Copied to
clipboard

G = C2xC52:5C8order 400 = 24·52

Direct product of C2 and C52:5C8

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C2xC52:5C8, C102.7C4, C10:2(C5:C8), (C5xC10):5C8, C52:14(C2xC8), (C2xC10).7F5, C10.25(C2xF5), C52:6C4.9C4, C22.2(C52:C4), C52:6C4.24C22, C5:3(C2xC5:C8), C2.3(C2xC52:C4), (C5xC10).38(C2xC4), (C2xC52:6C4).10C2, SmallGroup(400,160)

Series: Derived Chief Lower central Upper central

C1C52 — C2xC52:5C8
C1C5C52C5xC10C52:6C4C52:5C8 — C2xC52:5C8
C52 — C2xC52:5C8
C1C22

Generators and relations for C2xC52:5C8
 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, C2xC4, C10, C10, C2xC8, Dic5, C2xC10, C2xC10, C52, C5:C8, C2xDic5, C5xC10, C5xC10, C2xC5:C8, C52:6C4, C102, C52:5C8, C2xC52:6C4, C2xC52:5C8
Quotients: C1, C2, C4, C22, C8, C2xC4, C2xC8, F5, C5:C8, C2xF5, C2xC5:C8, C52:C4, C52:5C8, C2xC52:C4, C2xC52:5C8

Smallest permutation representation of C2xC52:5C8
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 2A2B2C4A4B4C4D5A···5F8A···8H10A···10R
order122244445···58···810···10
size1111252525254···425···254···4

40 irreducible representations

dim111111444444
type++++-++-+
imageC1C2C2C4C4C8F5C5:C8C2xF5C52:C4C52:5C8C2xC52:C4
kernelC2xC52:5C8C52:5C8C2xC52:6C4C52:6C4C102C5xC10C2xC10C10C10C22C2C2
# reps121228242484

Matrix representation of C2xC52:5C8 in GL6(F41)

100000
0400000
001000
000100
000010
000001
,
100000
010000
00354000
00364000
00304034
00202077
,
100000
010000
00354000
00364000
00202077
00303440
,
300000
0400000
0000401
002133934
002632380
002533380

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] >;

C2xC52:5C8 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

׿
x
:
Z
F
o
wr
Q
<