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G = C12.28D12order 288 = 25·32

28th non-split extension by C12 of D12 acting via D12/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.28D12, C62.34C23, (C4×Dic3)⋊4S3, (C2×Dic6)⋊3S3, (C6×Dic6)⋊7C2, C6.76(C2×D12), (C3×C12).77D4, (Dic3×C12)⋊9C2, C6.9(C4○D12), (C2×C12).278D6, C33(C427S3), C12.75(C3⋊D4), (C6×C12).94C22, C6.9(Q83S3), (C2×Dic3).95D6, C325(C4.4D4), C6.D1221C2, C4.10(C3⋊D12), C31(C12.23D4), C2.12(D6.6D6), (C6×Dic3).108C22, (C2×C4).75S32, C22.91(C2×S32), (C3×C6).85(C2×D4), C6.12(C2×C3⋊D4), (C3×C6).21(C4○D4), C2.16(C2×C3⋊D12), (C2×C6).53(C22×S3), (C2×C12⋊S3).12C2, (C22×C3⋊S3).11C22, SmallGroup(288,512)

Series: Derived Chief Lower central Upper central

C1C62 — C12.28D12
C1C3C32C3×C6C62C6×Dic3C6.D12 — C12.28D12
C32C62 — C12.28D12
C1C22C2×C4

Generators and relations for C12.28D12
 G = < a,b,c | a12=b12=c2=1, bab-1=a5, cac=a-1, cbc=a6b-1 >

Subgroups: 866 in 179 conjugacy classes, 52 normal (22 characteristic)
C1, C2, C2 [×2], C2 [×2], C3 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×8], C6 [×2], C6 [×4], C6 [×3], C2×C4, C2×C4 [×4], D4 [×2], Q8 [×2], C23 [×2], C32, Dic3 [×4], C12 [×4], C12 [×6], D6 [×20], C2×C6 [×2], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, C3⋊S3 [×2], C3×C6, C3×C6 [×2], Dic6 [×2], D12 [×8], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×5], C3×Q8 [×2], C22×S3 [×6], C4.4D4, C3×Dic3 [×4], C3×C12 [×2], C2×C3⋊S3 [×6], C62, C4×Dic3, D6⋊C4 [×8], C4×C12, C2×Dic6, C2×D12 [×3], C6×Q8, C3×Dic6 [×2], C6×Dic3 [×4], C12⋊S3 [×2], C6×C12, C22×C3⋊S3 [×2], C427S3, C12.23D4, C6.D12 [×4], Dic3×C12, C6×Dic6, C2×C12⋊S3, C12.28D12
Quotients: C1, C2 [×7], C22 [×7], S3 [×2], D4 [×2], C23, D6 [×6], C2×D4, C4○D4 [×2], D12 [×2], C3⋊D4 [×2], C22×S3 [×2], C4.4D4, S32, C2×D12, C4○D12 [×2], Q83S3 [×2], C2×C3⋊D4, C3⋊D12 [×2], C2×S32, C427S3, C12.23D4, D6.6D6 [×2], C2×C3⋊D12, C12.28D12

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

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

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

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

48 conjugacy classes

class 1 2A2B2C2D2E3A3B3C4A4B4C4D4E4F4G4H6A···6F6G6H6I12A12B12C12D12E···12J12K···12R12S12T12U12V
order122222333444444446···66661212121212···1212···1212121212
size1111363622422666612122···244422224···46···612121212

48 irreducible representations

dim1111122222222244444
type++++++++++++++++
imageC1C2C2C2C2S3S3D4D6D6C4○D4D12C3⋊D4C4○D12S32Q83S3C3⋊D12C2×S32D6.6D6
kernelC12.28D12C6.D12Dic3×C12C6×Dic6C2×C12⋊S3C4×Dic3C2×Dic6C3×C12C2×Dic3C2×C12C3×C6C12C12C6C2×C4C6C4C22C2
# reps1411111242444812214

Matrix representation of C12.28D12 in GL8(𝔽13)

01000000
120000000
00100000
00010000
00001000
00000100
0000001212
00000010
,
50000000
05000000
00010000
001200000
000011200
00001000
00000010
0000001212
,
10000000
012000000
001200000
00010000
000012000
000012100
00000010
0000001212

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

C12.28D12 in GAP, Magma, Sage, TeX

C_{12}._{28}D_{12}
% in TeX

G:=Group("C12.28D12");
// GroupNames label

G:=SmallGroup(288,512);
// by ID

G=gap.SmallGroup(288,512);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,176,422,100,1356,9414]);
// Polycyclic

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

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