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G = C4×C5⋊D4order 160 = 25·5

Direct product of C4 and C5⋊D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4×C5⋊D4, C208D4, C23.22D10, C55(C4×D4), D105(C2×C4), C222(C4×D5), (C22×C4)⋊2D5, (C22×C20)⋊9C2, Dic53(C2×C4), C10.41(C2×D4), (C4×Dic5)⋊16C2, C2.5(C4○D20), (C2×C4).103D10, C23.D514C2, D10⋊C418C2, C10.17(C4○D4), C10.D418C2, (C2×C20).77C22, C10.32(C22×C4), (C2×C10).46C23, C22.24(C22×D5), (C22×C10).38C22, (C2×Dic5).39C22, (C22×D5).27C22, (C2×C4×D5)⋊14C2, C2.20(C2×C4×D5), (C2×C10)⋊8(C2×C4), C2.3(C2×C5⋊D4), (C2×C5⋊D4).9C2, SmallGroup(160,149)

Series: Derived Chief Lower central Upper central

C1C10 — C4×C5⋊D4
C1C5C10C2×C10C22×D5C2×C5⋊D4 — C4×C5⋊D4
C5C10 — C4×C5⋊D4
C1C2×C4C22×C4

Generators and relations for C4×C5⋊D4
 G = < a,b,c,d | a4=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 264 in 94 conjugacy classes, 45 normal (29 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×6], C5, C2×C4 [×2], C2×C4 [×7], D4 [×4], C23, C23, D5 [×2], C10 [×3], C10 [×2], C42, C22⋊C4 [×2], C4⋊C4, C22×C4, C22×C4, C2×D4, Dic5 [×2], Dic5 [×2], C20 [×2], C20, D10 [×2], D10 [×2], C2×C10, C2×C10 [×2], C2×C10 [×2], C4×D4, C4×D5 [×2], C2×Dic5 [×3], C5⋊D4 [×4], C2×C20 [×2], C2×C20 [×2], C22×D5, C22×C10, C4×Dic5, C10.D4, D10⋊C4, C23.D5, C2×C4×D5, C2×C5⋊D4, C22×C20, C4×C5⋊D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×2], C23, D5, C22×C4, C2×D4, C4○D4, D10 [×3], C4×D4, C4×D5 [×2], C5⋊D4 [×2], C22×D5, C2×C4×D5, C4○D20, C2×C5⋊D4, C4×C5⋊D4

Smallest permutation representation of C4×C5⋊D4
On 80 points
Generators in S80
(1 31 11 21)(2 32 12 22)(3 33 13 23)(4 34 14 24)(5 35 15 25)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 71 51 61)(42 72 52 62)(43 73 53 63)(44 74 54 64)(45 75 55 65)(46 76 56 66)(47 77 57 67)(48 78 58 68)(49 79 59 69)(50 80 60 70)
(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 41 6 46)(2 45 7 50)(3 44 8 49)(4 43 9 48)(5 42 10 47)(11 51 16 56)(12 55 17 60)(13 54 18 59)(14 53 19 58)(15 52 20 57)(21 61 26 66)(22 65 27 70)(23 64 28 69)(24 63 29 68)(25 62 30 67)(31 71 36 76)(32 75 37 80)(33 74 38 79)(34 73 39 78)(35 72 40 77)
(1 11)(2 15)(3 14)(4 13)(5 12)(6 16)(7 20)(8 19)(9 18)(10 17)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)(41 56)(42 60)(43 59)(44 58)(45 57)(46 51)(47 55)(48 54)(49 53)(50 52)(61 76)(62 80)(63 79)(64 78)(65 77)(66 71)(67 75)(68 74)(69 73)(70 72)

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

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

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

C4×C5⋊D4 is a maximal subgroup of
C55(C8×D4)  D104M4(2)  Dic52M4(2)  C52C826D4  C4032D4  C40⋊D4  C4018D4  C42.277D10  C24.24D10  C24.27D10  C24.30D10  C24.31D10  C10.82+ 1+4  C10.2- 1+4  C10.102+ 1+4  C10.52- 1+4  C10.112+ 1+4  C10.62- 1+4  C42.93D10  C42.94D10  C42.95D10  C42.97D10  C42.98D10  C42.102D10  C42.104D10  C4×D4×D5  C4211D10  C42.108D10  C4212D10  C42.228D10  C4216D10  C42.229D10  C42.113D10  C42.114D10  C4217D10  C42.118D10  Dic1019D4  Dic1020D4  C10.342+ 1+4  D2019D4  C10.402+ 1+4  C10.732- 1+4  D2020D4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C10.452+ 1+4  C10.1152+ 1+4  D2021D4  D2022D4  Dic1021D4  Dic1022D4  C10.1182+ 1+4  C10.522+ 1+4  C10.532+ 1+4  C10.202- 1+4  C10.212- 1+4  C10.222- 1+4  C10.232- 1+4  C10.772- 1+4  C10.612+ 1+4  C10.622+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.842- 1+4  C10.662+ 1+4  C10.672+ 1+4  C24.72D10  C24.42D10  C10.452- 1+4  C10.1042- 1+4  (C2×C20)⋊15D4  C10.1452+ 1+4  C10.1072- 1+4  (C2×C20)⋊17D4  C10.1482+ 1+4  C1517(C4×D4)  C1522(C4×D4)  C1528(C4×D4)
C4×C5⋊D4 is a maximal quotient of
C10.92(C4×D4)  (C2×C42)⋊D5  C24.3D10  C24.4D10  C24.12D10  C24.13D10  C10.96(C4×D4)  C10.97(C4×D4)  D105(C4⋊C4)  C10.90(C4×D4)  C42.48D10  C42.51D10  C42.56D10  C42.59D10  C4032D4  C40⋊D4  C4018D4  C40.93D4  C40.50D4  C24.62D10  C24.65D10  C1517(C4×D4)  C1522(C4×D4)  C1528(C4×D4)

52 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E4F4G···4L5A5B10A···10N20A···20P
order122222224444444···45510···1020···20
size111122101011112210···10222···22···2

52 irreducible representations

dim11111111122222222
type++++++++++++
imageC1C2C2C2C2C2C2C2C4D4D5C4○D4D10D10C5⋊D4C4×D5C4○D20
kernelC4×C5⋊D4C4×Dic5C10.D4D10⋊C4C23.D5C2×C4×D5C2×C5⋊D4C22×C20C5⋊D4C20C22×C4C10C2×C4C23C4C22C2
# reps11111111822242888

Matrix representation of C4×C5⋊D4 in GL3(𝔽41) generated by

900
0320
0032
,
100
0640
010
,
4000
0236
02118
,
100
0400
0351
G:=sub<GL(3,GF(41))| [9,0,0,0,32,0,0,0,32],[1,0,0,0,6,1,0,40,0],[40,0,0,0,23,21,0,6,18],[1,0,0,0,40,35,0,0,1] >;

C4×C5⋊D4 in GAP, Magma, Sage, TeX

C_4\times C_5\rtimes D_4
% in TeX

G:=Group("C4xC5:D4");
// GroupNames label

G:=SmallGroup(160,149);
// by ID

G=gap.SmallGroup(160,149);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,217,50,4613]);
// Polycyclic

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

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