G = C5⋊3(C23⋊C8) order 320 = 26·5
The semidirect product of C5 and C23⋊C8 acting via C23⋊C8/C22⋊C8=C2
metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C5⋊3(C23⋊C8), C22⋊C8⋊1D5, (C22×D5)⋊1C8, C22.3(C8×D5), (C2×C4).107D20, (C2×C20).438D4, (C23×D5).1C4, C23.40(C4×D5), (C22×C4).1D10, C20.55D4⋊20C2, C2.5(D10⋊1C8), C10.18(C22⋊C8), C10.25(C23⋊C4), C10.9(C4.D4), C22.3(C8⋊D5), (C2×C10).10M4(2), (C22×Dic5).1C4, C2.1(C20.46D4), (C22×C20).322C22, C2.1(C23.1D10), C22.32(D10⋊C4), (C5×C22⋊C8)⋊1C2, (C2×C10).16(C2×C8), (C2×C4).209(C5⋊D4), (C22×C10).94(C2×C4), (C2×D10⋊C4).23C2, (C2×C10).104(C22⋊C4), SmallGroup(320,26)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C5⋊3(C23⋊C8)
G = < a,b,c,d,e | a5=b2=c2=d2=e8=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bcd, ece-1=cd=dc, de=ed >
Subgroups: 518 in 98 conjugacy classes, 31 normal (29 characteristic)
C1, C2, C2, C4, C22, C22, C5, C8, C2×C4, C2×C4, C23, C23, D5, C10, C10, C22⋊C4, C2×C8, C22×C4, C22×C4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22⋊C8, C22⋊C8, C2×C22⋊C4, C5⋊2C8, C40, C2×Dic5, C2×C20, C2×C20, C22×D5, C22×D5, C22×C10, C23⋊C8, C2×C5⋊2C8, D10⋊C4, C2×C40, C22×Dic5, C22×C20, C23×D5, C20.55D4, C5×C22⋊C8, C2×D10⋊C4, C5⋊3(C23⋊C8)
Quotients: C1, C2, C4, C22, C8, C2×C4, D4, D5, C22⋊C4, C2×C8, M4(2), D10, C22⋊C8, C23⋊C4, C4.D4, C4×D5, D20, C5⋊D4, C23⋊C8, C8×D5, C8⋊D5, D10⋊C4, C23.1D10, D10⋊1C8, C20.46D4, C5⋊3(C23⋊C8)
►On 80 points
Generators in S80
(1 77 54 44 30)(2 78 55 45 31)(3 79 56 46 32)(4 80 49 47 25)(5 73 50 48 26)(6 74 51 41 27)(7 75 52 42 28)(8 76 53 43 29)(9 71 21 63 34)(10 72 22 64 35)(11 65 23 57 36)(12 66 24 58 37)(13 67 17 59 38)(14 68 18 60 39)(15 69 19 61 40)(16 70 20 62 33)
(1 5)(3 14)(4 11)(7 10)(8 15)(12 16)(17 59)(18 46)(19 43)(20 58)(21 63)(22 42)(23 47)(24 62)(25 65)(26 77)(27 74)(28 72)(29 69)(30 73)(31 78)(32 68)(33 66)(34 71)(35 75)(36 80)(37 70)(38 67)(39 79)(40 76)(41 51)(44 50)(45 55)(48 54)(49 57)(52 64)(53 61)(56 60)
(1 12)(2 6)(3 14)(4 8)(5 16)(7 10)(9 13)(11 15)(17 21)(18 56)(19 23)(20 50)(22 52)(24 54)(25 29)(26 33)(27 31)(28 35)(30 37)(32 39)(34 38)(36 40)(41 45)(42 64)(43 47)(44 58)(46 60)(48 62)(49 53)(51 55)(57 61)(59 63)(65 69)(66 77)(67 71)(68 79)(70 73)(72 75)(74 78)(76 80)
(1 16)(2 9)(3 10)(4 11)(5 12)(6 13)(7 14)(8 15)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 49)(24 50)(25 36)(26 37)(27 38)(28 39)(29 40)(30 33)(31 34)(32 35)(41 59)(42 60)(43 61)(44 62)(45 63)(46 64)(47 57)(48 58)(65 80)(66 73)(67 74)(68 75)(69 76)(70 77)(71 78)(72 79)
(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)| (1,77,54,44,30)(2,78,55,45,31)(3,79,56,46,32)(4,80,49,47,25)(5,73,50,48,26)(6,74,51,41,27)(7,75,52,42,28)(8,76,53,43,29)(9,71,21,63,34)(10,72,22,64,35)(11,65,23,57,36)(12,66,24,58,37)(13,67,17,59,38)(14,68,18,60,39)(15,69,19,61,40)(16,70,20,62,33), (1,5)(3,14)(4,11)(7,10)(8,15)(12,16)(17,59)(18,46)(19,43)(20,58)(21,63)(22,42)(23,47)(24,62)(25,65)(26,77)(27,74)(28,72)(29,69)(30,73)(31,78)(32,68)(33,66)(34,71)(35,75)(36,80)(37,70)(38,67)(39,79)(40,76)(41,51)(44,50)(45,55)(48,54)(49,57)(52,64)(53,61)(56,60), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15)(17,21)(18,56)(19,23)(20,50)(22,52)(24,54)(25,29)(26,33)(27,31)(28,35)(30,37)(32,39)(34,38)(36,40)(41,45)(42,64)(43,47)(44,58)(46,60)(48,62)(49,53)(51,55)(57,61)(59,63)(65,69)(66,77)(67,71)(68,79)(70,73)(72,75)(74,78)(76,80), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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( (1,77,54,44,30)(2,78,55,45,31)(3,79,56,46,32)(4,80,49,47,25)(5,73,50,48,26)(6,74,51,41,27)(7,75,52,42,28)(8,76,53,43,29)(9,71,21,63,34)(10,72,22,64,35)(11,65,23,57,36)(12,66,24,58,37)(13,67,17,59,38)(14,68,18,60,39)(15,69,19,61,40)(16,70,20,62,33), (1,5)(3,14)(4,11)(7,10)(8,15)(12,16)(17,59)(18,46)(19,43)(20,58)(21,63)(22,42)(23,47)(24,62)(25,65)(26,77)(27,74)(28,72)(29,69)(30,73)(31,78)(32,68)(33,66)(34,71)(35,75)(36,80)