G = C32⋊5(C4⋊C8) order 288 = 25·32
2nd semidirect product of C32 and C4⋊C8 acting via C4⋊C8/C2×C4=C4
Aliases: C32⋊5(C4⋊C8), (C6×C12).8C4, C3⋊Dic3⋊7C8, C3⋊Dic3.9Q8, C62.7(C2×C4), C3⋊Dic3.38D4, (C3×C6).10M4(2), C2.2(C62.C4), (C3×C6).11(C2×C8), (C3×C6).16(C4⋊C4), (C2×C4).4(C32⋊C4), C2.5(C3⋊S3⋊3C8), C2.2(C4⋊(C32⋊C4)), (C2×C3⋊Dic3).16C4, (C4×C3⋊Dic3).17C2, (C2×C32⋊2C8).3C2, C22.12(C2×C32⋊C4), (C2×C3⋊Dic3).111C22, SmallGroup(288,427)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C32 — C3×C6 — C3⋊Dic3 — C2×C3⋊Dic3 — C2×C32⋊2C8 — C32⋊5(C4⋊C8) |
Generators and relations for C32⋊5(C4⋊C8)
G = < a,b,c,d | a3=b3=c4=d8=1, dbd-1=ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dcd-1=c-1 >
Subgroups: 272 in 68 conjugacy classes, 22 normal (18 characteristic)
C1, C2, C3, 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, C32⋊5(C4⋊C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, Q8, C4⋊C4, C2×C8, M4(2), C4⋊C8, C32⋊C4, C2×C32⋊C4, C3⋊S3⋊3C8, C4⋊(C32⋊C4), C62.C4, C32⋊5(C4⋊C8)
►On 96 points
Generators in S96
(2 68 23)(4 17 70)(6 72 19)(8 21 66)(10 27 34)(12 36 29)(14 31 38)(16 40 25)(42 88 90)(44 92 82)(46 84 94)(48 96 86)(50 74 60)(52 62 76)(54 78 64)(56 58 80)
(1 67 22)(2 68 23)(3 24 69)(4 17 70)(5 71 18)(6 72 19)(7 20 65)(8 21 66)(9 26 33)(10 27 34)(11 35 28)(12 36 29)(13 30 37)(14 31 38)(15 39 32)(16 40 25)(41 87 89)(42 88 90)(43 91 81)(44 92 82)(45 83 93)(46 84 94)(47 95 85)(48 96 86)(49 73 59)(50 74 60)(51 61 75)(52 62 76)(53 77 63)(54 78 64)(55 57 79)(56 58 80)
(1 75 47 13)(2 14 48 76)(3 77 41 15)(4 16 42 78)(5 79 43 9)(6 10 44 80)(7 73 45 11)(8 12 46 74)(17 25 88 54)(18 55 81 26)(19 27 82 56)(20 49 83 28)(21 29 84 50)(22 51 85 30)(23 31 86 52)(24 53 87 32)(33 71 57 91)(34 92 58 72)(35 65 59 93)(36 94 60 66)(37 67 61 95)(38 96 62 68)(39 69 63 89)(40 90 64 70)
(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)
G:=sub<Sym(96)| (2,68,23)(4,17,70)(6,72,19)(8,21,66)(10,27,34)(12,36,29)(14,31,38)(16,40,25)(42,88,90)(44,92,82)(46,84,94)(48,96,86)(50,74,60)(52,62,76)(54,78,64)(56,58,80), (1,67,22)(2,68,23)(3,24,69)(4,17,70)(5,71,18)(6,72,19)(7,20,65)(8,21,66)(9,26,33)(10,27,34)(11,35,28)(12,36,29)(13,30,37)(14,31,38)(15,39,32)(16,40,25)(41,87,89)(42,88,90)(43,91,81)(44,92,82)(45,83,93)(46,84,94)(47,95,85)(48,96,86)(49,73,59)(50,74,60)(51,61,75)(52,62,76)(53,77,63)(54,78,64)(55,57,79)(56,58,80), (1,75,47,13)(2,14,48,76)(3,77,41,15)(4,16,42,78)(5,79,43,9)(6,10,44,80)(7,73,45,11)(8,12,46,74)(17,25,88,54)(18,55,81,26)(19,27,82,56)(20,49,83,28)(21,29,84,50)(22,51,85,30)(23,31,86,52)(24,53,87,32)(33,71,57,91)(34,92,58,72)(35,65,59,93)(36,94,60,66)(37,67,61,95)(38,96,62,68)(39,69,63,89)(40,90,64,70), (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)>;
