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G = C2×C23⋊F5order 320 = 26·5

Direct product of C2 and C23⋊F5

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C23⋊F5, C243F5, C234(C2×F5), (C23×C10)⋊6C4, C102(C23⋊C4), D10.12(C2×D4), C22⋊F53C22, (C22×Dic5)⋊9C4, (C22×D5).75D4, D10.19(C22⋊C4), C22.17(C22×F5), (C23×D5).90C22, C22.52(C22⋊F5), (C22×D5).150C23, C53(C2×C23⋊C4), (C2×C5⋊D4)⋊16C4, (C2×C22⋊F5)⋊5C2, (C22×C10)⋊8(C2×C4), (C2×Dic5)⋊4(C2×C4), C2.38(C2×C22⋊F5), C10.37(C2×C22⋊C4), (C2×C10).90(C22×C4), (C22×D5).59(C2×C4), (C22×C5⋊D4).14C2, (C2×C10).61(C22⋊C4), (C2×C5⋊D4).155C22, SmallGroup(320,1134)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C2×C23⋊F5
C1C5C10D10C22×D5C22⋊F5C2×C22⋊F5 — C2×C23⋊F5
C5C10C2×C10 — C2×C23⋊F5
C1C22C23C24

Generators and relations for C2×C23⋊F5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e3 >

Subgroups: 1034 in 210 conjugacy classes, 52 normal (20 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×6], C22 [×3], C22 [×22], C5, C2×C4 [×12], D4 [×8], C23, C23 [×2], C23 [×12], D5 [×4], C10, C10 [×2], C10 [×4], C22⋊C4 [×6], C22×C4 [×3], C2×D4 [×8], C24, C24, Dic5 [×2], F5 [×4], D10 [×4], D10 [×8], C2×C10 [×3], C2×C10 [×10], C23⋊C4 [×4], C2×C22⋊C4 [×2], C22×D4, C2×Dic5 [×2], C2×Dic5 [×2], C5⋊D4 [×8], C2×F5 [×8], C22×D5 [×2], C22×D5 [×4], C22×D5 [×2], C22×C10, C22×C10 [×2], C22×C10 [×4], C2×C23⋊C4, C22⋊F5 [×4], C22⋊F5 [×2], C22×Dic5, C2×C5⋊D4 [×4], C2×C5⋊D4 [×4], C22×F5 [×2], C23×D5, C23×C10, C23⋊F5 [×4], C2×C22⋊F5 [×2], C22×C5⋊D4, C2×C23⋊F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C23⋊C4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C23⋊C4, C22⋊F5 [×2], C22×F5, C23⋊F5 [×2], C2×C22⋊F5, C2×C23⋊F5

Smallest permutation representation of C2×C23⋊F5
On 80 points
Generators in S80
(1 11)(2 12)(3 13)(4 14)(5 15)(6 18)(7 19)(8 20)(9 16)(10 17)(21 33)(22 34)(23 35)(24 31)(25 32)(26 38)(27 39)(28 40)(29 36)(30 37)(41 52)(42 53)(43 54)(44 55)(45 51)(46 57)(47 58)(48 59)(49 60)(50 56)(61 72)(62 73)(63 74)(64 75)(65 71)(66 77)(67 78)(68 79)(69 80)(70 76)
(1 24)(2 25)(3 21)(4 22)(5 23)(6 26)(7 27)(8 28)(9 29)(10 30)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 80)
(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 80)
(1 9)(2 10)(3 6)(4 7)(5 8)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 46)(42 47)(43 48)(44 49)(45 50)(51 56)(52 57)(53 58)(54 59)(55 60)(61 66)(62 67)(63 68)(64 69)(65 70)(71 76)(72 77)(73 78)(74 79)(75 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 53 11 42)(2 55 15 45)(3 52 14 43)(4 54 13 41)(5 51 12 44)(6 57 19 48)(7 59 18 46)(8 56 17 49)(9 58 16 47)(10 60 20 50)(21 77 39 63)(22 79 38 61)(23 76 37 64)(24 78 36 62)(25 80 40 65)(26 72 34 68)(27 74 33 66)(28 71 32 69)(29 73 31 67)(30 75 35 70)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A···4J 5 10A···10O
order1222222222224···4510···10
size111122441010101020···2044···4

38 irreducible representations

dim1111111244444
type+++++++++
imageC1C2C2C2C4C4C4D4F5C23⋊C4C2×F5C22⋊F5C23⋊F5
kernelC2×C23⋊F5C23⋊F5C2×C22⋊F5C22×C5⋊D4C22×Dic5C2×C5⋊D4C23×C10C22×D5C24C10C23C22C2
# reps1421242412348

Matrix representation of C2×C23⋊F5 in GL6(𝔽41)

4000000
0400000
0040000
0004000
0000400
0000040
,
010000
100000
00243500
0071700
003501835
00406623
,
4000000
0400000
001000
000100
003821400
003821040
,
100000
010000
0040000
0004000
0000400
0000040
,
100000
010000
0040100
0033700
00740406
002343535
,
0400000
100000
0000401
00640396
0032010
00101910

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,24,7,35,40,0,0,35,17,0,6,0,0,0,0,18,6,0,0,0,0,35,23],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,38,38,0,0,0,1,21,21,0,0,0,0,40,0,0,0,0,0,0,40],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,33,7,2,0,0,1,7,40,34,0,0,0,0,40,35,0,0,0,0,6,35],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,6,3,10,0,0,0,40,20,19,0,0,40,39,1,1,0,0,1,6,0,0] >;

C2×C23⋊F5 in GAP, Magma, Sage, TeX

C_2\times C_2^3\rtimes F_5
% in TeX

G:=Group("C2xC2^3:F5");
// GroupNames label

G:=SmallGroup(320,1134);
// by ID

G=gap.SmallGroup(320,1134);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,422,297,1684,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=b*c*d,f*c*f^-1=c*d=d*c,c*e=e*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

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