], C6×Q8 [×5], C3×C4○D4 [×2], C3×C4○D4 [×10], C3×Dic6 [×9], S3×C12 [×6], C3×D12, C6×Dic3 [×6], C3×C3⋊D4 [×6], C6×C12 [×3], D4×C32 [×3], Q8×C32, Q8○D12, C3×2- 1+4, C6×Dic6 [×3], C3×C4○D12 [×3], C3×D42S3 [×6], C3×S3×Q8 [×2], C32×C4○D4, C3×Q8○D12
Quotients: C1, C2 [×15], C3, C22 [×35], S3, C6 [×15], C23 [×15], D6 [×7], C2×C6 [×35], C24, C3×S3, C22×S3 [×7], C22×C6 [×15], 2- 1+4, S3×C6 [×7], S3×C23, C23×C6, S3×C2×C6 [×7], Q8○D12, C3×2- 1+4, S3×C22×C6, C3×Q8○D12

Smallest permutation representation of C3×Q8○D12
On 48 points
Generators in S48
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 23 7 17)(2 24 8 18)(3 13 9 19)(4 14 10 20)(5 15 11 21)(6 16 12 22)(25 43 31 37)(26 44 32 38)(27 45 33 39)(28 46 34 40)(29 47 35 41)(30 48 36 42)
(1 20 7 14)(2 21 8 15)(3 22 9 16)(4 23 10 17)(5 24 11 18)(6 13 12 19)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 37 34 43)(29 38 35 44)(30 39 36 45)
(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 30)(8 29)(9 28)(10 27)(11 26)(12 25)(13 40)(14 39)(15 38)(16 37)(17 48)(18 47)(19 46)(20 45)(21 44)(22 43)(23 42)(24 41)

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

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

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

81 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C3D3E4A4B4C4D4E···4J6A6B6C···6K6L···6T6U6V6W6X12A···12N12O···12W12X···12AI
order12222223333344444···4666···66···6666612···1212···1212···12
size11222661122222226···6112···24···466662···24···46···6

81 irreducible representations

dim111111111111222222224444
type++++++++++--
imageC1C2C2C2C2C2C3C6C6C6C6C6S3D6D6D6C3×S3S3×C6S3×C6S3×C62- 1+4Q8○D12C3×2- 1+4C3×Q8○D12
kernelC3×Q8○D12C6×Dic6C3×C4○D12C3×D42S3C3×S3×Q8C32×C4○D4Q8○D12C2×Dic6C4○D12D42S3S3×Q8C3×C4○D4C3×C4○D4C2×C12C3×D4C3×Q8C4○D4C2×C4D4Q8C32C3C3C1
# reps1336212661242133126621224

Matrix representation of C3×Q8○D12 in GL4(𝔽13) generated by

3000
0300
0030
0003
,
0100
12000
0001
00120
,
0800
8000
0008
0080
,
6000
0600
00110
00011
,
0020
0002
7000
0700
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,3,0,0,0,0,3],[0,12,0,0,1,0,0,0,0,0,0,12,0,0,1,0],[0,8,0,0,8,0,0,0,0,0,0,8,0,0,8,0],[6,0,0,0,0,6,0,0,0,0,11,0,0,0,0,11],[0,0,7,0,0,0,0,7,2,0,0,0,0,2,0,0] >;

C3×Q8○D12 in GAP, Magma, Sage, TeX

C_3\times Q_8\circ D_{12}
% in TeX

G:=Group("C3xQ8oD12");
// GroupNames label

G:=SmallGroup(288,1000);
// by ID

G=gap.SmallGroup(288,1000);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,344,555,268,1571,192,9414]);
// Polycyclic

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

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