Copied to
clipboard

G = (C2×Q8)⋊F5order 320 = 26·5

2nd semidirect product of C2×Q8 and F5 acting via F5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Q8)⋊2F5, (Q8×C10)⋊2C4, (C4×Dic5)⋊6C4, C52(C423C4), C2.9(C23⋊F5), (C22×D5).13D4, C10.18(C23⋊C4), D10.D4.2C2, C20.23D4.3C2, (C2×D20).41C22, C22.21(C22⋊F5), (C2×C4).3(C2×F5), (C2×C20).16(C2×C4), (C2×C10).41(C22⋊C4), SmallGroup(320,266)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×Q8)⋊F5
C1C5C10C2×C10C22×D5C2×D20D10.D4 — (C2×Q8)⋊F5
C5C10C2×C10C2×C20 — (C2×Q8)⋊F5
C1C2C22C2×C4C2×Q8

Generators and relations for (C2×Q8)⋊F5
 G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ebe-1=ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, cd=dc, ece-1=b-1c, ede-1=d3 >

Subgroups: 458 in 70 conjugacy classes, 18 normal (14 characteristic)
C1, C2, C2 [×3], C4 [×5], C22, C22 [×4], C5, C2×C4, C2×C4 [×4], D4, Q8, C23 [×2], D5 [×2], C10, C10, C42, C22⋊C4 [×4], C2×D4, C2×Q8, Dic5, C20 [×2], F5 [×2], D10 [×4], C2×C10, C23⋊C4 [×2], C4.4D4, D20, C2×Dic5, C2×C20, C2×C20, C5×Q8, C2×F5 [×2], C22×D5 [×2], C423C4, C4×Dic5, D10⋊C4 [×2], C22⋊F5 [×2], C2×D20, Q8×C10, D10.D4 [×2], C20.23D4, (C2×Q8)⋊F5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], C22⋊C4, F5, C23⋊C4, C2×F5, C423C4, C22⋊F5, C23⋊F5, (C2×Q8)⋊F5

Character table of (C2×Q8)⋊F5

 class 12A2B2C2D4A4B4C4D4E4F4G4H510A10B10C20A20B20C20D20E20F
 size 1122020482020404040404444888888
ρ111111111111111111111111    trivial
ρ2111111111-1-1-1-11111111111    linear of order 2
ρ3111111-1-1-1-111-1111111-1-1-1-1    linear of order 2
ρ4111111-1-1-11-1-11111111-1-1-1-1    linear of order 2
ρ5111-1-11-111-i-iii111111-1-1-1-1    linear of order 4
ρ6111-1-11-111ii-i-i111111-1-1-1-1    linear of order 4
ρ7111-1-111-1-1i-ii-i1111111111    linear of order 4
ρ8111-1-111-1-1-ii-ii1111111111    linear of order 4
ρ92222-2-200000002222-2-20000    orthogonal lifted from D4
ρ10222-22-200000002222-2-20000    orthogonal lifted from D4
ρ1144-400000000004-4-44000000    orthogonal lifted from C23⋊C4
ρ12444004-4000000-1-1-1-1-1-11111    orthogonal lifted from C2×F5
ρ134440044000000-1-1-1-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ1444400-40000000-1-1-1-111-555-5    orthogonal lifted from C22⋊F5
ρ1544400-40000000-1-1-1-1115-5-55    orthogonal lifted from C22⋊F5
ρ164-4000002i-2i0000400-4000000    complex lifted from C423C4
ρ1744-40000000000-111-15-554+2ζ52+154+2ζ53+152+2ζ5+153+2ζ5+1    complex lifted from C23⋊F5
ρ184-400000-2i2i0000400-4000000    complex lifted from C423C4
ρ1944-40000000000-111-15-553+2ζ5+152+2ζ5+154+2ζ53+154+2ζ52+1    complex lifted from C23⋊F5
ρ2044-40000000000-111-1-5554+2ζ53+153+2ζ5+154+2ζ52+152+2ζ5+1    complex lifted from C23⋊F5
ρ2144-40000000000-111-1-5552+2ζ5+154+2ζ52+153+2ζ5+154+2ζ53+1    complex lifted from C23⋊F5
ρ228-800000000000-2-25252000000    orthogonal faithful
ρ238-800000000000-225-252000000    orthogonal faithful

Smallest permutation representation of (C2×Q8)⋊F5
On 80 points
Generators in S80
(1 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 56 6 51)(2 57 7 52)(3 58 8 53)(4 59 9 54)(5 60 10 55)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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)
(2 3 5 4)(7 8 10 9)(11 16)(12 18 15 19)(13 20 14 17)(21 36 26 31)(22 38 30 34)(23 40 29 32)(24 37 28 35)(25 39 27 33)(41 61 51 76)(42 63 55 79)(43 65 54 77)(44 62 53 80)(45 64 52 78)(46 66 56 71)(47 68 60 74)(48 70 59 72)(49 67 58 75)(50 69 57 73)

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

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

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

Matrix representation of (C2×Q8)⋊F5 in GL8(𝔽41)

040000000
400000000
000400000
004000000
000040000
000004000
000000400
000000040
,
357050000
735500000
25326340000
32253460000
0000223803
0000019383
0000338190
0000303822
,
28223920000
22282390000
101113190000
111019130000
000014323132
00002852222
000019193613
0000910927
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
10000000
040000000
2233010000
8194000000
00000010
00001000
00000001
00000100

G:=sub<GL(8,GF(41))| [0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[35,7,25,32,0,0,0,0,7,35,32,25,0,0,0,0,0,5,6,34,0,0,0,0,5,0,34,6,0,0,0,0,0,0,0,0,22,0,3,3,0,0,0,0,38,19,38,0,0,0,0,0,0,38,19,38,0,0,0,0,3,3,0,22],[28,22,10,11,0,0,0,0,22,28,11,10,0,0,0,0,39,2,13,19,0,0,0,0,2,39,19,13,0,0,0,0,0,0,0,0,14,28,19,9,0,0,0,0,32,5,19,10,0,0,0,0,31,22,36,9,0,0,0,0,32,22,13,27],[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],[1,0,22,8,0,0,0,0,0,40,33,19,0,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0] >;

(C2×Q8)⋊F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes F_5
% in TeX

G:=Group("(C2xQ8):F5");
// GroupNames label

G:=SmallGroup(320,266);
// by ID

G=gap.SmallGroup(320,266);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,1571,570,297,136,1684,6278,3156]);
// Polycyclic

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

Export

Character table of (C2×Q8)⋊F5 in TeX

׿
×
𝔽