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G = (C2×D4)⋊6F5  order 320 = 26·5

4th semidirect product of C2×D4 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×D4)⋊6F5, (D4×C10)⋊8C4, D4.8(C2×F5), D4⋊2D5⋊7C4, D4⋊F5⋊5C2, D10.4(C2×D4), (C4×D5).35D4, (C4×F5)⋊1C22, C4.F5⋊2C22, (C2×Dic10)⋊9C4, D5⋊M4(2)⋊2C2, C4.16(C22×F5), C20.16(C22×C4), Dic10.7(C2×C4), (C22×D5).66D4, (C4×D5).38C23, C4.12(C22⋊F5), C20.12(C22⋊C4), Dic5.108(C2×D4), (C2×Dic5).260D4, C5⋊1(C42⋊C22), D10.12(C22⋊C4), D4⋊2D5.12C22, C22.27(C22⋊F5), D10.C23⋊2C2, Dic5.43(C22⋊C4), (C5×D4).8(C2×C4), (C2×C4).34(C2×F5), (C2×C20).52(C2×C4), (C4×D5).22(C2×C4), C2.17(C2×C22⋊F5), C10.16(C2×C22⋊C4), (C2×C4×D5).199C22, (C2×D4⋊2D5).14C2, (C2×C10).56(C22⋊C4), SmallGroup(320,1107)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — (C2×D4)⋊6F5
Chief series
C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D5⋊M4(2) — (C2×D4)⋊6F5
Lower central
C5 — C10 — C20 — (C2×D4)⋊6F5
Upper central
C1 — C2 — C2×C4 — C2×D4

Generators and relations for (C2×D4)⋊6F5
 G = < a,b,c,d,e | a2=b4=c2=d5=e4=1, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc=ebe-1=b-1, bd=db, cd=dc, ece-1=abc, ede-1=d3 >

Subgroups: 634 in 154 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C2×C8, M4(2), C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, Dic5, Dic5, C20, F5, D10, D10, C2×C10, C2×C10, C4≀C2, C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊C8, Dic10, Dic10, C4×D5, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×D4, C2×F5, C22×D5, C22×C10, C42⋊C22, D5⋊C8, C4.F5, C4×F5, C4⋊F5, C22.F5, C22⋊F5, C2×Dic10, C2×C4×D5, D4⋊2D5, D4⋊2D5, C22×Dic5, C2×C5⋊D4, D4×C10, D4⋊F5, D5⋊M4(2), D10.C23, C2×D4⋊2D5, (C2×D4)⋊6F5
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22⋊C4, C22×C4, C2×D4, F5, C2×C22⋊C4, C2×F5, C42⋊C22, C22⋊F5, C22×F5, C2×C22⋊F5, (C2×D4)⋊6F5

Smallest permutation representation of (C2×D4)⋊6F5
►On 80 points
Generators in S80
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 46)(12 47)(13 48)(14 49)(15 50)(16 41)(17 42)(18 43)(19 44)(20 45)(21 76)(22 77)(23 78)(24 79)(25 80)(26 71)(27 72)(28 73)(29 74)(30 75)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F···4K 5 8A8B8C8D10A10B10C10D10E10F10G20A20B
order1222222444444···458888101010101010102020
size11244101022551020···20420202020444888888

32 irreducible representations

dim111111112224444448
type+++++++++++++-
imageC1C2C2C2C2C4C4C4D4D4D4F5C2×F5C2×F5C42⋊C22C22⋊F5C22⋊F5(C2×D4)⋊6F5
kernel(C2×D4)⋊6F5D4⋊F5D5⋊M4(2)D10.C23C2×D4⋊2D5C2×Dic10D4⋊2D5D4×C10C4×D5C2×Dic5C22×D5C2×D4C2×C4D4C5C4C22C1
# reps141112422111122222

Matrix representation of (C2×D4)⋊6F5 ►in GL8(𝔽41)

130400000
7032320000
4002800000
29700000
000040000
000004000
000000400
000000040
,
60500000
203910000
903500000
6401900000
000040000
000004000
000000400
000000040
,
278000000
3214000000
14159230000
0149320000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
3503600000
3603890000
70600000
2992200000
000000400
000040000
000000040
000004000

G:=sub<GL(8,GF(41))| [13,7,40,2,0,0,0,0,0,0,0,9,0,0,0,0,4,32,28,7,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[6,2,9,6,0,0,0,0,0,0,0,40,0,0,0,0,5,39,35,19,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[27,32,14,0,0,0,0,0,8,14,15,14,0,0,0,0,0,0,9,9,0,0,0,0,0,0,23,32,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[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],[35,36,7,29,0,0,0,0,0,0,0,9,0,0,0,0,36,38,6,22,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0] >;
 

(C2×D4)⋊6F5 in GAP, Magma, Sage, TeX 

(C_2\times D_4)\rtimes_6F_5
 
% in TeX
 
G:=Group("(C2xD4):6F5");
 
// GroupNames label
 
G:=SmallGroup(320,1107);
 
// by ID
 
G=gap.SmallGroup(320,1107);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,136,1684,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^4=1,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b^2,c*b*c=e*b*e^-1=b^-1,b*d=d*b,c*d=d*c,e*c*e^-1=a*b*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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