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G = C127D12order 288 = 25·32

1st semidirect product of C12 and D12 acting via D12/D6=C2

metabelian, supersoluble, monomial

Aliases: D63D12, C127D12, C62.79C23, (C3×C12)⋊5D4, (S3×C6)⋊10D4, C4⋊Dic39S3, C6.22(S3×D4), C121(C3⋊D4), C6.23(C2×D12), C2.24(S3×D12), C42(C3⋊D12), C33(C12⋊D4), C31(C127D4), (C2×C12).139D6, C327(C4⋊D4), C6.16(C4○D12), C6.D123C2, (C2×Dic3).32D6, (C22×S3).68D6, (C6×C12).106C22, C6.17(Q83S3), C2.18(D6.6D6), (C6×Dic3).16C22, (S3×C2×C4)⋊2S3, (S3×C2×C12)⋊4C2, (C2×C4).82S32, C6.16(C2×C3⋊D4), (C2×C3⋊D12)⋊3C2, (C3×C4⋊Dic3)⋊18C2, C22.117(C2×S32), (C3×C6).105(C2×D4), (C2×C12⋊S3)⋊11C2, (S3×C2×C6).81C22, (C3×C6).49(C4○D4), C2.19(C2×C3⋊D12), (C2×C6).98(C22×S3), (C22×C3⋊S3).23C22, SmallGroup(288,557)

Series: Derived Chief Lower central Upper central

C1C62 — C127D12
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C127D12
C32C62 — C127D12
C1C22C2×C4

Generators and relations for C127D12
 G = < a,b,c | a12=b12=c2=1, bab-1=cac=a-1, cbc=b-1 >

Subgroups: 1058 in 215 conjugacy classes, 56 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C3 [×2], C3, C4 [×2], C4 [×3], C22, C22 [×10], S3 [×10], C6 [×6], C6 [×5], C2×C4, C2×C4 [×5], D4 [×6], C23 [×3], C32, Dic3 [×3], C12 [×4], C12 [×5], D6 [×2], D6 [×22], C2×C6 [×2], C2×C6 [×5], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×2], D12 [×12], C2×Dic3, C2×Dic3 [×2], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×6], C22×S3, C22×S3 [×6], C22×C6, C4⋊D4, C3×Dic3 [×3], C3×C12 [×2], S3×C6 [×2], S3×C6 [×2], C2×C3⋊S3 [×6], C62, C4⋊Dic3, D6⋊C4 [×4], C3×C4⋊C4, S3×C2×C4, C2×D12 [×5], C2×C3⋊D4 [×2], C22×C12, C3⋊D12 [×4], S3×C12 [×2], C6×Dic3, C6×Dic3 [×2], C12⋊S3 [×2], C6×C12, S3×C2×C6, C22×C3⋊S3 [×2], C12⋊D4, C127D4, C6.D12 [×2], C3×C4⋊Dic3, C2×C3⋊D12 [×2], S3×C2×C12, C2×C12⋊S3, C127D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×4], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12 [×2], C4○D12, S3×D4, Q83S3, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C12⋊D4, C127D4, D6.6D6, S3×D12, C2×C3⋊D12, C127D12

Smallest permutation representation of C127D12
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 18 42 27 5 14 46 35 9 22 38 31)(2 17 43 26 6 13 47 34 10 21 39 30)(3 16 44 25 7 24 48 33 11 20 40 29)(4 15 45 36 8 23 37 32 12 19 41 28)
(1 45)(2 44)(3 43)(4 42)(5 41)(6 40)(7 39)(8 38)(9 37)(10 48)(11 47)(12 46)(13 20)(14 19)(15 18)(16 17)(21 24)(22 23)(25 30)(26 29)(27 28)(31 36)(32 35)(33 34)

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A12B12C12D12E···12J12K12L12M12N12O12P12Q12R
order122222223334444446···666666661212121212···121212121212121212
size1111663636224226612122···2444666622224···4666612121212

48 irreducible representations

dim1111112222222222224444444
type++++++++++++++++++++++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C3⋊D4D12C4○D12S32S3×D4Q83S3C3⋊D12C2×S32D6.6D6S3×D12
kernelC127D12C6.D12C3×C4⋊Dic3C2×C3⋊D12S3×C2×C12C2×C12⋊S3C4⋊Dic3S3×C2×C4C3×C12S3×C6C2×Dic3C2×C12C22×S3C3×C6C12C12D6C6C2×C4C6C6C4C22C2C2
# reps1212111122321244441112122

Matrix representation of C127D12 in GL8(𝔽13)

43000000
39000000
001200000
000120000
000012000
000001200
0000001212
00000010
,
910000000
54000000
00010000
001200000
000012100
000012000
00000010
0000001212
,
43000000
89000000
00100000
000120000
00001000
000011200
00000010
0000001212

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

C127D12 in GAP, Magma, Sage, TeX

C_{12}\rtimes_7D_{12}
% in TeX

G:=Group("C12:7D12");
// GroupNames label

G:=SmallGroup(288,557);
// by ID

G=gap.SmallGroup(288,557);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,100,1356,9414]);
// Polycyclic

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

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