G:=Group( (2,68,23)(4,17,70)(6,72,19)(8,21,66)(10,27,34)(12,36,29)(14,31,38)(16,40,25)(42,88,90)(44,92,82)(46,84,94)(48,96,86)(50,74,60)(52,62,76)(54,78,64)(56,58,80), (1,67,22)(2,68,23)(3,24,69)(4,17,70)(5,71,18)(6,72,19)(7,20,65)(8,21,66)(9,26,33)(10,27,34)(11,35,28)(12,36,29)(13,30,37)(14,31,38)(15,39,32)(16,40,25)(41,87,89)(42,88,90)(43,91,81)(44,92,82)(45,83,93)(46,84,94)(47,95,85)(48,96,86)(49,73,59)(50,74,60)(51,61,75)(52,62,76)(53,77,63)(54,78,64)(55,57,79)(56,58,80), (1,75,47,13)(2,14,48,76)(3,77,41,15)(4,16,42,78)(5,79,43,9)(6,10,44,80)(7,73,45,11)(8,12,46,74)(17,25,88,54)(18,55,81,26)(19,27,82,56)(20,49,83,28)(21,29,84,50)(22,51,85,30)(23,31,86,52)(24,53,87,32)(33,71,57,91)(34,92,58,72)(35,65,59,93)(36,94,60,66)(37,67,61,95)(38,96,62,68)(39,69,63,89)(40,90,64,70), (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) );
G=PermutationGroup([[(2,68,23),(4,17,70),(6,72,19),(8,21,66),(10,27,34),(12,36,29),(14,31,38),(16,40,25),(42,88,90),(44,92,82),(46,84,94),(48,96,86),(50,74,60),(52,62,76),(54,78,64),(56,58,80)], [(1,67,22),(2,68,23),(3,24,69),(4,17,70),(5,71,18),(6,72,19),(7,20,65),(8,21,66),(9,26,33),(10,27,34),(11,35,28),(12,36,29),(13,30,37),(14,31,38),(15,39,32),(16,40,25),(41,87,89),(42,88,90),(43,91,81),(44,92,82),(45,83,93),(46,84,94),(47,95,85),(48,96,86),(49,73,59),(50,74,60),(51,61,75),(52,62,76),(53,77,63),(54,78,64),(55,57,79),(56,58,80)], [(1,75,47,13),(2,14,48,76),(3,77,41,15),(4,16,42,78),(5,79,43,9),(6,10,44,80),(7,73,45,11),(8,12,46,74),(17,25,88,54),(18,55,81,26),(19,27,82,56),(20,49,83,28),(21,29,84,50),(22,51,85,30),(23,31,86,52),(24,53,87,32),(33,71,57,91),(34,92,58,72),(35,65,59,93),(36,94,60,66),(37,67,61,95),(38,96,62,68),(39,69,63,89),(40,90,64,70)], [(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)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 8A | ··· | 8H | 12A | ··· | 12H |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 4 | ··· | 4 | 18 | ··· | 18 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D4 | Q8 | M4(2) | C32⋊C4 | C2×C32⋊C4 | C3⋊S3⋊3C8 | C4⋊(C32⋊C4) | C62.C4 |
| kernel | C32⋊5(C4⋊C8) | C4×C3⋊Dic3 | C2×C32⋊2C8 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C3⋊Dic3 | C3⋊Dic3 | C3×C6 | C2×C4 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 |
Matrix representation of C32⋊5(C4⋊C8) ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 0 | 0 | 1 | 72 |
| 46 | 7 | 0 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 42 | 28 | 0 | 0 |
| 0 | 0 | 70 | 31 | 0 | 0 |
| 0 | 0 | 0 | 0 | 42 | 28 |
| 0 | 0 | 0 | 0 | 70 | 31 |
| 64 | 15 | 0 | 0 | 0 | 0 |
| 14 | 9 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 34 | 47 | 0 | 0 |
| 0 | 0 | 8 | 39 | 0 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[46,0,0,0,0,0,7,27,0,0,0,0,0,0,42,70,0,0,0,0,28,31,0,0,0,0,0,0,42,70,0,0,0,0,28,31],[64,14,0,0,0,0,15,9,0,0,0,0,0,0,0,0,34,8,0,0,0,0,47,39,0,0,1,0,0,0,0,0,0,1,0,0] >;
C32⋊5(C4⋊C8) in GAP, Magma, Sage, TeX
C_3^2\rtimes_5(C_4\rtimes C_8)
% in TeX
G:=Group("C3^2:5(C4:C8)"); // GroupNames label
G:=SmallGroup(288,427);
// by ID
G=gap.SmallGroup(288,427);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,176,100,9413,691,12550,2372]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^4=d^8=1,d*b*d^-1=a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations