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G = C12.28D12  order 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, C3⋊3(C42⋊7S3), C12.75(C3⋊D4), (C6×C12).94C22, C6.9(Q8⋊3S3), (C2×Dic3).95D6, C32⋊5(C4.4D4), C6.D12⋊21C2, C4.10(C3⋊D12), C3⋊1(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

Derived series
C1 — C62 — C12.28D12
Chief series
C1 — C3 — C32 — C3×C6 — C62 — C6×Dic3 — C6.D12 — C12.28D12
Lower central
C32 — C62 — C12.28D12
Upper central
C1 — C22 — C2×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, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C4.4D4, C3×Dic3, C3×C12, C2×C3⋊S3, C62, C4×Dic3, D6⋊C4, C4×C12, C2×Dic6, C2×D12, C6×Q8, C3×Dic6, C6×Dic3, C12⋊S3, C6×C12, C22×C3⋊S3, C42⋊7S3, C12.23D4, C6.D12, Dic3×C12, C6×Dic6, C2×C12⋊S3, C12.28D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4.4D4, S32, C2×D12, C4○D12, Q8⋊3S3, C2×C3⋊D4, C3⋊D12, C2×S32, C42⋊7S3, C12.23D4, D6.6D6, 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 26 48 22 5 34 40 18 9 30 44 14)(2 31 37 15 6 27 41 23 10 35 45 19)(3 36 38 20 7 32 42 16 11 28 46 24)(4 29 39 13 8 25 43 21 12 33 47 17)

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

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○D12S32Q8⋊3S3C3⋊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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