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G = C22×C5⋊D4order 160 = 25·5

Direct product of C22 and C5⋊D4

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

Aliases: C22×C5⋊D4, C243D5, C234D10, D103C23, C10.15C24, Dic52C23, (C2×C10)⋊9D4, C103(C2×D4), C53(C22×D4), (C2×C10)⋊3C23, (C23×C10)⋊4C2, (C23×D5)⋊5C2, C2.15(C23×D5), C222(C22×D5), (C22×C10)⋊7C22, (C22×Dic5)⋊9C2, (C22×D5)⋊7C22, (C2×Dic5)⋊11C22, SmallGroup(160,227)

Series: Derived Chief Lower central Upper central

C1C10 — C22×C5⋊D4
C1C5C10D10C22×D5C23×D5 — C22×C5⋊D4
C5C10 — C22×C5⋊D4
C1C23C24

Generators and relations for C22×C5⋊D4
 G = < a,b,c,d,e | a2=b2=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 680 in 236 conjugacy classes, 105 normal (11 characteristic)
C1, C2, C2 [×6], C2 [×8], C4 [×4], C22 [×11], C22 [×28], C5, C2×C4 [×6], D4 [×16], C23, C23 [×6], C23 [×14], D5 [×4], C10, C10 [×6], C10 [×4], C22×C4, C2×D4 [×12], C24, C24, Dic5 [×4], D10 [×4], D10 [×12], C2×C10 [×11], C2×C10 [×12], C22×D4, C2×Dic5 [×6], C5⋊D4 [×16], C22×D5 [×6], C22×D5 [×4], C22×C10, C22×C10 [×6], C22×C10 [×4], C22×Dic5, C2×C5⋊D4 [×12], C23×D5, C23×C10, C22×C5⋊D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C24, D10 [×7], C22×D4, C5⋊D4 [×4], C22×D5 [×7], C2×C5⋊D4 [×6], C23×D5, C22×C5⋊D4

Smallest permutation representation of C22×C5⋊D4
On 80 points
Generators in S80
(1 46)(2 47)(3 48)(4 49)(5 50)(6 41)(7 42)(8 43)(9 44)(10 45)(11 56)(12 57)(13 58)(14 59)(15 60)(16 51)(17 52)(18 53)(19 54)(20 55)(21 66)(22 67)(23 68)(24 69)(25 70)(26 61)(27 62)(28 63)(29 64)(30 65)(31 76)(32 77)(33 78)(34 79)(35 80)(36 71)(37 72)(38 73)(39 74)(40 75)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 21)(7 22)(8 23)(9 24)(10 25)(11 36)(12 37)(13 38)(14 39)(15 40)(16 31)(17 32)(18 33)(19 34)(20 35)(41 66)(42 67)(43 68)(44 69)(45 70)(46 61)(47 62)(48 63)(49 64)(50 65)(51 76)(52 77)(53 78)(54 79)(55 80)(56 71)(57 72)(58 73)(59 74)(60 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 51 6 56)(2 55 7 60)(3 54 8 59)(4 53 9 58)(5 52 10 57)(11 46 16 41)(12 50 17 45)(13 49 18 44)(14 48 19 43)(15 47 20 42)(21 71 26 76)(22 75 27 80)(23 74 28 79)(24 73 29 78)(25 72 30 77)(31 66 36 61)(32 70 37 65)(33 69 38 64)(34 68 39 63)(35 67 40 62)
(1 61)(2 65)(3 64)(4 63)(5 62)(6 66)(7 70)(8 69)(9 68)(10 67)(11 76)(12 80)(13 79)(14 78)(15 77)(16 71)(17 75)(18 74)(19 73)(20 72)(21 41)(22 45)(23 44)(24 43)(25 42)(26 46)(27 50)(28 49)(29 48)(30 47)(31 56)(32 60)(33 59)(34 58)(35 57)(36 51)(37 55)(38 54)(39 53)(40 52)

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

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

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

C22×C5⋊D4 is a maximal subgroup of
C24.12D10  C24.13D10  C23.45D20  C24.14D10  C232D20  C24.16D10  C24.65D10  C24.21D10  C24.24D10  C24.27D10  C233D20  C243D10  C244D10  C24.33D10  C24.34D10  C248D10  C22×D4×D5
C22×C5⋊D4 is a maximal quotient of
C24.72D10  C248D10  C24.41D10  C24.42D10  C10.442- 1+4  C10.452- 1+4  C20.C24  C10.1042- 1+4  C10.1052- 1+4  (C2×C20)⋊15D4  C10.1452+ 1+4  C10.1462+ 1+4  C10.1072- 1+4  (C2×C20)⋊17D4  C10.1472+ 1+4  C10.1482+ 1+4  D20.32C23  D20.33C23  D20.34C23  D20.35C23

52 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A4B4C4D5A5B10A···10AD
order12···22222222244445510···10
size11···122221010101010101010222···2

52 irreducible representations

dim111112222
type++++++++
imageC1C2C2C2C2D4D5D10C5⋊D4
kernelC22×C5⋊D4C22×Dic5C2×C5⋊D4C23×D5C23×C10C2×C10C24C23C22
# reps111211421416

Matrix representation of C22×C5⋊D4 in GL4(𝔽41) generated by

40000
0100
0010
0001
,
40000
04000
0010
0001
,
1000
0100
00740
00840
,
1000
04000
00318
00438
,
40000
0100
00341
00347
G:=sub<GL(4,GF(41))| [40,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,7,8,0,0,40,40],[1,0,0,0,0,40,0,0,0,0,3,4,0,0,18,38],[40,0,0,0,0,1,0,0,0,0,34,34,0,0,1,7] >;

C22×C5⋊D4 in GAP, Magma, Sage, TeX

C_2^2\times C_5\rtimes D_4
% in TeX

G:=Group("C2^2xC5:D4");
// GroupNames label

G:=SmallGroup(160,227);
// by ID

G=gap.SmallGroup(160,227);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,579,4613]);
// Polycyclic

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

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