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G = C23.11D10order 160 = 25·5

1st non-split extension by C23 of D10 acting via D10/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C23.11D10, (C2×Dic5)⋊4C4, (C4×Dic5)⋊9C2, (C2×C4).26D10, C22⋊C4.3D5, C22.6(C4×D5), C53(C42⋊C2), C10.D47C2, C23.D5.1C2, C10.20(C4○D4), C2.1(D42D5), (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

C1C10 — C23.11D10
C1C5C10C2×C10C2×Dic5C22×Dic5 — C23.11D10
C5C10 — C23.11D10
C1C22C22⋊C4

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, D42D5 [×2], C23.11D10

Smallest permutation representation of C23.11D10
On 80 points
Generators in S80
(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⋊C45D5  Dic5.C42  C20⋊C8⋊C2  C23.(C2×F5)  C24.24D10  C24.31D10  C42.87D10  D5×C42⋊C2  C42.96D10  C4×D42D5  C42.105D10  C42.108D10  C4216D10  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⋊C428D10  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  Dic52C42  C52(C428C4)  C52(C425C4)  C2.(C4×D20)  C4⋊Dic515C4  Dic5.14M4(2)  Dic5.9M4(2)  C408C4⋊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 2A2B2C2D2E4A4B4C4D4E4F4G4H4I···4N5A5B10A···10F10G10H10I10J20A···20H
order122222444444444···45510···101010101020···20
size1111222222555510···10222···244444···4

40 irreducible representations

dim1111111222224
type+++++++++-
imageC1C2C2C2C2C2C4D5C4○D4D10D10C4×D5D42D5
kernelC23.11D10C4×Dic5C10.D4C23.D5C5×C22⋊C4C22×Dic5C2×Dic5C22⋊C4C10C2×C4C23C22C2
# reps1221118244284

Matrix representation of C23.11D10 in GL5(𝔽41)

400000
01000
004000
00010
00001
,
400000
040000
004000
00010
00001
,
10000
040000
004000
00010
00001
,
320000
00100
040000
00006
000347
,
320000
003200
09000
000236
0001739

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

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