Copied to
clipboard

G = (C22×C4)⋊7F5  order 320 = 26·5

3rd semidirect product of C22×C4 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C22×C4)⋊7F5, (C22×C20)⋊11C4, D10.92(C2×D4), C5⋊(C23.34D4), C23.49(C2×F5), D10.3Q8⋊4C2, D10.26(C4○D4), (C22×Dic5)⋊18C4, (C22×D5).144D4, D10.40(C22⋊C4), (C22×F5).6C22, C22.89(C22×F5), C10.18(C42⋊C2), C22.25(C22⋊F5), D5.4(C22.D4), (C22×D5).276C23, (C23×D5).134C22, C2.18(D10.C23), (C2×C4×D5)⋊18C4, (C2×C4).112(C2×F5), (D5×C22×C4).24C2, (C2×C20).112(C2×C4), (C2×C22⋊F5).6C2, C2.12(C2×C22⋊F5), C10.11(C2×C22⋊C4), (C2×C4×D5).367C22, (C22×C10).71(C2×C4), (C2×C10).71(C22×C4), (C2×C10).52(C22⋊C4), (C2×Dic5).190(C2×C4), (C22×D5).127(C2×C4), SmallGroup(320,1102)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — (C22×C4)⋊7F5
Chief series
C1 — C5 — D5 — D10 — C22×D5 — C22×F5 — C2×C22⋊F5 — (C22×C4)⋊7F5
Lower central
C5 — C2×C10 — (C22×C4)⋊7F5
Upper central
C1 — C22 — C22×C4

Generators and relations for (C22×C4)⋊7F5
 G = < a,b,c,d,e | a2=b2=c4=d5=e4=1, ab=ba, ece-1=ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=bc2, cd=dc, ede-1=d3 >

Subgroups: 954 in 218 conjugacy classes, 60 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2×C4, C2×C4, C23, C23, D5, D5, C10, C10, C10, C22⋊C4, C22×C4, C22×C4, C24, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C2×C10, C2.C42, C2×C22⋊C4, C23×C4, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C2×F5, C22×D5, C22×D5, C22×D5, C22×C10, C23.34D4, C22⋊F5, C2×C4×D5, C2×C4×D5, C22×Dic5, C22×C20, C22×F5, C23×D5, D10.3Q8, C2×C22⋊F5, D5×C22×C4, (C22×C4)⋊7F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, C4○D4, F5, C2×C22⋊C4, C42⋊C2, C22.D4, C2×F5, C23.34D4, C22⋊F5, C22×F5, D10.C23, C2×C22⋊F5, (C22×C4)⋊7F5

Smallest permutation representation of (C22×C4)⋊7F5
►On 80 points
Generators in S80
(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 41)(2 42)(3 43)(4 44)(5 45)(6 46)(7 47)(8 48)(9 49)(10 50)(11 51)(12 52)(13 53)(14 54)(15 55)(16 56)(17 57)(18 58)(19 59)(20 60)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I···4P 5 10A···10G20A···20H
order122222222222444444444···4510···1020···20
size1111225555101022221010101020···2044···44···4

44 irreducible representations

dim11111112244444
type+++++++++
imageC1C2C2C2C4C4C4D4C4○D4F5C2×F5C2×F5C22⋊F5D10.C23
kernel(C22×C4)⋊7F5D10.3Q8C2×C22⋊F5D5×C22×C4C2×C4×D5C22×Dic5C22×C20C22×D5D10C22×C4C2×C4C23C22C2
# reps14214224812148

Matrix representation of (C22×C4)⋊7F5 ►in GL8(𝔽41)

400000000
040000000
004000000
000400000
00001000
00000100
00000010
00000001
,
118000000
040000000
004000000
000400000
000040000
000004000
000000400
000000040
,
322000000
09000000
004020000
00010000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
4023000000
321000000
0032180000
003290000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(41))| [40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,18,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,0,2,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[40,32,0,0,0,0,0,0,23,1,0,0,0,0,0,0,0,0,32,32,0,0,0,0,0,0,18,9,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;
 

(C22×C4)⋊7F5 in GAP, Magma, Sage, TeX 

(C_2^2\times C_4)\rtimes_7F_5
 
% in TeX
 
G:=Group("(C2^2xC4):7F5");
 
// GroupNames label
 
G:=SmallGroup(320,1102);
 
// by ID
 
G=gap.SmallGroup(320,1102);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,422,184,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^2=c^4=d^5=e^4=1,a*b=b*a,e*c*e^-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^2,c*d=d*c,e*d*e^-1=d^3>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