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G = C12.71D12order 288 = 25·32

2nd non-split extension by C12 of D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: C12.71D12, C6.3(D6⋊C4), (C2×C12).74D6, (C3×C12).31D4, C62.30(C2×C4), C4.Dic3.3S3, C12.27(C3⋊D4), (C6×C12).23C22, C4.13(C3⋊D12), C31(C12.47D4), C324(C4.10D4), C2.4(C6.D12), C22.3(C6.D6), (C2×C4).4S32, (C2×C6).11(C4×S3), (C2×C3⋊Dic3).1C4, (C3×C4.Dic3).2C2, (C2×C324Q8).7C2, (C3×C6).28(C22⋊C4), SmallGroup(288,209)

Series: Derived Chief Lower central Upper central

C1C62 — C12.71D12
C1C3C32C3×C6C3×C12C6×C12C3×C4.Dic3 — C12.71D12
C32C3×C6C62 — C12.71D12
C1C2C2×C4

Generators and relations for C12.71D12
 G = < a,b,c | a12=1, b12=c2=a6, bab-1=cac-1=a-1, cbc-1=a9b11 >

Subgroups: 354 in 95 conjugacy classes, 32 normal (10 characteristic)
C1, C2, C2, C3 [×2], C3, C4 [×2], C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C32, Dic3 [×8], C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, M4(2) [×2], C2×Q8, C3×C6, C3×C6, C3⋊C8 [×2], C24 [×2], Dic6 [×8], C2×Dic3 [×6], C2×C12 [×2], C2×C12, C4.10D4, C3⋊Dic3 [×2], C3×C12 [×2], C62, C4.Dic3 [×2], C3×M4(2) [×2], C2×Dic6 [×3], C3×C3⋊C8 [×2], C324Q8 [×2], C2×C3⋊Dic3 [×2], C6×C12, C12.47D4 [×2], C3×C4.Dic3 [×2], C2×C324Q8, C12.71D12
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C2×C4, D4 [×2], D6 [×2], C22⋊C4, C4×S3 [×2], D12 [×2], C3⋊D4 [×2], C4.10D4, S32, D6⋊C4 [×2], C6.D6, C3⋊D12 [×2], C12.47D4 [×2], C6.D12, C12.71D12

Smallest permutation representation of C12.71D12
On 48 points
Generators in S48
(1 11 21 7 17 3 13 23 9 19 5 15)(2 16 6 20 10 24 14 4 18 8 22 12)(25 35 45 31 41 27 37 47 33 43 29 39)(26 40 30 44 34 48 38 28 42 32 46 36)
(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 27 13 39)(2 32 14 44)(3 25 15 37)(4 30 16 42)(5 47 17 35)(6 28 18 40)(7 45 19 33)(8 26 20 38)(9 43 21 31)(10 48 22 36)(11 41 23 29)(12 46 24 34)

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

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

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

39 conjugacy classes

class 1 2A2B3A3B3C4A4B4C4D6A6B6C···6G8A8B8C8D12A12B12C12D12E···12J24A···24H
order1223334444666···688881212121212···1224···24
size112224223636224···41212121222224···412···12

39 irreducible representations

dim1111222222444444
type+++++++-+++--
imageC1C2C2C4S3D4D6D12C3⋊D4C4×S3C4.10D4S32C3⋊D12C6.D6C12.47D4C12.71D12
kernelC12.71D12C3×C4.Dic3C2×C324Q8C2×C3⋊Dic3C4.Dic3C3×C12C2×C12C12C12C2×C6C32C2×C4C4C22C3C1
# reps1214222444112144

Matrix representation of C12.71D12 in GL4(𝔽73) generated by

596600
76600
005966
00766
,
002210
006151
681400
19500
,
681400
19500
001419
00559
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,59,7,0,0,66,66],[0,0,68,19,0,0,14,5,22,61,0,0,10,51,0,0],[68,19,0,0,14,5,0,0,0,0,14,5,0,0,19,59] >;

C12.71D12 in GAP, Magma, Sage, TeX

C_{12}._{71}D_{12}
% in TeX

G:=Group("C12.71D12");
// GroupNames label

G:=SmallGroup(288,209);
// by ID

G=gap.SmallGroup(288,209);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,204,219,100,675,346,1356,9414]);
// Polycyclic

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

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