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G = C12.80D12  order 288 = 25·32

11st non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.80D12, C62.25D4, C32⋊5C4≀C2, C12⋊S3⋊7C4, C12.20(C4×S3), C6.6(D6⋊C4), (C4×Dic3)⋊2S3, (C2×C6).53D12, C4.Dic3⋊1S3, C32⋊4Q8⋊7C4, (Dic3×C12)⋊5C2, (C2×C12).287D6, (C3×C12).111D4, C3⋊1(C42⋊4S3), C3⋊1(D12⋊C4), C12.79(C3⋊D4), (C6×C12).32C22, C4.3(C6.D6), C4.29(C3⋊D12), C12.59D6.2C2, C2.7(C6.D12), C22.2(C3⋊D12), (C2×C4).57S32, (C3×C12).30(C2×C4), (C2×C6).11(C3⋊D4), (C3×C4.Dic3)⋊16C2, (C3×C6).37(C22⋊C4), SmallGroup(288,218)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C12 — C12.80D12
Chief series
C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C3×C4.Dic3 — C12.80D12
Lower central
C32 — C3×C6 — C3×C12 — C12.80D12
Upper central
C1 — C4 — C2×C4

Generators and relations for C12.80D12
 G = < a,b,c | a12=b12=c2=1, bab-1=cac=a5, cbc=a3b-1 >

Subgroups: 474 in 108 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, M4(2), C4○D4, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C4≀C2, C3×Dic3, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C4.Dic3, C4×Dic3, C4×C12, C3×M4(2), C4○D12, C3×C3⋊C8, C6×Dic3, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, C42⋊4S3, D12⋊C4, C3×C4.Dic3, Dic3×C12, C12.59D6, C12.80D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, C4×S3, D12, C3⋊D4, C4≀C2, S32, D6⋊C4, C6.D6, C3⋊D12, C42⋊4S3, D12⋊C4, C6.D12, C12.80D12

Smallest permutation representation of C12.80D12
►On 48 points
Generators in S48
(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 32 46 13 9 36 42 17 5 28 38 21)(2 25 47 18 10 29 43 22 6 33 39 14)(3 30 48 23 11 34 44 15 7 26 40 19)(4 35 37 16 12 27 45 20 8 31 41 24)

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

48 conjugacy classes

class 1 2A2B2C3A3B3C4A4B4C4D4E4F4G4H6A6B6C6D6E6F6G6H8A8B12A···12F12G···12K12L···12S24A24B24C24D
order122233344444444666666668812···1212···1212···1224242424
size112362241126666362222444412122···24···46···612121212

48 irreducible representations

dim111111222222222222444444
type+++++++++++++++
imageC1C2C2C2C4C4S3S3D4D4D6C4×S3D12C3⋊D4D12C3⋊D4C4≀C2C42⋊4S3S32C6.D6C3⋊D12C3⋊D12D12⋊C4C12.80D12
kernelC12.80D12C3×C4.Dic3Dic3×C12C12.59D6C32⋊4Q8C12⋊S3C4.Dic3C4×Dic3C3×C12C62C2×C12C12C12C12C2×C6C2×C6C32C3C2×C4C4C4C22C3C1
# reps111122111124222248111124

Matrix representation of C12.80D12 ►in GL6(𝔽73)

4600000
0460000
001000
000100
0000072
0000172
,
0450000
14450000
0017200
001000
000001
000010
,
7220000
010000
0072000
0072100
000001
000010

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[0,14,0,0,0,0,45,45,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[72,0,0,0,0,0,2,1,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
 

C12.80D12 in GAP, Magma, Sage, TeX 

C_{12}._{80}D_{12}
 
% in TeX
 
G:=Group("C12.80D12");
 
// GroupNames label
 
G:=SmallGroup(288,218);
 
// by ID
 
G=gap.SmallGroup(288,218);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,92,100,675,80,1356,9414]);
 
// Polycyclic
 
G:=Group<a,b,c|a^12=b^12=c^2=1,b*a*b^-1=c*a*c=a^5,c*b*c=a^3*b^-1>;
 
// generators/relations
 

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