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G = (C2×Q8)⋊6F5  order 320 = 26·5

4th semidirect product of C2×Q8 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Q8)⋊6F5, (Q8×C10)⋊6C4, Q8.8(C2×F5), (C2×D20)⋊12C4, Q8⋊2F5⋊5C2, Q8⋊2D5⋊7C4, D20.8(C2×C4), D10.8(C2×D4), (C4×D5).43D4, (C4×F5)⋊2C22, C4.F5⋊4C22, D5⋊M4(2)⋊4C2, C4.20(C22×F5), C20.20(C22×C4), (C22×D5).70D4, (C4×D5).42C23, C4.19(C22⋊F5), C20.19(C22⋊C4), Dic5.112(C2×D4), (C2×Dic5).262D4, C5⋊2(C42⋊C22), D10.15(C22⋊C4), Q8⋊2D5.12C22, C22.29(C22⋊F5), D10.C23⋊4C2, Dic5.47(C22⋊C4), (C2×C4).40(C2×F5), (C5×Q8).8(C2×C4), (C2×C20).62(C2×C4), (C4×D5).26(C2×C4), C2.29(C2×C22⋊F5), C10.28(C2×C22⋊C4), (C2×C4×D5).207C22, (C2×Q8⋊2D5).10C2, (C2×C10).60(C22⋊C4), SmallGroup(320,1122)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — (C2×Q8)⋊6F5
Chief series
C1 — C5 — C10 — Dic5 — C4×D5 — C4.F5 — D5⋊M4(2) — (C2×Q8)⋊6F5
Lower central
C5 — C10 — C20 — (C2×Q8)⋊6F5

Subgroups: 698 in 154 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×6], C22, C22 [×7], C5, C8 [×2], C2×C4, C2×C4 [×12], D4 [×7], Q8 [×2], Q8, C23 [×2], D5 [×4], C10, C10, C42 [×2], C22⋊C4, C4⋊C4, C2×C8, M4(2) [×3], C22×C4 [×2], C2×D4 [×2], C2×Q8, C4○D4 [×6], Dic5 [×2], C20 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×5], C2×C10, C4≀C2 [×4], C42⋊C2, C2×M4(2), C2×C4○D4, C5⋊C8 [×2], C4×D5 [×4], C4×D5 [×4], D20 [×2], D20 [×5], C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×F5 [×2], C22×D5, C22×D5, C42⋊C22, D5⋊C8, C4.F5 [×2], C4×F5 [×2], C4⋊F5, C22.F5, C22⋊F5, C2×C4×D5, C2×C4×D5, C2×D20, C2×D20, Q8⋊2D5 [×4], Q8⋊2D5 [×2], Q8×C10, Q8⋊2F5 [×4], D5⋊M4(2), D10.C23, C2×Q8⋊2D5, (C2×Q8)⋊6F5

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C2×F5 [×3], C42⋊C22, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×Q8)⋊6F5

Generators and relations
 G = < a,b,c,d,e | a2=b4=d5=e4=1, c2=b2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, ebe-1=b-1c, cd=dc, ce=ec, ede-1=d3 >

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

10000000
01000000
00100000
00010000
00001000
00000100
000000400
000000040
,
19038380000
322300000
032230000
38380190000
000004000
00001000
00000009
00000090
,
400000000
040000000
004000000
000400000
00009000
000003200
000000320
00000009
,
404040400000
10000000
01000000
00100000
00001000
00000100
00000010
00000001
,
32018180000
18180320000
23142300000
9272790000
00000001
000000400
00000100
00001000

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[19,3,0,38,0,0,0,0,0,22,3,38,0,0,0,0,38,3,22,0,0,0,0,0,38,0,3,19,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0],[40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9],[40,1,0,0,0,0,0,0,40,0,1,0,0,0,0,0,40,0,0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[32,18,23,9,0,0,0,0,0,18,14,27,0,0,0,0,18,0,23,27,0,0,0,0,18,32,0,9,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K 5 8A8B8C8D10A10B10C20A···20F
order1222222444444444445888810101020···20
size1121010202022445510202020204202020204448···8

32 irreducible representations

dim111111112224444448
type++++++++++++++
imageC1C2C2C2C2C4C4C4D4D4D4F5C2×F5C2×F5C42⋊C22C22⋊F5C22⋊F5(C2×Q8)⋊6F5
kernel(C2×Q8)⋊6F5Q8⋊2F5D5⋊M4(2)D10.C23C2×Q8⋊2D5C2×D20Q8⋊2D5Q8×C10C4×D5C2×Dic5C22×D5C2×Q8C2×C4Q8C5C4C22C1
# reps141112422111122222

In GAP, Magma, Sage, TeX 

(C_2\times Q_8)\rtimes_6F_5
 
% in TeX
 
G:=Group("(C2xQ8):6F5");
 
// GroupNames label
 
G:=SmallGroup(320,1122);
 
// by ID
 
G=gap.SmallGroup(320,1122);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,136,1684,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^4=1,c^2=b^2,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,e*b*e^-1=b^-1*c,c*d=d*c,c*e=e*c,e*d*e^-1=d^3>;
 
// generators/relations
 

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