Copied to
clipboard

## G = C2×C52⋊5C8order 400 = 24·52

### Direct product of C2 and C52⋊5C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊5C8
 Chief series C1 — C5 — C52 — C5×C10 — C52⋊6C4 — C52⋊5C8 — C2×C52⋊5C8
 Lower central C52 — C2×C52⋊5C8
 Upper central C1 — C22

Generators and relations for C2×C525C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >

Subgroups: 316 in 60 conjugacy classes, 26 normal (10 characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, C2×C4, C10, C10, C2×C8, Dic5, C2×C10, C2×C10, C52, C5⋊C8, C2×Dic5, C5×C10, C5×C10, C2×C5⋊C8, C526C4, C102, C525C8, C2×C526C4, C2×C525C8
Quotients: C1, C2, C4, C22, C8, C2×C4, C2×C8, F5, C5⋊C8, C2×F5, C2×C5⋊C8, C52⋊C4, C525C8, C2×C52⋊C4, C2×C525C8

Smallest permutation representation of C2×C525C8
On 80 points
Generators in S80
(1 69)(2 70)(3 71)(4 72)(5 65)(6 66)(7 67)(8 68)(9 19)(10 20)(11 21)(12 22)(13 23)(14 24)(15 17)(16 18)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(31 33)(32 34)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(47 79)(48 80)(49 63)(50 64)(51 57)(52 58)(53 59)(54 60)(55 61)(56 62)
(1 17 37 75 57)(2 38 58 18 76)(3 59 77 39 19)(4 78 20 60 40)(5 21 33 79 61)(6 34 62 22 80)(7 63 73 35 23)(8 74 24 64 36)(9 71 53 45 29)(10 54 30 72 46)(11 31 47 55 65)(12 48 66 32 56)(13 67 49 41 25)(14 50 26 68 42)(15 27 43 51 69)(16 44 70 28 52)
(1 17 37 75 57)(2 76 18 58 38)(3 59 77 39 19)(4 40 60 20 78)(5 21 33 79 61)(6 80 22 62 34)(7 63 73 35 23)(8 36 64 24 74)(9 71 53 45 29)(10 46 72 30 54)(11 31 47 55 65)(12 56 32 66 48)(13 67 49 41 25)(14 42 68 26 50)(15 27 43 51 69)(16 52 28 70 44)
(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)

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

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

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

40 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A ··· 5F 8A ··· 8H 10A ··· 10R order 1 2 2 2 4 4 4 4 5 ··· 5 8 ··· 8 10 ··· 10 size 1 1 1 1 25 25 25 25 4 ··· 4 25 ··· 25 4 ··· 4

40 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 4 4 4 type + + + + - + + - + image C1 C2 C2 C4 C4 C8 F5 C5⋊C8 C2×F5 C52⋊C4 C52⋊5C8 C2×C52⋊C4 kernel C2×C52⋊5C8 C52⋊5C8 C2×C52⋊6C4 C52⋊6C4 C102 C5×C10 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 2 2 8 2 4 2 4 8 4

Matrix representation of C2×C525C8 in GL6(𝔽41)

 1 0 0 0 0 0 0 40 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
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 40 0 0 0 0 36 40 0 0 0 0 3 0 40 34 0 0 20 20 7 7
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 35 40 0 0 0 0 36 40 0 0 0 0 20 20 7 7 0 0 3 0 34 40
,
 3 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 1 0 0 21 3 39 34 0 0 26 32 38 0 0 0 25 33 38 0

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,36,3,20,0,0,40,40,0,20,0,0,0,0,40,7,0,0,0,0,34,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,35,36,20,3,0,0,40,40,20,0,0,0,0,0,7,34,0,0,0,0,7,40],[3,0,0,0,0,0,0,40,0,0,0,0,0,0,0,21,26,25,0,0,0,3,32,33,0,0,40,39,38,38,0,0,1,34,0,0] >;

C2×C525C8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_5C_8
% in TeX

G:=Group("C2xC5^2:5C8");
// GroupNames label

G:=SmallGroup(400,160);
// by ID

G=gap.SmallGroup(400,160);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,1444,496,5765,2897]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^5=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations

׿
×
𝔽