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G = C12.80D12order 288 = 25·32

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

metabelian, supersoluble, monomial

Aliases: C12.80D12, C62.25D4, C32:5C4wrC2, C12:S3:7C4, C12.20(C4xS3), C6.6(D6:C4), (C4xDic3):2S3, (C2xC6).53D12, C4.Dic3:1S3, C32:4Q8:7C4, (Dic3xC12):5C2, (C2xC12).287D6, (C3xC12).111D4, C3:1(C42:4S3), C3:1(D12:C4), C12.79(C3:D4), (C6xC12).32C22, C4.3(C6.D6), C4.29(C3:D12), C12.59D6.2C2, C2.7(C6.D12), C22.2(C3:D12), (C2xC4).57S32, (C3xC12).30(C2xC4), (C2xC6).11(C3:D4), (C3xC4.Dic3):16C2, (C3xC6).37(C22:C4), SmallGroup(288,218)

Series: Derived Chief Lower central Upper central

C1C3xC12 — C12.80D12
C1C3C32C3xC6C3xC12C6xC12C3xC4.Dic3 — C12.80D12
C32C3xC6C3xC12 — C12.80D12
C1C4C2xC4

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, C2xC4, C2xC4, D4, Q8, C32, Dic3, C12, C12, D6, C2xC6, C2xC6, C42, M4(2), C4oD4, C3:S3, C3xC6, C3xC6, C3:C8, C24, Dic6, C4xS3, D12, C2xDic3, C3:D4, C2xC12, C2xC12, C4wrC2, C3xDic3, C3:Dic3, C3xC12, C2xC3:S3, C62, C4.Dic3, C4xDic3, C4xC12, C3xM4(2), C4oD12, C3xC3:C8, C6xDic3, C32:4Q8, C4xC3:S3, C12:S3, C32:7D4, C6xC12, C42:4S3, D12:C4, C3xC4.Dic3, Dic3xC12, C12.59D6, C12.80D12
Quotients: C1, C2, C4, C22, S3, C2xC4, D4, D6, C22:C4, C4xS3, D12, C3:D4, C4wrC2, 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+++++++++++++++
imageC1C2C2C2C4C4S3S3D4D4D6C4xS3D12C3:D4D12C3:D4C4wrC2C42:4S3S32C6.D6C3:D12C3:D12D12:C4C12.80D12
kernelC12.80D12C3xC4.Dic3Dic3xC12C12.59D6C32:4Q8C12:S3C4.Dic3C4xDic3C3xC12C62C2xC12C12C12C12C2xC6C2xC6C32C3C2xC4C4C4C22C3C1
# 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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