metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C23.11D10, (C2×Dic5)⋊4C4, (C4×Dic5)⋊9C2, (C2×C4).26D10, C22⋊C4.3D5, C22.6(C4×D5), C5⋊3(C42⋊C2), C10.D4⋊7C2, C23.D5.1C2, C10.20(C4○D4), C2.1(D4⋊2D5), (C2×C20).50C22, C10.18(C22×C4), (C2×C10).18C23, Dic5.20(C2×C4), (C22×C10).7C22, (C22×Dic5).2C2, C22.12(C22×D5), (C2×Dic5).60C22, C2.7(C2×C4×D5), (C2×C10).24(C2×C4), (C5×C22⋊C4).3C2, SmallGroup(160,98)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C23.11D10
G = < a,b,c,d,e | a2=b2=c2=1, d10=b, e2=cb=bc, ab=ba, dad-1=eae-1=ac=ca, bd=db, be=eb, cd=dc, ce=ec, ede-1=d9 >
Subgroups: 184 in 76 conjugacy classes, 41 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×8], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×8], C23, C10, C10 [×2], C10 [×2], C42 [×2], C22⋊C4, C22⋊C4, C4⋊C4 [×2], C22×C4, Dic5 [×4], Dic5 [×2], C20 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C42⋊C2, C2×Dic5 [×2], C2×Dic5 [×6], C2×C20 [×2], C22×C10, C4×Dic5 [×2], C10.D4 [×2], C23.D5, C5×C22⋊C4, C22×Dic5, C23.11D10
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], C23, D5, C22×C4, C4○D4 [×2], D10 [×3], C42⋊C2, C4×D5 [×2], C22×D5, C2×C4×D5, D4⋊2D5 [×2], C23.11D10
►On 80 points
(2 62)(4 64)(6 66)(8 68)(10 70)(12 72)(14 74)(16 76)(18 78)(20 80)(22 45)(24 47)(26 49)(28 51)(30 53)(32 55)(34 57)(36 59)(38 41)(40 43)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 61)(2 62)(3 63)(4 64)(5 65)(6 66)(7 67)(8 68)(9 69)(10 70)(11 71)(12 72)(13 73)(14 74)(15 75)(16 76)(17 77)(18 78)(19 79)(20 80)(21 44)(22 45)(23 46)(24 47)(25 48)(26 49)(27 50)(28 51)(29 52)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 41)(39 42)(40 43)
(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)
(1 41 71 28)(2 50 72 37)(3 59 73 26)(4 48 74 35)(5 57 75 24)(6 46 76 33)(7 55 77 22)(8 44 78 31)(9 53 79 40)(10 42 80 29)(11 51 61 38)(12 60 62 27)(13 49 63 36)(14 58 64 25)(15 47 65 34)(16 56 66 23)(17 45 67 32)(18 54 68 21)(19 43 69 30)(20 52 70 39)
G:=sub<Sym(80)| (2,62)(4,64)(6,66)(8,68)(10,70)(12,72)(14,74)(16,76)(18,78)(20,80)(22,45)(24,47)(26,49)(28,51)(30,53)(32,55)(34,57)(36,59)(38,41)(40,43), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43), (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), (1,41,71,28)(2,50,72,37)(3,59,73,26)(4,48,74,35)(5,57,75,24)(6,46,76,33)(7,55,77,22)(8,44,78,31)(9,53,79,40)(10,42,80,29)(11,51,61,38)(12,60,62,27)(13,49,63,36)(14,58,64,25)(15,47,65,34)(16,56,66,23)(17,45,67,32)(18,54,68,21)(19,43,69,30)(20,52,70,39)>;
G:=Group( (2,62)(4,64)(6,66)(8,68)(10,70)(12,72)(14,74)(16,76)(18,78)(20,80)(22,45)(24,47)(26,49)(28,51)(30,53)(32,55)(34,57)(36,59)(38,41)(40,43), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,31)(22,32)(23,33)(24,34)(25,35)(26,36)(27,37)(28,38)(29,39)(30,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,61)(2,62)(3,63)(4,64)(5,65)(6,66)(7,67)(8,68)(9,69)(10,70)(11,71)(12,72)(13,73)(14,74)(15,75)(16,76)(17,77)(18,78)(19,79)(20,80)(21,44)(22,45)(23,46)(24,47)(25,48)(26,49)(27,50)(28,51)(29,52)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,41)(39,42)(40,43), (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), (1,41,71,28)(2,50,72,37)(3,59,73,26)(4,48,74,35)(5,57,75,24)(6,46,76,33)(7,55,77,22)(8,44,78,31)(9,53,79,40)(10,42,80,29)(11,51,61,38)(12,60,62,27)(13,49,63,36)(14,58,64,25)(15,47,65,34)(16,56,66,23)(17,45,67,32)(18,54,68,21)(19,43,69,30)(20,52,70,39) );
