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## G = C2×C52⋊3C8order 400 = 24·52

### Direct product of C2 and C52⋊3C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C2×C52⋊3C8
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊3C8 — C2×C52⋊3C8
 Lower central C52 — C2×C52⋊3C8
 Upper central C1 — C22

Generators and relations for C2×C523C8
G = < a,b,c,d | a2=b5=c5=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c2 >

Smallest permutation representation of C2×C523C8
On 80 points
Generators in S80
(1 64)(2 57)(3 58)(4 59)(5 60)(6 61)(7 62)(8 63)(9 40)(10 33)(11 34)(12 35)(13 36)(14 37)(15 38)(16 39)(17 29)(18 30)(19 31)(20 32)(21 25)(22 26)(23 27)(24 28)(41 66)(42 67)(43 68)(44 69)(45 70)(46 71)(47 72)(48 65)(49 74)(50 75)(51 76)(52 77)(53 78)(54 79)(55 80)(56 73)
(1 67 11 75 25)(2 26 76 12 68)(3 69 13 77 27)(4 28 78 14 70)(5 71 15 79 29)(6 30 80 16 72)(7 65 9 73 31)(8 32 74 10 66)(17 60 46 38 54)(18 55 39 47 61)(19 62 48 40 56)(20 49 33 41 63)(21 64 42 34 50)(22 51 35 43 57)(23 58 44 36 52)(24 53 37 45 59)
(1 75 67 25 11)(2 68 12 76 26)(3 13 27 69 77)(4 28 78 14 70)(5 79 71 29 15)(6 72 16 80 30)(7 9 31 65 73)(8 32 74 10 66)(17 38 60 54 46)(18 61 47 39 55)(19 48 56 62 40)(20 49 33 41 63)(21 34 64 50 42)(22 57 43 35 51)(23 44 52 58 36)(24 53 37 45 59)
(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)| (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73), (1,67,11,75,25)(2,26,76,12,68)(3,69,13,77,27)(4,28,78,14,70)(5,71,15,79,29)(6,30,80,16,72)(7,65,9,73,31)(8,32,74,10,66)(17,60,46,38,54)(18,55,39,47,61)(19,62,48,40,56)(20,49,33,41,63)(21,64,42,34,50)(22,51,35,43,57)(23,58,44,36,52)(24,53,37,45,59), (1,75,67,25,11)(2,68,12,76,26)(3,13,27,69,77)(4,28,78,14,70)(5,79,71,29,15)(6,72,16,80,30)(7,9,31,65,73)(8,32,74,10,66)(17,38,60,54,46)(18,61,47,39,55)(19,48,56,62,40)(20,49,33,41,63)(21,34,64,50,42)(22,57,43,35,51)(23,44,52,58,36)(24,53,37,45,59), (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( (1,64)(2,57)(3,58)(4,59)(5,60)(6,61)(7,62)(8,63)(9,40)(10,33)(11,34)(12,35)(13,36)(14,37)(15,38)(16,39)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)(41,66)(42,67)(43,68)(44,69)(45,70)(46,71)(47,72)(48,65)(49,74)(50,75)(51,76)(52,77)(53,78)(54,79)(55,80)(56,73), (1,67,11,75,25)(2,26,76,12,68)(3,69,13,77,27)(4,28,78,14,70)(5,71,15,79,29)(6,30,80,16,72)(7,65,9,73,31)(8,32,74,10,66)(17,60,46,38,54)(18,55,39,47,61)(19,62,48,40,56)(20,49,33,41,63)(21,64,42,34,50)(22,51,35,43,57)(23,58,44,36,52)(24,53,37,45,59), (1,75,67,25,11)(2,68,12,76,26)(3,13,27,69,77)(4,28,78,14,70)(5,79,71,29,15)(6,72,16,80,30)(7,9,31,65,73)(8,32,74,10,66)(17,38,60,54,46)(18,61,47,39,55)(19,48,56,62,40)(20,49,33,41,63)(21,34,64,50,42)(22,57,43,35,51)(23,44,52,58,36)(24,53,37,45,59), (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([[(1,64),(2,57),(3,58),(4,59),(5,60),(6,61),(7,62),(8,63),(9,40),(10,33),(11,34),(12,35),(13,36),(14,37),(15,38),(16,39),(17,29),(18,30),(19,31),(20,32),(21,25),(22,26),(23,27),(24,28),(41,66),(42,67),(43,68),(44,69),(45,70),(46,71),(47,72),(48,65),(49,74),(50,75),(51,76),(52,77),(53,78),(54,79),(55,80),(56,73)], [(1,67,11,75,25),(2,26,76,12,68),(3,69,13,77,27),(4,28,78,14,70),(5,71,15,79,29),(6,30,80,16,72),(7,65,9,73,31),(8,32,74,10,66),(17,60,46,38,54),(18,55,39,47,61),(19,62,48,40,56),(20,49,33,41,63),(21,64,42,34,50),(22,51,35,43,57),(23,58,44,36,52),(24,53,37,45,59)], [(1,75,67,25,11),(2,68,12,76,26),(3,13,27,69,77),(4,28,78,14,70),(5,79,71,29,15),(6,72,16,80,30),(7,9,31,65,73),(8,32,74,10,66),(17,38,60,54,46),(18,61,47,39,55),(19,48,56,62,40),(20,49,33,41,63),(21,34,64,50,42),(22,57,43,35,51),(23,44,52,58,36),(24,53,37,45,59)], [(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)]])

