Copied to
clipboard

G = C20.23D4order 160 = 25·5

23rd non-split extension by C20 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.23D4, (C2×Q8)⋊4D5, (Q8×C10)⋊4C2, (C4×Dic5)⋊7C2, C10.58(C2×D4), (C2×C4).57D10, C54(C4.4D4), (C2×D20).10C2, C4.11(C5⋊D4), D10⋊C416C2, C10.37(C4○D4), (C2×C10).59C23, (C2×C20).40C22, C2.9(Q82D5), C22.65(C22×D5), (C2×Dic5).44C22, (C22×D5).13C22, C2.22(C2×C5⋊D4), SmallGroup(160,168)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.23D4
C1C5C10C2×C10C22×D5C2×D20 — C20.23D4
C5C2×C10 — C20.23D4
C1C22C2×Q8

Generators and relations for C20.23D4
 G = < a,b,c | a20=b4=c2=1, bab-1=a9, cac=a-1, cbc=a10b-1 >

Subgroups: 272 in 76 conjugacy classes, 33 normal (13 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×4], C22, C22 [×6], C5, C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], D5 [×2], C10, C10 [×2], C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], C20 [×2], D10 [×6], C2×C10, C4.4D4, D20 [×2], C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5 [×2], C4×Dic5, D10⋊C4 [×4], C2×D20, Q8×C10, C20.23D4
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, C2×D4, C4○D4 [×2], D10 [×3], C4.4D4, C5⋊D4 [×2], C22×D5, Q82D5 [×2], C2×C5⋊D4, C20.23D4

Smallest permutation representation of C20.23D4
On 80 points
Generators in S80
(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 59 62 33)(2 48 63 22)(3 57 64 31)(4 46 65 40)(5 55 66 29)(6 44 67 38)(7 53 68 27)(8 42 69 36)(9 51 70 25)(10 60 71 34)(11 49 72 23)(12 58 73 32)(13 47 74 21)(14 56 75 30)(15 45 76 39)(16 54 77 28)(17 43 78 37)(18 52 79 26)(19 41 80 35)(20 50 61 24)
(2 20)(3 19)(4 18)(5 17)(6 16)(7 15)(8 14)(9 13)(10 12)(21 41)(22 60)(23 59)(24 58)(25 57)(26 56)(27 55)(28 54)(29 53)(30 52)(31 51)(32 50)(33 49)(34 48)(35 47)(36 46)(37 45)(38 44)(39 43)(40 42)(61 63)(64 80)(65 79)(66 78)(67 77)(68 76)(69 75)(70 74)(71 73)

G:=sub<Sym(80)| (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), (1,59,62,33)(2,48,63,22)(3,57,64,31)(4,46,65,40)(5,55,66,29)(6,44,67,38)(7,53,68,27)(8,42,69,36)(9,51,70,25)(10,60,71,34)(11,49,72,23)(12,58,73,32)(13,47,74,21)(14,56,75,30)(15,45,76,39)(16,54,77,28)(17,43,78,37)(18,52,79,26)(19,41,80,35)(20,50,61,24), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73)>;

G:=Group( (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), (1,59,62,33)(2,48,63,22)(3,57,64,31)(4,46,65,40)(5,55,66,29)(6,44,67,38)(7,53,68,27)(8,42,69,36)(9,51,70,25)(10,60,71,34)(11,49,72,23)(12,58,73,32)(13,47,74,21)(14,56,75,30)(15,45,76,39)(16,54,77,28)(17,43,78,37)(18,52,79,26)(19,41,80,35)(20,50,61,24), (2,20)(3,19)(4,18)(5,17)(6,16)(7,15)(8,14)(9,13)(10,12)(21,41)(22,60)(23,59)(24,58)(25,57)(26,56)(27,55)(28,54)(29,53)(30,52)(31,51)(32,50)(33,49)(34,48)(35,47)(36,46)(37,45)(38,44)(39,43)(40,42)(61,63)(64,80)(65,79)(66,78)(67,77)(68,76)(69,75)(70,74)(71,73) );

