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G = C42⋊28D10order 320 = 26·5

28th semidirect product of C42 and D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C42⋊28D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — C23⋊D10 — C42⋊28D10
 Lower central C5 — C2×C10 — C42⋊28D10
 Upper central C1 — C22 — C4⋊1D4

Generators and relations for C4228D10
G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2, C2, C4, C22, C22, C5, C2×C4, C2×C4, D4, C23, C23, C23, D5, C10, C10, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×D4, C24, Dic5, C20, D10, C2×C10, C2×C10, C22≀C2, C4⋊D4, C22.D4, C422C2, C41D4, C2×Dic5, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C22.54C24, C10.D4, D10⋊C4, C23.D5, C4×C20, C22×Dic5, C2×C5⋊D4, D4×C10, C23×D5, C422D5, C23.18D10, C23⋊D10, Dic5⋊D4, C5×C41D4, C4228D10
Quotients: C1, C2, C22, C23, D5, C24, D10, 2+ 1+4, C22×D5, C22.54C24, C23×D5, D46D10, C4228D10

Smallest permutation representation of C4228D10
On 80 points
Generators in S80
(1 70 20 65)(2 66 16 61)(3 62 17 67)(4 68 18 63)(5 64 19 69)(6 77 15 72)(7 73 11 78)(8 79 12 74)(9 75 13 80)(10 71 14 76)(21 32 53 43)(22 44 54 33)(23 34 55 45)(24 46 56 35)(25 36 57 47)(26 48 58 37)(27 38 59 49)(28 50 60 39)(29 40 51 41)(30 42 52 31)
(1 28 9 23)(2 24 10 29)(3 30 6 25)(4 26 7 21)(5 22 8 27)(11 53 18 58)(12 59 19 54)(13 55 20 60)(14 51 16 56)(15 57 17 52)(31 72 47 67)(32 68 48 73)(33 74 49 69)(34 70 50 75)(35 76 41 61)(36 62 42 77)(37 78 43 63)(38 64 44 79)(39 80 45 65)(40 66 46 71)
(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 12)(2 11)(3 15)(4 14)(5 13)(6 17)(7 16)(8 20)(9 19)(10 18)(21 24)(22 23)(25 30)(26 29)(27 28)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 58)(52 57)(53 56)(54 55)(59 60)(61 63)(64 70)(65 69)(66 68)(71 73)(74 80)(75 79)(76 78)

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

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

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

47 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D ··· 4I 5A 5B 10A ··· 10F 10G ··· 10N 20A ··· 20L order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 4 4 4 4 20 20 4 4 4 20 ··· 20 2 2 2 ··· 2 8 ··· 8 4 ··· 4

47 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 4 4 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 D5 D10 D10 2+ 1+4 D4⋊6D10 kernel C42⋊28D10 C42⋊2D5 C23.18D10 C23⋊D10 Dic5⋊D4 C5×C4⋊1D4 C4⋊1D4 C42 C2×D4 C10 C2 # reps 1 2 3 3 6 1 2 2 12 3 12

Matrix representation of C4228D10 in GL8(𝔽41)

 18 6 0 0 0 0 0 0 35 23 0 0 0 0 0 0 0 0 18 6 0 0 0 0 0 0 35 23 0 0 0 0 0 0 0 0 26 0 7 7 0 0 0 0 39 30 36 12 0 0 0 0 13 12 13 2 0 0 0 0 25 29 2 13
,
 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 0 0 24 34 38 3 0 0 0 0 6 17 38 0 0 0 0 0 0 28 18 6 0 0 0 0 13 28 35 23
,
 1 6 0 0 0 0 0 0 35 6 0 0 0 0 0 0 0 0 40 35 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 0 35 3 16 0 0 0 0 7 35 16 28 0 0 0 0 0 0 40 7 0 0 0 0 0 0 34 7
,
 6 1 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 6 1 0 0 0 0 0 0 6 35 0 0 0 0 0 0 0 0 6 35 1 40 0 0 0 0 40 35 1 0 0 0 0 0 0 0 7 40 0 0 0 0 0 0 7 34

G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;

C4228D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{28}D_{10}
% in TeX

G:=Group("C4^2:28D10");
// GroupNames label

G:=SmallGroup(320,1392);
// by ID

G=gap.SmallGroup(320,1392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=c^10=d^2=1,a*b=b*a,c*a*c^-1=a^-1,d*a*d=a^-1*b^2,c*b*c^-1=b^-1,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations

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