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G = C12.82D12order 288 = 25·32

13rd non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.82D12, C62.4Q8, C3⋊C8.1Dic3, C12.91(C4×S3), (C2×C6).7Dic6, (C3×C12).114D4, (C2×C12).288D6, C6.4(C4⋊Dic3), C4.13(S3×Dic3), C31(C24.C4), C4.Dic3.1S3, C12.81(C3⋊D4), C6.7(Dic3⋊C4), (C6×C12).37C22, C325(C8.C4), C12.12(C2×Dic3), C4.31(C3⋊D12), C31(C12.53D4), C12.58D6.3C2, C2.5(Dic3⋊Dic3), C22.1(C322Q8), (C3×C3⋊C8).1C4, (C2×C4).60S32, (C2×C3⋊C8).5S3, (C6×C3⋊C8).11C2, (C3×C6).24(C4⋊C4), (C3×C12).35(C2×C4), (C3×C4.Dic3).3C2, SmallGroup(288,225)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C12.82D12
C1C3C32C3×C6C3×C12C6×C12C6×C3⋊C8 — C12.82D12
C32C3×C6C3×C12 — C12.82D12
C1C4C2×C4

Generators and relations for C12.82D12
 G = < a,b,c | a12=1, b12=a6, c2=a9, bab-1=cac-1=a5, cbc-1=a6b11 >

Subgroups: 178 in 70 conjugacy classes, 34 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×4], C8 [×4], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, M4(2) [×2], C3×C6, C3×C6, C3⋊C8 [×2], C3⋊C8 [×5], C24 [×3], C2×C12 [×2], C2×C12, C8.C4, C3×C12 [×2], C62, C2×C3⋊C8, C4.Dic3, C4.Dic3 [×3], C2×C24, C3×M4(2), C3×C3⋊C8 [×2], C3×C3⋊C8, C324C8, C6×C12, C24.C4, C12.53D4, C6×C3⋊C8, C3×C4.Dic3, C12.58D6, C12.82D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4, Q8, Dic3 [×2], D6 [×2], C4⋊C4, Dic6 [×2], C4×S3, D12, C2×Dic3, C3⋊D4, C8.C4, S32, Dic3⋊C4, C4⋊Dic3, S3×Dic3, C3⋊D12, C322Q8, C24.C4, C12.53D4, Dic3⋊Dic3, C12.82D12

Smallest permutation representation of C12.82D12
On 48 points
Generators in S48
(1 23 21 19 17 15 13 11 9 7 5 3)(2 16 6 20 10 24 14 4 18 8 22 12)(25 27 29 31 33 35 37 39 41 43 45 47)(26 36 46 32 42 28 38 48 34 44 30 40)
(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 28 7 46 13 40 19 34)(2 27 8 45 14 39 20 33)(3 26 9 44 15 38 21 32)(4 25 10 43 16 37 22 31)(5 48 11 42 17 36 23 30)(6 47 12 41 18 35 24 29)

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

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

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

48 conjugacy classes

class 1 2A2B3A3B3C4A4B4C6A6B6C6D6E6F6G6H8A8B8C8D8E8F8G8H12A···12F12G···12K24A···24H24I24J24K24L
order122333444666666668888888812···1212···1224···2424242424
size112224112222244446666121236362···24···46···612121212

48 irreducible representations

dim11111222222222222444444
type+++++++--++-+-+-
imageC1C2C2C2C4S3S3D4Q8Dic3D6C4×S3D12C3⋊D4Dic6C8.C4C24.C4S32S3×Dic3C3⋊D12C322Q8C12.53D4C12.82D12
kernelC12.82D12C6×C3⋊C8C3×C4.Dic3C12.58D6C3×C3⋊C8C2×C3⋊C8C4.Dic3C3×C12C62C3⋊C8C2×C12C12C12C12C2×C6C32C3C2×C4C4C4C22C3C1
# reps11114111122222448111124

Matrix representation of C12.82D12 in GL6(𝔽73)

4600000
0460000
000100
0072100
0000720
0000072
,
1000000
0220000
000100
001000
0000721
0000720
,
0220000
2200000
0004600
0046000
0000270
00002746

G:=sub<GL(6,GF(73))| [46,0,0,0,0,0,0,46,0,0,0,0,0,0,0,72,0,0,0,0,1,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[10,0,0,0,0,0,0,22,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,22,0,0,0,0,22,0,0,0,0,0,0,0,0,46,0,0,0,0,46,0,0,0,0,0,0,0,27,27,0,0,0,0,0,46] >;

C12.82D12 in GAP, Magma, Sage, TeX

C_{12}._{82}D_{12}
% in TeX

G:=Group("C12.82D12");
// GroupNames label

G:=SmallGroup(288,225);
// by ID

G=gap.SmallGroup(288,225);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,346,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^6*b^11>;
// generators/relations

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