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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, (C22×C10).68(C2×C4), (C2×C10).68(C22×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
Upper central
C1 — C4 — C2×C4 — C22×C4

Generators and relations for (C4×D5).D4
 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 >

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

Smallest permutation representation of (C4×D5).D4
►On 80 points
Generators in S80
(1 7 5 3)(2 4 6 8)(9 79 13 75)(10 76 14 80)(11 73 15 77)(12 78 16 74)(17 43 21 47)(18 48 22 44)(19 45 23 41)(20 42 24 46)(25 72 29 68)(26 69 30 65)(27 66 31 70)(28 71 32 67)(33 55 37 51)(34 52 38 56)(35 49 39 53)(36 54 40 50)(57 63 61 59)(58 60 62 64)

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

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

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

Matrix representation of (C4×D5).D4 ►in 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] >;
 

(C4×D5).D4 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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