G=PermutationGroup([(2,62),(4,64),(6,66),(8,68),(10,70),(12,72),(14,74),(16,76),(18,78),(20,80),(22,45),(24,47),(26,49),(28,51),(30,53),(32,55),(34,57),(36,59),(38,41),(40,43)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20),(21,31),(22,32),(23,33),(24,34),(25,35),(26,36),(27,37),(28,38),(29,39),(30,40),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(49,59),(50,60),(61,71),(62,72),(63,73),(64,74),(65,75),(66,76),(67,77),(68,78),(69,79),(70,80)], [(1,61),(2,62),(3,63),(4,64),(5,65),(6,66),(7,67),(8,68),(9,69),(10,70),(11,71),(12,72),(13,73),(14,74),(15,75),(16,76),(17,77),(18,78),(19,79),(20,80),(21,44),(22,45),(23,46),(24,47),(25,48),(26,49),(27,50),(28,51),(29,52),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,41),(39,42),(40,43)], [(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)], [(1,41,71,28),(2,50,72,37),(3,59,73,26),(4,48,74,35),(5,57,75,24),(6,46,76,33),(7,55,77,22),(8,44,78,31),(9,53,79,40),(10,42,80,29),(11,51,61,38),(12,60,62,27),(13,49,63,36),(14,58,64,25),(15,47,65,34),(16,56,66,23),(17,45,67,32),(18,54,68,21),(19,43,69,30),(20,52,70,39)])
C23.11D10 is a maximal subgroup of
C22⋊C4.F5 C23⋊C4⋊5D5 Dic5.C42 C20⋊C8⋊C2 C23.(C2×F5) C24.24D10 C24.31D10 C42.87D10 D5×C42⋊C2 C42.96D10 C4×D4⋊2D5 C42.105D10 C42.108D10 C42⋊16D10 C24.56D10 C24.32D10 C24.34D10 C4⋊C4.178D10 C10.342+ 1+4 C10.442+ 1+4 C10.452+ 1+4 (Q8×Dic5)⋊C2 C10.502+ 1+4 C10.532+ 1+4 C10.772- 1+4 C10.792- 1+4 C4⋊C4.197D10 C10.802- 1+4 C4⋊C4⋊28D10 C10.642+ 1+4 C10.842- 1+4 C42.137D10 C42.138D10 C42.139D10 C42.234D10 C42.159D10 C42.160D10 C42.189D10 C42.162D10 D6.(C4×D5) (S3×Dic5)⋊C4 (C6×Dic5)⋊7C4 C23.48(S3×D5) C23.15D30
C23.11D10 is a maximal quotient of
Dic5.15C42 Dic5⋊2C42 C5⋊2(C42⋊8C4) C5⋊2(C42⋊5C4) C2.(C4×D20) C4⋊Dic5⋊15C4 Dic5.14M4(2) Dic5.9M4(2) C40⋊8C4⋊C2 C22⋊C4×Dic5 C24.44D10 C23.42D20 C24.3D10 C24.4D10 D6.(C4×D5) (S3×Dic5)⋊C4 (C6×Dic5)⋊7C4 C23.48(S3×D5) C23.15D30
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4N | 5A | 5B | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 20A | ··· | 20H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 5 | 5 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 5 | 5 | 5 | 5 | 10 | ··· | 10 | 2 | 2 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | D10 | C4×D5 | D4⋊2D5 |
| kernel | C23.11D10 | C4×Dic5 | C10.D4 | C23.D5 | C5×C22⋊C4 | C22×Dic5 | C2×Dic5 | C22⋊C4 | C10 | C2×C4 | C23 | C22 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 8 | 2 | 4 | 4 | 2 | 8 | 4 |
Matrix representation of C23.11D10 ►in GL5(𝔽41)
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 |
| 0 | 0 | 0 | 34 | 7 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 32 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 36 |
| 0 | 0 | 0 | 17 | 39 |
G:=sub<GL(5,GF(41))| [40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[40,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,1],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,0,34,0,0,0,6,7],[32,0,0,0,0,0,0,9,0,0,0,32,0,0,0,0,0,0,2,17,0,0,0,36,39] >;
C23.11D10 in GAP, Magma, Sage, TeX
C_2^3._{11}D_{10} % in TeX
G:=Group("C2^3.11D10"); // GroupNames label
G:=SmallGroup(160,98);
// by ID
G=gap.SmallGroup(160,98);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,96,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^10=b,e^2=c*b=b*c,a*b=b*a,d*a*d^-1=e*a*e^-1=a*c=c*a,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^9>;
// generators/relations