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G = (C4×D5).D4  order 320 = 26·5

54th non-split extension by C4×D5 of D4 acting via D4/C2=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C4×D5).54D4, C23.F5⋊5C2, (C22×C4).7F5, (C2×D20).22C4, Dic5.2(C2×D4), C23.21(C2×F5), D5⋊M4(2)⋊11C2, (C22×C20).15C4, C4.32(C22⋊F5), C20.45(C22⋊C4), Dic5.D4⋊5C2, D10.10(C22⋊C4), C22.10(C22×F5), C22.F5.2C22, (C2×Dic5).172C23, (C2×Dic10).229C22, C5⋊1(M4(2).8C22), (C2×C4×D5).8C4, (C2×C4).55(C2×F5), (C2×C5⋊D4).25C4, C10.8(C2×C22⋊C4), (C2×C20).151(C2×C4), C2.11(C2×C22⋊F5), (C2×C4○D20).27C2, (C2×Dic5).7(C2×C4), (C2×C4×D5).285C22, (C2×C10).68(C22×C4), (C22×C10).68(C2×C4), (C22×D5).54(C2×C4), (C2×C5⋊D4).154C22, SmallGroup(320,1099)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — (C4×D5).D4
Chief series
C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — D5⋊M4(2) — (C4×D5).D4
Lower central
C5 — C10 — C2×C10 — (C4×D5).D4

Subgroups: 618 in 150 conjugacy classes, 48 normal (30 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×4], C22, C22 [×7], C5, C8 [×4], C2×C4 [×2], C2×C4 [×10], D4 [×6], Q8 [×2], C23, C23 [×2], D5 [×3], C10, C10 [×2], C2×C8 [×2], M4(2) [×6], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], Dic5, C20 [×2], C20, D10 [×2], D10 [×3], C2×C10, C2×C10 [×2], C4.D4 [×2], C4.10D4 [×2], C2×M4(2) [×2], C2×C4○D4, C5⋊C8 [×4], Dic10 [×2], C4×D5 [×4], C4×D5 [×2], D20 [×2], C2×Dic5 [×2], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, M4(2).8C22, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C4○D20 [×4], C2×C5⋊D4 [×2], C22×C20, Dic5.D4 [×2], C23.F5 [×2], D5⋊M4(2) [×2], C2×C4○D20, (C4×D5).D4

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], M4(2).8C22, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C4×D5).D4

Generators and relations
 G = < a,b,c,d,e | a4=b5=c2=1, d4=a2, e2=ab-1c, ab=ba, ac=ca, dad-1=eae-1=a-1, cbc=b-1, dbd-1=ebe-1=b3, dcd-1=b2c, ece-1=a2b2c, ede-1=a-1b-1cd3 >

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

90000000
09000000
003200000
000320000
00001000
00000100
00000010
00000001
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
400000000
01000000
00100000
000400000
000000140
000001040
000010040
000000040
,
00010000
00100000
09000000
320000000
0000338360
0000288033
0000330828
0000036833
,
00100000
00010000
320000000
09000000
000083350
0000133308
0000803313
000005338

G:=sub<GL(8,GF(41))| [9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[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],[40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,40,40,40],[0,0,0,32,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,33,28,33,0,0,0,0,0,8,8,0,36,0,0,0,0,36,0,8,8,0,0,0,0,0,33,28,33],[0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,8,13,8,0,0,0,0,0,33,33,0,5,0,0,0,0,5,0,33,33,0,0,0,0,0,8,13,8] >;
 

38 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G 5 8A···8H10A···10G20A···20H
order1222222444444458···810···1020···20
size11241010201124101020420···204···44···4

38 irreducible representations

dim1111111112444444
type++++++++++
imageC1C2C2C2C2C4C4C4C4D4F5C2×F5C2×F5M4(2).8C22C22⋊F5(C4×D5).D4
kernel(C4×D5).D4Dic5.D4C23.F5D5⋊M4(2)C2×C4○D20C2×C4×D5C2×D20C2×C5⋊D4C22×C20C4×D5C22×C4C2×C4C23C5C4C1
# reps1222122224121248

In GAP, Magma, Sage, TeX 

(C_4\times D_5).D_4
 
% in TeX
 
G:=Group("(C4xD5).D4");
 
// GroupNames label
 
G:=SmallGroup(320,1099);
 
// by ID
 
G=gap.SmallGroup(320,1099);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,297,136,1684,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^4=b^5=c^2=1,d^4=a^2,e^2=a*b^-1*c,a*b=b*a,a*c=c*a,d*a*d^-1=e*a*e^-1=a^-1,c*b*c=b^-1,d*b*d^-1=e*b*e^-1=b^3,d*c*d^-1=b^2*c,e*c*e^-1=a^2*b^2*c,e*d*e^-1=a^-1*b^-1*c*d^3>;
 
// generators/relations
 

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