52 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C ··· 5G 8A ··· 8H 10A ··· 10F 10G ··· 10U 20A ··· 20H order 1 2 2 2 4 4 4 4 5 5 5 ··· 5 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 5 5 5 5 2 2 4 ··· 4 25 ··· 25 2 ··· 2 4 ··· 4 10 ··· 10

52 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 4 4 type + + + + - + - + - + image C1 C2 C2 C4 C4 C8 D5 Dic5 D10 Dic5 C5⋊2C8 F5 C5⋊C8 C2×F5 D5.D5 C52⋊3C8 C2×D5.D5 kernel C2×C52⋊3C8 C52⋊3C8 C10×Dic5 C5×Dic5 C102 C5×C10 C2×Dic5 Dic5 Dic5 C2×C10 C10 C2×C10 C10 C10 C22 C2 C2 # reps 1 2 1 2 2 8 2 2 2 2 8 1 2 1 4 8 4

Matrix representation of C2×C523C8 in GL6(𝔽41)

 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
,
 35 6 0 0 0 0 35 40 0 0 0 0 0 0 10 0 0 0 0 0 0 10 0 0 0 0 29 32 37 0 0 0 33 40 0 37
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 16 0 0 0 0 0 0 18 0 0 0 0 18 21 37 0 0 0 20 17 0 10
,
 0 38 0 0 0 0 38 0 0 0 0 0 0 0 7 36 15 0 0 0 32 4 0 15 0 0 32 1 34 5 0 0 12 26 9 37

G:=sub<GL(6,GF(41))| [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],[35,35,0,0,0,0,6,40,0,0,0,0,0,0,10,0,29,33,0,0,0,10,32,40,0,0,0,0,37,0,0,0,0,0,0,37],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,0,18,20,0,0,0,18,21,17,0,0,0,0,37,0,0,0,0,0,0,10],[0,38,0,0,0,0,38,0,0,0,0,0,0,0,7,32,32,12,0,0,36,4,1,26,0,0,15,0,34,9,0,0,0,15,5,37] >;

C2×C523C8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_3C_8
% in TeX

G:=Group("C2xC5^2:3C8");
// GroupNames label

G:=SmallGroup(400,146);
// by ID

G=gap.SmallGroup(400,146);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,50,1924,8645,2897]);
// 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^-1,d*c*d^-1=c^2>;
// generators/relations

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