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G = (C3×C12)⋊4C8  order 288 = 25·32

2nd semidirect product of C3×C12 and C8 acting via C8/C2=C4

metabelian, soluble, monomial

Aliases: (C3×C12)⋊4C8, C32⋊4(C4⋊C8), C4⋊(C32⋊2C8), (C6×C12).5C4, C3⋊Dic3.8Q8, C62.4(C2×C4), C3⋊Dic3.37D4, (C3×C6).1M4(2), C2.1(C32⋊M4(2)), (C3×C6).25(C2×C8), (C3×C6).15(C4⋊C4), (C2×C4).6(C32⋊C4), C2.4(C2×C32⋊2C8), C22.9(C2×C32⋊C4), C2.1(C4⋊(C32⋊C4)), (C4×C3⋊Dic3).15C2, (C2×C3⋊Dic3).14C4, (C2×C32⋊2C8).2C2, (C2×C3⋊Dic3).108C22, SmallGroup(288,424)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — (C3×C12)⋊4C8
Chief series
C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — (C3×C12)⋊4C8
Lower central
C32 — C3×C6 — (C3×C12)⋊4C8
Upper central
C1 — C22 — C2×C4

Generators and relations for (C3×C12)⋊4C8
 G = < a,b,c | a3=b12=c8=1, ab=ba, cac-1=a-1b4, cbc-1=ab7 >

Subgroups: 272 in 68 conjugacy classes, 24 normal (18 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C2×C8, C3×C6, C2×Dic3, C2×C12, C4⋊C8, C3⋊Dic3, C3⋊Dic3, C3×C12, C62, C4×Dic3, C32⋊2C8, C2×C3⋊Dic3, C6×C12, C4×C3⋊Dic3, C2×C32⋊2C8, (C3×C12)⋊4C8
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, C32⋊C4, C32⋊2C8, C2×C32⋊C4, C32⋊M4(2), C4⋊(C32⋊C4), C2×C32⋊2C8, (C3×C12)⋊4C8

Smallest permutation representation of (C3×C12)⋊4C8
►On 96 points
Generators in S96
(25 29 33)(26 30 34)(27 31 35)(28 32 36)(49 53 57)(50 54 58)(51 55 59)(52 56 60)(61 65 69)(62 66 70)(63 67 71)(64 68 72)(85 89 93)(86 90 94)(87 91 95)(88 92 96)

(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)

(1 72 23 57 41 29 78 95)(2 67 16 56 42 36 83 94)(3 62 21 55 43 31 76 93)(4 69 14 54 44 26 81 92)(5 64 19 53 45 33 74 91)(6 71 24 52 46 28 79 90)(7 66 17 51 47 35 84 89)(8 61 22 50 48 30 77 88)(9 68 15 49 37 25 82 87)(10 63 20 60 38 32 75 86)(11 70 13 59 39 27 80 85)(12 65 18 58 40 34 73 96)
 
G:=sub<Sym(96)| (25,29,33)(26,30,34)(27,31,35)(28,32,36)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,72,23,57,41,29,78,95)(2,67,16,56,42,36,83,94)(3,62,21,55,43,31,76,93)(4,69,14,54,44,26,81,92)(5,64,19,53,45,33,74,91)(6,71,24,52,46,28,79,90)(7,66,17,51,47,35,84,89)(8,61,22,50,48,30,77,88)(9,68,15,49,37,25,82,87)(10,63,20,60,38,32,75,86)(11,70,13,59,39,27,80,85)(12,65,18,58,40,34,73,96)>;
 
G:=Group( (25,29,33)(26,30,34)(27,31,35)(28,32,36)(49,53,57)(50,54,58)(51,55,59)(52,56,60)(61,65,69)(62,66,70)(63,67,71)(64,68,72)(85,89,93)(86,90,94)(87,91,95)(88,92,96), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,72,23,57,41,29,78,95)(2,67,16,56,42,36,83,94)(3,62,21,55,43,31,76,93)(4,69,14,54,44,26,81,92)(5,64,19,53,45,33,74,91)(6,71,24,52,46,28,79,90)(7,66,17,51,47,35,84,89)(8,61,22,50,48,30,77,88)(9,68,15,49,37,25,82,87)(10,63,20,60,38,32,75,86)(11,70,13,59,39,27,80,85)(12,65,18,58,40,34,73,96) );
 
G=PermutationGroup([[(25,29,33),(26,30,34),(27,31,35),(28,32,36),(49,53,57),(50,54,58),(51,55,59),(52,56,60),(61,65,69),(62,66,70),(63,67,71),(64,68,72),(85,89,93),(86,90,94),(87,91,95),(88,92,96)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,72,23,57,41,29,78,95),(2,67,16,56,42,36,83,94),(3,62,21,55,43,31,76,93),(4,69,14,54,44,26,81,92),(5,64,19,53,45,33,74,91),(6,71,24,52,46,28,79,90),(7,66,17,51,47,35,84,89),(8,61,22,50,48,30,77,88),(9,68,15,49,37,25,82,87),(10,63,20,60,38,32,75,86),(11,70,13,59,39,27,80,85),(12,65,18,58,40,34,73,96)]])
 

36 conjugacy classes

class 1 2A2B2C3A3B4A4B4C4D4E4F4G4H6A···6F8A···8H12A···12H
order122233444444446···68···812···12
size11114422999918184···418···184···4

36 irreducible representations

dim11111122244444
type++++-+-+
imageC1C2C2C4C4C8D4Q8M4(2)C32⋊C4C32⋊2C8C2×C32⋊C4C32⋊M4(2)C4⋊(C32⋊C4)
kernel(C3×C12)⋊4C8C4×C3⋊Dic3C2×C32⋊2C8C2×C3⋊Dic3C6×C12C3×C12C3⋊Dic3C3⋊Dic3C3×C6C2×C4C4C22C2C2
# reps11222811224244

Matrix representation of (C3×C12)⋊4C8 ►in GL8(𝔽73)

10000000
01000000
00100000
00010000
00001000
00000100
000072727272
00000010
,
171000000
172000000
007220000
007210000
000072100
000072000
00000727272
00001010
,
100000000
1063000000
005990000
0027140000
000000721
000072727172
0000652010
0000206210

G:=sub<GL(8,GF(73))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[1,1,0,0,0,0,0,0,71,72,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,72,72,0,1,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,1,0,0,0,0,0,0,72,0],[10,10,0,0,0,0,0,0,0,63,0,0,0,0,0,0,0,0,59,27,0,0,0,0,0,0,9,14,0,0,0,0,0,0,0,0,0,72,65,20,0,0,0,0,0,72,20,62,0,0,0,0,72,71,1,1,0,0,0,0,1,72,0,0] >;
 

(C3×C12)⋊4C8 in GAP, Magma, Sage, TeX 

(C_3\times C_{12})\rtimes_4C_8
 
% in TeX
 
G:=Group("(C3xC12):4C8");
 
// GroupNames label
 
G:=SmallGroup(288,424);
 
// by ID
 
G=gap.SmallGroup(288,424);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,100,9413,691,12550,2372]);
 
// Polycyclic
 
G:=Group<a,b,c|a^3=b^12=c^8=1,a*b=b*a,c*a*c^-1=a^-1*b^4,c*b*c^-1=a*b^7>;
 
// generators/relations
 

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