G=PermutationGroup([(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)], [(1,59,62,33),(2,48,63,22),(3,57,64,31),(4,46,65,40),(5,55,66,29),(6,44,67,38),(7,53,68,27),(8,42,69,36),(9,51,70,25),(10,60,71,34),(11,49,72,23),(12,58,73,32),(13,47,74,21),(14,56,75,30),(15,45,76,39),(16,54,77,28),(17,43,78,37),(18,52,79,26),(19,41,80,35),(20,50,61,24)], [(2,20),(3,19),(4,18),(5,17),(6,16),(7,15),(8,14),(9,13),(10,12),(21,41),(22,60),(23,59),(24,58),(25,57),(26,56),(27,55),(28,54),(29,53),(30,52),(31,51),(32,50),(33,49),(34,48),(35,47),(36,46),(37,45),(38,44),(39,43),(40,42),(61,63),(64,80),(65,79),(66,78),(67,77),(68,76),(69,75),(70,74),(71,73)])

C20.23D4 is a maximal subgroup of
(C2×C4).D20  (C2×Q8).F5  (C2×Q8)⋊F5  D20.5D4  Q8⋊C4⋊D5  Dic10.11D4  Q8⋊Dic5⋊C2  D20.12D4  (C5×D4).D4  C40.43D4  C409D4  (C2×Q16)⋊D5  C40.37D4  C40.28D4  D20.39D4  2- 1+42D5  C42.122D10  C42.131D10  C42.133D10  C42.136D10  C22⋊Q825D5  D2021D4  Dic1022D4  C10.532+ 1+4  C10.222- 1+4  C10.242- 1+4  C10.562+ 1+4  C10.262- 1+4  C42.138D10  D5×C4.4D4  C4218D10  C4220D10  C42.143D10  C4222D10  C42.171D10  C42.240D10  C42.177D10  C42.178D10  C42.179D10  C42.180D10  C10.442- 1+4  C10.452- 1+4  C10.1452+ 1+4  C10.1462+ 1+4  (C2×C20)⋊17D4  C60.88D4  C60.70D4  (C2×Dic6)⋊D5  C60.23D4
C20.23D4 is a maximal quotient of
C20.48(C4⋊C4)  (C2×C20).55D4  (C2×D20)⋊22C4  C10.90(C4×D4)  (C2×C4)⋊3D20  (C2×C20).290D4  C42.70D10  C42.216D10  C42.71D10  C20.D8  C42.82D10  C20.11Q16  (Q8×C10)⋊17C4  (C22×D5)⋊Q8  C60.88D4  C60.70D4  (C2×Dic6)⋊D5  C60.23D4

34 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B10A···10F20A···20L
order122222444444445510···1020···20
size11112020224410101010222···24···4

34 irreducible representations

dim11111222224
type+++++++++
imageC1C2C2C2C2D4D5C4○D4D10C5⋊D4Q82D5
kernelC20.23D4C4×Dic5D10⋊C4C2×D20Q8×C10C20C2×Q8C10C2×C4C4C2
# reps11411224684

Matrix representation of C20.23D4 in GL4(𝔽41) generated by

0100
40000
003540
003640
,
9000
0900
002117
001520
,
1000
04000
0061
00635
G:=sub<GL(4,GF(41))| [0,40,0,0,1,0,0,0,0,0,35,36,0,0,40,40],[9,0,0,0,0,9,0,0,0,0,21,15,0,0,17,20],[1,0,0,0,0,40,0,0,0,0,6,6,0,0,1,35] >;

C20.23D4 in GAP, Magma, Sage, TeX

C_{20}._{23}D_4
% in TeX

G:=Group("C20.23D4");
// GroupNames label

G:=SmallGroup(160,168);
// by ID

G=gap.SmallGroup(160,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,103,218,188,86,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=b^4=c^2=1,b*a*b^-1=a^9,c*a*c=a^-1,c*b*c=a^10*b^-1>;
// generators/relations

׿
×
𝔽