direct product, metabelian, supersoluble, monomial, A-group
Aliases: C2×C52⋊3C8, C102.3C4, Dic5.14D10, Dic5.2Dic5, C10⋊3(C5⋊C8), C10⋊(C5⋊2C8), (C5×C10)⋊3C8, C52⋊12(C2×C8), (C2×C10).10F5, C10.39(C2×F5), (C2×Dic5).4D5, (C5×Dic5).9C4, C10.5(C2×Dic5), (C2×C10).1Dic5, (C10×Dic5).8C2, C22.2(D5.D5), (C5×Dic5).18C22, C5⋊5(C2×C5⋊C8), C5⋊2(C2×C5⋊2C8), C2.3(C2×D5.D5), (C5×C10).24(C2×C4), SmallGroup(400,146)
Series: Derived ►Chief ►Lower central ►Upper central
| C52 — C2×C52⋊3C8 |
Generators and relations for C2×C52⋊3C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c2 >
Smallest permutation representation of C2×C52⋊3C8
►On 80 points
Generators in S80
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)
(1 67 11 75 25)(2 26 76 12 68)(3 69 13 77 27)(4 28 78 14 70)(5 71 15 79 29)(6 30 80 16 72)(7 65 9 73 31)(8 32 74 10 66)(17 60 46 38 54)(18 55 39 47 61)(19 62 48 40 56)(20 49 33 41 63)(21 64 42 34 50)(22 51 35 43 57)(23 58 44 36 52)(24 53 37 45 59)
(1 75 67 25 11)(2 68 12 76 26)(3 13 27 69 77)(4 28 78 14 70)(5 79 71 29 15)(6 72 16 80 30)(7 9 31 65 73)(8 32 74 10 66)(17 38 60 54 46)(18 61 47 39 55)(19 48 56 62 40)(20 49 33 41 63)(21 34 64 50 42)(22 57 43 35 51)(23 44 52 58 36)(24 53 37 45 59)
(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)
G:=sub<Sym(80)| (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73), (1,67,11,75,25)(2,26,76,12,68)(3,69,13,77,27)(4,28,78,14,70)(5,71,15,79,29)(6,30,80,16,72)(7,65,9,73,31)(8,32,74,10,66)(17,60,46,38,54)(18,55,39,47,61)(19,62,48,40,56)(20,49,33,41,63)(21,64,42,34,50)(22,51,35,43,57)(23,58,44,36,52)(24,53,37,45,59), (1,75,67,25,11)(2,68,12,76,26)(3,13,27,69,77)(4,28,78,14,70)(5,79,71,29,15)(6,72,16,80,30)(7,9,31,65,73)(8,32,74,10,66)(17,38,60,54,46)(18,61,47,39,55)(19,48,56,62,40)(20,49,33,41,63)(21,34,64,50,42)(22,57,43,35,51)(23,44,52,58,36)(24,53,37,45,59), (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)>;
G:=Group( (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73), (1,67,11,75,25)(2,26,76,12,68)(3,69,13,77,27)(4,28,78,14,70)(5,71,15,79,29)(6,30,80,16,72)(7,65,9,73,31)(8,32,74,10,66)(17,60,46,38,54)(18,55,39,47,61)(19,62,48,40,56)(20,49,33,41,63)(21,64,42,34,50)(22,51,35,43,57)(23,58,44,36,52)(24,53,37,45,59), (1,75,67,25,11)(2,68,12,76,26)(3,13,27,69,77)(4,28,78,14,70)(5,79,71,29,15)(6,72,16,80,30)(7,9,31,65,73)(8,32,74,10,66)(17,38,60,54,46)(18,61,47,39,55)(19,48,56,62,40)(20,49,33,41,63)(21,34,64,50,42)(22,57,43,35,51)(23,44,52,58,36)(24,53,37,45,59), (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) );
G=PermutationGroup([[(1,64),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73)], [(1,67,11,75,25),(2,26,76,12,68),(3,69,13,77,27),(4,28,78,14,70),(5,71,15,79,29),(6,30,80,16,72),(7,65,9,73,31),(8,32,74,10,66),(17,60,46,38,54),(18,55,39,47,61),(19,62,48,40,56),(20,49,33,41,63),(21,64,42,34,50),(22,51,35,43,57),(23,58,44,36,52),(24,53,37,45,59)], [(1,75,67,25,11),(2,68,12,76,26),(3,13,27,69,77),(4,28,78,14,70),(5,79,71,29,15),(6,72,16,80,30),(7,9,31,65,73),(8,32,74,10,66),(17,38,60,54,46),(18,61,47,39,55),(19,48,56,62,40),(20,49,33,41,63),(21,34,64,50,42),(22,57,43,35,51),(23,44,52,58,36),(24,53,37,45,59)], [(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)]])
52 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 5A | 5B | 5C | ··· | 5G | 8A | ··· | 8H | 10A | ··· | 10F | 10G | ··· | 10U | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 5 | 5 | 5 | ··· | 5 | 8 | ··· | 8 | 10 | ··· | 10 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 5 | 5 | 5 | 5 | 2 | 2 | 4 | ··· | 4 | 25 | ··· | 25 | 2 | ··· | 2 | 4 | ··· | 4 | 10 | ··· | 10 |
52 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | - | + | - | + | - | + | |||||||
| image | C1 | C2 | C2 | C4 | C4 | C8 | D5 | Dic5 | D10 | Dic5 | C5⋊2C8 | F5 | C5⋊C8 | C2×F5 | D5.D5 | C52⋊3C8 | C2×D5.D5 |
| kernel | C2×C52⋊3C8 | C52⋊3C8 | C10×Dic5 | C5×Dic5 | C102 | C5×C10 | C2×Dic5 | Dic5 | Dic5 | C2×C10 | C10 | C2×C10 | C10 | C10 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 8 | 1 | 2 | 1 | 4 | 8 | 4 |
Matrix representation of C2×C52⋊3C8 ►in GL6(𝔽41)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 35 | 6 | 0 | 0 | 0 | 0 |
| 35 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 0 | 0 | 0 |
| 0 | 0 | 0 | 10 | 0 | 0 |
| 0 | 0 | 29 | 32 | 37 | 0 |
| 0 | 0 | 33 | 40 | 0 | 37 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 18 | 21 | 37 | 0 |
| 0 | 0 | 20 | 17 | 0 | 10 |
| 0 | 38 | 0 | 0 | 0 | 0 |
| 38 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 36 | 15 | 0 |
| 0 | 0 | 32 | 4 | 0 | 15 |
| 0 | 0 | 32 | 1 | 34 | 5 |
| 0 | 0 | 12 | 26 | 9 | 37 |
G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[35,35,0,0,0,0,6,40,0,0,0,0,0,0,10,0,29,33,0,0,0,10,32,40,0,0,0,0,37,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,18,20,0,0,0,18,21,17,0,0,0,0,37,0,0,0,0,0,0,10],[0,38,0,0,0,0,38,0,0,0,0,0,0,0,7,32,32,12,0,0,36,4,1,26,0,0,15,0,34,9,0,0,0,15,5,37] >;
C2×C52⋊3C8 in GAP, Magma, Sage, TeX
C_2\times C_5^2\rtimes_3C_8
% in TeX
G:=Group("C2xC5^2:3C8"); // GroupNames label
G:=SmallGroup(400,146);
// by ID
G=gap.SmallGroup(400,146);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,1924,8645,2897]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^5=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^2>;
// generators/relations
Export