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G = D12.25D6order 288 = 25·32

10th non-split extension by D12 of D6 acting via D6/S3=C2

metabelian, supersoluble, monomial

Aliases: D12.25D6, Dic6.25D6, C3242- 1+4, (S3×Q8)⋊7S3, Q8.18S32, Q83S37S3, C34(Q8○D12), (C4×S3).16D6, (C3×Q8).46D6, (S3×Dic6)⋊11C2, D6.D68C2, (C3×C6).22C24, C6.22(S3×C23), D12⋊S311C2, D125S311C2, (S3×C6).24C23, C12.34(C22×S3), (C3×C12).34C23, D6.22(C22×S3), (S3×C12).33C22, C33(Q8.15D6), D6⋊S3.9C22, C3⋊D12.3C22, (C3×D12).30C22, C3⋊Dic3.24C23, (S3×Dic3).4C22, Dic3.11(C22×S3), (C3×Dic6).30C22, (C3×Dic3).15C23, C322Q8.10C22, (Q8×C32).21C22, C324Q8.22C22, (C3×S3×Q8)⋊7C2, C4.34(C2×S32), (Q8×C3⋊S3)⋊6C2, C2.24(C22×S32), (C3×Q83S3)⋊7C2, (C4×C3⋊S3).44C22, (C2×C3⋊S3).46C23, SmallGroup(288,963)

Series: Derived Chief Lower central Upper central

C1C3×C6 — D12.25D6
C1C3C32C3×C6S3×C6S3×Dic3S3×Dic6 — D12.25D6
C32C3×C6 — D12.25D6
C1C2Q8

Generators and relations for D12.25D6
 G = < a,b,c,d | a12=b2=1, c6=d2=a6, bab=a-1, cac-1=a7, ad=da, cbc-1=a6b, bd=db, dcd-1=a6c5 >

Subgroups: 1026 in 311 conjugacy classes, 108 normal (24 characteristic)
C1, C2, C2 [×5], C3 [×2], C3, C4 [×3], C4 [×7], C22 [×5], S3 [×8], C6 [×2], C6 [×5], C2×C4 [×15], D4 [×10], Q8, Q8 [×9], C32, Dic3, Dic3 [×3], Dic3 [×9], C12 [×6], C12 [×7], D6, D6 [×3], D6 [×3], C2×C6 [×4], C2×Q8 [×5], C4○D4 [×10], C3×S3 [×4], C3⋊S3, C3×C6, Dic6 [×3], Dic6 [×15], C4×S3 [×6], C4×S3 [×15], D12 [×3], D12 [×4], C2×Dic3 [×6], C3⋊D4 [×10], C2×C12 [×6], C3×D4 [×3], C3×Q8 [×2], C3×Q8 [×4], 2- 1+4, C3×Dic3, C3×Dic3 [×3], C3⋊Dic3 [×3], C3×C12 [×3], S3×C6, S3×C6 [×3], C2×C3⋊S3, C2×Dic6 [×3], C4○D12 [×9], D42S3 [×6], S3×Q8, S3×Q8 [×6], Q83S3, Q83S3 [×3], C6×Q8, C3×C4○D4, S3×Dic3 [×6], D6⋊S3 [×3], C3⋊D12, C3⋊D12 [×3], C322Q8 [×3], C3×Dic6 [×3], S3×C12 [×6], C3×D12 [×3], C324Q8 [×3], C4×C3⋊S3 [×3], Q8×C32, Q8.15D6, Q8○D12, S3×Dic6 [×3], D125S3 [×3], D12⋊S3 [×3], D6.D6 [×3], C3×S3×Q8, C3×Q83S3, Q8×C3⋊S3, D12.25D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×2], C23 [×15], D6 [×14], C24, C22×S3 [×14], 2- 1+4, S32, S3×C23 [×2], C2×S32 [×3], Q8.15D6, Q8○D12, C22×S32, D12.25D6

Smallest permutation representation of D12.25D6
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 47)(2 46)(3 45)(4 44)(5 43)(6 42)(7 41)(8 40)(9 39)(10 38)(11 37)(12 48)(13 28)(14 27)(15 26)(16 25)(17 36)(18 35)(19 34)(20 33)(21 32)(22 31)(23 30)(24 29)
(1 21 3 23 5 13 7 15 9 17 11 19)(2 16 4 18 6 20 8 22 10 24 12 14)(25 38 35 48 33 46 31 44 29 42 27 40)(26 45 36 43 34 41 32 39 30 37 28 47)
(1 32 7 26)(2 33 8 27)(3 34 9 28)(4 35 10 29)(5 36 11 30)(6 25 12 31)(13 45 19 39)(14 46 20 40)(15 47 21 41)(16 48 22 42)(17 37 23 43)(18 38 24 44)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F3A3B3C4A4B4C4D4E4F4G4H4I4J6A6B6C6D6E6F6G6H12A···12F12G12H12I12J12K12L12M12N
order122222233344444444446666666612···121212121212121212
size116666182242226666181818224661212124···466888121212

42 irreducible representations

dim11111111222222444448
type++++++++++++++-++--
imageC1C2C2C2C2C2C2C2S3S3D6D6D6D62- 1+4S32C2×S32Q8.15D6Q8○D12D12.25D6
kernelD12.25D6S3×Dic6D125S3D12⋊S3D6.D6C3×S3×Q8C3×Q83S3Q8×C3⋊S3S3×Q8Q83S3Dic6C4×S3D12C3×Q8C32Q8C4C3C3C1
# reps13333111113632113221

Matrix representation of D12.25D6 in GL6(𝔽13)

1200000
0120000
000058
000050
005800
005000
,
1200000
0120000
000092
0000114
0041100
002900
,
0120000
110000
005000
000500
000080
000008
,
010000
100000
000037
0000610
003700
0061000

G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,5,5,0,0,0,0,8,0,0,0,5,5,0,0,0,0,8,0,0,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,4,2,0,0,0,0,11,9,0,0,9,11,0,0,0,0,2,4,0,0],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,8,0,0,0,0,0,0,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,6,0,0,0,0,7,10,0,0,3,6,0,0,0,0,7,10,0,0] >;

D12.25D6 in GAP, Magma, Sage, TeX

D_{12}._{25}D_6
% in TeX

G:=Group("D12.25D6");
// GroupNames label

G:=SmallGroup(288,963);
// by ID

G=gap.SmallGroup(288,963);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,219,100,675,185,80,1356,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^12=b^2=1,c^6=d^2=a^6,b*a*b=a^-1,c*a*c^-1=a^7,a*d=d*a,c*b*c^-1=a^6*b,b*d=d*b,d*c*d^-1=a^6*c^5>;
// generators/relations

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