(37,70)(38,67)(39,79)(40,76)(41,51)(44,50)(45,55)(48,54)(49,57)(52,64)(53,61)(56,60), (1,12)(2,6)(3,14)(4,8)(5,16)(7,10)(9,13)(11,15)(17,21)(18,56)(19,23)(20,50)(22,52)(24,54)(25,29)(26,33)(27,31)(28,35)(30,37)(32,39)(34,38)(36,40)(41,45)(42,64)(43,47)(44,58)(46,60)(48,62)(49,53)(51,55)(57,61)(59,63)(65,69)(66,77)(67,71)(68,79)(70,73)(72,75)(74,78)(76,80), (1,16)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)(8,15)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,49)(24,50)(25,36)(26,37)(27,38)(28,39)(29,40)(30,33)(31,34)(32,35)(41,59)(42,60)(43,61)(44,62)(45,63)(46,64)(47,57)(48,58)(65,80)(66,73)(67,74)(68,75)(69,76)(70,77)(71,78)(72,79), (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([[(1,77,54,44,30),(2,78,55,45,31),(3,79,56,46,32),(4,80,49,47,25),(5,73,50,48,26),(6,74,51,41,27),(7,75,52,42,28),(8,76,53,43,29),(9,71,21,63,34),(10,72,22,64,35),(11,65,23,57,36),(12,66,24,58,37),(13,67,17,59,38),(14,68,18,60,39),(15,69,19,61,40),(16,70,20,62,33)], [(1,5),(3,14),(4,11),(7,10),(8,15),(12,16),(17,59),(18,46),(19,43),(20,58),(21,63),(22,42),(23,47),(24,62),(25,65),(26,77),(27,74),(28,72),(29,69),(30,73),(31,78),(32,68),(33,66),(34,71),(35,75),(36,80),(37,70),(38,67),(39,79),(40,76),(41,51),(44,50),(45,55),(48,54),(49,57),(52,64),(53,61),(56,60)], [(1,12),(2,6),(3,14),(4,8),(5,16),(7,10),(9,13),(11,15),(17,21),(18,56),(19,23),(20,50),(22,52),(24,54),(25,29),(26,33),(27,31),(28,35),(30,37),(32,39),(34,38),(36,40),(41,45),(42,64),(43,47),(44,58),(46,60),(48,62),(49,53),(51,55),(57,61),(59,63),(65,69),(66,77),(67,71),(68,79),(70,73),(72,75),(74,78),(76,80)], [(1,16),(2,9),(3,10),(4,11),(5,12),(6,13),(7,14),(8,15),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,49),(24,50),(25,36),(26,37),(27,38),(28,39),(29,40),(30,33),(31,34),(32,35),(41,59),(42,60),(43,61),(44,62),(45,63),(46,64),(47,57),(48,58),(65,80),(66,73),(67,74),(68,75),(69,76),(70,77),(71,78),(72,79)], [(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)]])
62 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | 5A | 5B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H | 20I | 20J | 20K | 20L | 40A | ··· | 40P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 | 40 | ··· | 40 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 20 | 20 | 2 | 2 | 2 | 2 | 20 | 20 | 2 | 2 | 4 | 4 | 4 | 4 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
62 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | D4 | D5 | M4(2) | D10 | D20 | C5⋊D4 | C4×D5 | C8×D5 | C8⋊D5 | C23⋊C4 | C4.D4 | C23.1D10 | C20.46D4 |
| kernel | C5⋊3(C23⋊C8) | C20.55D4 | C5×C22⋊C8 | C2×D10⋊C4 | C22×Dic5 | C23×D5 | C22×D5 | C2×C20 | C22⋊C8 | C2×C10 | C22×C4 | C2×C4 | C2×C4 | C23 | C22 | C22 | C10 | C10 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 1 | 1 | 4 | 4 |
Matrix representation of C5⋊3(C23⋊C8) ►in GL6(𝔽41)
| 17 | 32 | 0 | 0 | 0 | 0 |
| 9 | 17 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 40 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 40 | 6 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 6 | 40 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 6 | 40 |
| 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 | 40 | 0 |
| 0 | 0 | 0 | 0 | 0 | 40 |
| 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 39 | 28 | 0 | 0 |
| 0 | 0 | 13 | 2 | 0 | 0 |
G:=sub<GL(6,GF(41))| [17,9,0,0,0,0,32,17,0,0,0,0,0,0,0,40,0,0,0,0,1,6,0,0,0,0,0,0,0,40,0,0,0,0,1,6],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,39,13,0,0,0,0,28,2,0,0,1,0,0,0,0,0,0,1,0,0] >;
C5⋊3(C23⋊C8) in GAP, Magma, Sage, TeX
C_5\rtimes_3(C_2^3\rtimes C_8)
% in TeX
G:=Group("C5:3(C2^3:C8)"); // GroupNames label
G:=SmallGroup(320,26);
// by ID
G=gap.SmallGroup(320,26);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,758,100,570,12550]);
// Polycyclic
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b*c*d,e*c*e^-1=c*d=d*c,d*e=e*d>;
// generators/relations