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G = C122D12order 288 = 25·32

2nd semidirect product of C12 and D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C122D12, C62.86C23, (C3×C12)⋊8D4, (C6×D12)⋊8C2, C6.43(S3×D4), (C2×D12)⋊11S3, C123(C3⋊D4), C4⋊Dic315S3, C6.80(C2×D12), C43(C3⋊D12), C31(D63D4), C35(C12⋊D4), D6⋊Dic313C2, (C2×C12).142D6, C3210(C4⋊D4), (C2×Dic3).35D6, (C22×S3).21D6, C6.12(D42S3), C2.18(D6⋊D6), (C6×C12).109C22, C6.36(Q83S3), C2.19(D12⋊S3), (C6×Dic3).19C22, (C2×C4).120S32, (C2×C3⋊S3)⋊10D4, C6.18(C2×C3⋊D4), (C2×C3⋊D12)⋊6C2, (C3×C4⋊Dic3)⋊19C2, C22.124(C2×S32), (C3×C6).111(C2×D4), (S3×C2×C6).36C22, (C3×C6).53(C4○D4), C2.21(C2×C3⋊D12), (C2×C6).105(C22×S3), (C22×C3⋊S3).74C22, (C2×C3⋊Dic3).139C22, (C2×C4×C3⋊S3)⋊1C2, SmallGroup(288,564)

Series: Derived Chief Lower central Upper central

C1C62 — C122D12
C1C3C32C3×C6C62S3×C2×C6C2×C3⋊D12 — C122D12
C32C62 — C122D12
C1C22C2×C4

Generators and relations for C122D12
 G = < a,b,c | a12=b12=c2=1, bab-1=a-1, cac=a5, cbc=b-1 >

Subgroups: 946 in 215 conjugacy classes, 54 normal (32 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 [×6], C12 [×4], C12 [×4], D6 [×20], C2×C6 [×2], C2×C6 [×7], C22⋊C4 [×2], C4⋊C4, C22×C4, C2×D4 [×3], C3×S3 [×2], C3⋊S3 [×2], C3×C6 [×3], C4×S3 [×8], D12 [×6], C2×Dic3 [×2], C2×Dic3 [×3], C3⋊D4 [×4], C2×C12 [×2], C2×C12 [×3], C3×D4 [×2], C22×S3 [×2], C22×S3 [×3], C22×C6 [×2], C4⋊D4, C3×Dic3 [×2], C3⋊Dic3, C3×C12 [×2], S3×C6 [×6], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C4⋊Dic3, D6⋊C4 [×2], C6.D4 [×2], C3×C4⋊C4, S3×C2×C4 [×3], C2×D12, C2×D12 [×2], C2×C3⋊D4 [×2], C6×D4, C3⋊D12 [×4], C3×D12 [×2], C6×Dic3 [×2], C4×C3⋊S3 [×2], C2×C3⋊Dic3, C6×C12, S3×C2×C6 [×2], C22×C3⋊S3, C12⋊D4, D63D4, D6⋊Dic3 [×2], C3×C4⋊Dic3, C2×C3⋊D12 [×2], C6×D12, C2×C4×C3⋊S3, C122D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×4], C23, D6 [×6], C2×D4 [×2], C4○D4, D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4⋊D4, S32, C2×D12, S3×D4 [×2], D42S3, Q83S3, C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C12⋊D4, D63D4, D12⋊S3, D6⋊D6, C2×C3⋊D12, C122D12

Smallest permutation representation of C122D12
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 34 15 45 5 30 19 41 9 26 23 37)(2 33 16 44 6 29 20 40 10 25 24 48)(3 32 17 43 7 28 21 39 11 36 13 47)(4 31 18 42 8 27 22 38 12 35 14 46)
(1 15)(2 20)(3 13)(4 18)(5 23)(6 16)(7 21)(8 14)(9 19)(10 24)(11 17)(12 22)(26 30)(27 35)(29 33)(32 36)(37 45)(39 43)(40 48)(42 46)

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

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

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

42 conjugacy classes

class 1 2A2B2C2D2E2F2G3A3B3C4A4B4C4D4E4F6A···6F6G6H6I6J6K6L6M12A···12H12I12J12K12L
order122222223334444446···6666666612···1212121212
size11111212181822422121218182···2444121212124···412121212

42 irreducible representations

dim111111222222222244444444
type++++++++++++++++-+++
imageC1C2C2C2C2C2S3S3D4D4D6D6D6C4○D4D12C3⋊D4S32S3×D4D42S3Q83S3C3⋊D12C2×S32D12⋊S3D6⋊D6
kernelC122D12D6⋊Dic3C3×C4⋊Dic3C2×C3⋊D12C6×D12C2×C4×C3⋊S3C4⋊Dic3C2×D12C3×C12C2×C3⋊S3C2×Dic3C2×C12C22×S3C3×C6C12C12C2×C4C6C6C6C4C22C2C2
# reps121211112222224412112122

Matrix representation of C122D12 in GL8(𝔽13)

83000000
05000000
001200000
000120000
00001100
000012000
000000120
000000012
,
115000000
22000000
001210000
001200000
00001000
0000121200
0000001211
00000011
,
10000000
01000000
001200000
001210000
00001000
0000121200
000000120
00000011

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

C122D12 in GAP, Magma, Sage, TeX

C_{12}\rtimes_2D_{12}
% in TeX

G:=Group("C12:2D12");
// GroupNames label

G:=SmallGroup(288,564);
// by ID

G=gap.SmallGroup(288,564);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,422,219,100,1356,9414]);
// Polycyclic

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

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