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G = C5×C233D4order 320 = 26·5

Direct product of C5 and C233D4

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C233D4, C10.1512+ 1+4, C233(C5×D4), C4⋊D45C10, C243(C2×C10), (C22×C10)⋊6D4, C22≀C22C10, (C22×D4)⋊6C10, C22.2(D4×C10), (D4×C10)⋊35C22, (C23×C10)⋊3C22, (C2×C20).663C23, (C2×C10).354C24, (C22×C20)⋊47C22, C22.D42C10, C10.189(C22×D4), C2.3(C5×2+ 1+4), C22.28(C23×C10), (C22×C10).90C23, C23.36(C22×C10), C4⋊C43(C2×C10), (D4×C2×C10)⋊21C2, C2.13(D4×C2×C10), (C2×D4)⋊3(C2×C10), (C5×C4⋊D4)⋊32C2, C22⋊C43(C2×C10), (C5×C4⋊C4)⋊37C22, (C22×C4)⋊7(C2×C10), (C2×C10).90(C2×D4), (C5×C22≀C2)⋊12C2, (C2×C22⋊C4)⋊12C10, (C10×C22⋊C4)⋊32C2, (C5×C22⋊C4)⋊38C22, (C2×C4).21(C22×C10), (C5×C22.D4)⋊21C2, SmallGroup(320,1536)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C233D4
C1C2C22C2×C10C22×C10D4×C10C5×C4⋊D4 — C5×C233D4
C1C22 — C5×C233D4
C1C2×C10 — C5×C233D4

Generators and relations for C5×C233D4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, ece-1=fcf=cd=dc, de=ed, df=fd, fef=e-1 >

Subgroups: 642 in 346 conjugacy classes, 162 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×10], C4 [×8], C22, C22 [×6], C22 [×30], C5, C2×C4 [×8], C2×C4 [×6], D4 [×20], C23, C23 [×10], C23 [×10], C10, C10 [×2], C10 [×10], C22⋊C4 [×12], C4⋊C4 [×4], C22×C4 [×4], C2×D4 [×12], C2×D4 [×8], C24, C24 [×2], C20 [×8], C2×C10, C2×C10 [×6], C2×C10 [×30], C2×C22⋊C4, C22≀C2 [×4], C4⋊D4 [×4], C22.D4 [×4], C22×D4 [×2], C2×C20 [×8], C2×C20 [×6], C5×D4 [×20], C22×C10, C22×C10 [×10], C22×C10 [×10], C233D4, C5×C22⋊C4 [×12], C5×C4⋊C4 [×4], C22×C20 [×4], D4×C10 [×12], D4×C10 [×8], C23×C10, C23×C10 [×2], C10×C22⋊C4, C5×C22≀C2 [×4], C5×C4⋊D4 [×4], C5×C22.D4 [×4], D4×C2×C10 [×2], C5×C233D4
Quotients: C1, C2 [×15], C22 [×35], C5, D4 [×4], C23 [×15], C10 [×15], C2×D4 [×6], C24, C2×C10 [×35], C22×D4, 2+ 1+4 [×2], C5×D4 [×4], C22×C10 [×15], C233D4, D4×C10 [×6], C23×C10, D4×C2×C10, C5×2+ 1+4 [×2], C5×C233D4

Smallest permutation representation of C5×C233D4
On 80 points
Generators in S80
(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 36)(2 37)(3 38)(4 39)(5 40)(6 71)(7 72)(8 73)(9 74)(10 75)(11 56)(12 57)(13 58)(14 59)(15 60)(16 61)(17 62)(18 63)(19 64)(20 65)(21 41)(22 42)(23 43)(24 44)(25 45)(26 46)(27 47)(28 48)(29 49)(30 50)(31 51)(32 52)(33 53)(34 54)(35 55)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 71)(57 72)(58 73)(59 74)(60 75)(61 66)(62 67)(63 68)(64 69)(65 70)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 31)(22 32)(23 33)(24 34)(25 35)(36 46)(37 47)(38 48)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(56 66)(57 67)(58 68)(59 69)(60 70)(61 71)(62 72)(63 73)(64 74)(65 75)
(1 61 21 56)(2 62 22 57)(3 63 23 58)(4 64 24 59)(5 65 25 60)(6 51 76 46)(7 52 77 47)(8 53 78 48)(9 54 79 49)(10 55 80 50)(11 36 16 41)(12 37 17 42)(13 38 18 43)(14 39 19 44)(15 40 20 45)(26 71 31 66)(27 72 32 67)(28 73 33 68)(29 74 34 69)(30 75 35 70)
(1 56)(2 57)(3 58)(4 59)(5 60)(6 41)(7 42)(8 43)(9 44)(10 45)(11 46)(12 47)(13 48)(14 49)(15 50)(16 51)(17 52)(18 53)(19 54)(20 55)(21 61)(22 62)(23 63)(24 64)(25 65)(26 66)(27 67)(28 68)(29 69)(30 70)(31 71)(32 72)(33 73)(34 74)(35 75)(36 76)(37 77)(38 78)(39 79)(40 80)

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

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

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

110 conjugacy classes

class 1 2A2B2C2D···2I2J2K2L2M4A···4H5A5B5C5D10A···10L10M···10AJ10AK···10AZ20A···20AF
order12222···222224···4555510···1010···1010···1020···20
size11112···244444···411111···12···24···44···4

110 irreducible representations

dim1111111111112244
type++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D4C5×D42+ 1+4C5×2+ 1+4
kernelC5×C233D4C10×C22⋊C4C5×C22≀C2C5×C4⋊D4C5×C22.D4D4×C2×C10C233D4C2×C22⋊C4C22≀C2C4⋊D4C22.D4C22×D4C22×C10C23C10C2
# reps11444244161616841628

Matrix representation of C5×C233D4 in GL6(𝔽41)

1800000
0180000
001000
000100
000010
000001
,
4000000
0400000
0040100
000100
0000140
0000040
,
100000
010000
0040000
0004000
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
0400000
100000
000010
0000240
0040000
0039100
,
0400000
4000000
000010
000001
001000
000100

G:=sub<GL(6,GF(41))| [18,0,0,0,0,0,0,18,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,40,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,39,0,0,0,0,0,1,0,0,1,2,0,0,0,0,0,40,0,0],[0,40,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C233D4 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes_3D_4
% in TeX

G:=Group("C5xC2^3:3D4");
// GroupNames label

G:=SmallGroup(320,1536);
// by ID

G=gap.SmallGroup(320,1536);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1149,3446,891,2467]);
// Polycyclic

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

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