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G = C10.622+ 1+4order 320 = 26·5

62nd non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.622+ 1+4, C4⋊C431D10, (C22×C4)⋊6D10, C22⋊C418D10, (C2×D4).99D10, D10⋊Q830C2, C23⋊D10.3C2, C4⋊Dic540C22, C22.D48D5, D10.36(C4○D4), C20.17D421C2, (C2×C20).599C23, (C2×C10).203C24, (C22×C20)⋊39C22, (C4×Dic5)⋊56C22, D10.12D432C2, C23.D554C22, C2.64(D46D10), C57(C22.45C24), (C2×Dic10)⋊31C22, (D4×C10).141C22, C23.D1031C2, C10.D466C22, (C23×D5).60C22, C23.130(C22×D5), C22.224(C23×D5), (C22×C10).223C23, (C2×Dic5).105C23, (C22×D5).219C23, D10⋊C4.131C22, (C4×C5⋊D4)⋊48C2, C2.65(D5×C4○D4), C4⋊C47D533C2, (C5×C4⋊C4)⋊29C22, (D5×C22⋊C4)⋊15C2, C10.177(C2×C4○D4), (C2×C4×D5).121C22, (C2×C4).65(C22×D5), (C5×C22⋊C4)⋊25C22, (C5×C22.D4)⋊11C2, (C2×C5⋊D4).143C22, SmallGroup(320,1331)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.622+ 1+4
C1C5C10C2×C10C22×D5C23×D5D5×C22⋊C4 — C10.622+ 1+4
C5C2×C10 — C10.622+ 1+4
C1C22C22.D4

Generators and relations for C10.622+ 1+4
 G = < a,b,c,d,e | a10=b4=c2=e2=1, d2=b2, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=a5b-1, dbd-1=ebe=a5b, dcd-1=ece=a5c, ede=b2d >

Subgroups: 950 in 248 conjugacy classes, 95 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×18], C5, C2×C4, C2×C4 [×4], C2×C4 [×13], D4 [×5], Q8, C23 [×2], C23 [×7], D5 [×4], C10, C10 [×2], C10 [×2], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×2], C4⋊C4 [×6], C22×C4, C22×C4 [×4], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×6], C20 [×5], D10 [×4], D10 [×8], C2×C10, C2×C10 [×6], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4, C22.D4 [×2], C4.4D4, C422C2 [×2], Dic10, C4×D5 [×6], C2×Dic5 [×6], C5⋊D4 [×4], C2×C20, C2×C20 [×4], C2×C20, C5×D4, C22×D5 [×2], C22×D5 [×5], C22×C10 [×2], C22.45C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×6], C23.D5, C23.D5 [×4], C5×C22⋊C4, C5×C22⋊C4 [×2], C5×C4⋊C4 [×2], C2×Dic10, C2×C4×D5 [×4], C2×C5⋊D4 [×2], C22×C20, D4×C10, C23×D5, C23.D10 [×2], D5×C22⋊C4 [×2], D10.12D4 [×2], C4⋊C47D5 [×2], D10⋊Q8 [×2], C4×C5⋊D4 [×2], C20.17D4, C23⋊D10, C5×C22.D4, C10.622+ 1+4
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×4], C24, D10 [×7], C2×C4○D4 [×2], 2+ 1+4, C22×D5 [×7], C22.45C24, C23×D5, D46D10, D5×C4○D4 [×2], C10.622+ 1+4

Smallest permutation representation of C10.622+ 1+4
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 43 18 58)(2 44 19 59)(3 45 20 60)(4 46 11 51)(5 47 12 52)(6 48 13 53)(7 49 14 54)(8 50 15 55)(9 41 16 56)(10 42 17 57)(21 66 36 71)(22 67 37 72)(23 68 38 73)(24 69 39 74)(25 70 40 75)(26 61 31 76)(27 62 32 77)(28 63 33 78)(29 64 34 79)(30 65 35 80)
(21 26)(22 27)(23 28)(24 29)(25 30)(31 36)(32 37)(33 38)(34 39)(35 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 38 18 23)(2 37 19 22)(3 36 20 21)(4 35 11 30)(5 34 12 29)(6 33 13 28)(7 32 14 27)(8 31 15 26)(9 40 16 25)(10 39 17 24)(41 80 56 65)(42 79 57 64)(43 78 58 63)(44 77 59 62)(45 76 60 61)(46 75 51 70)(47 74 52 69)(48 73 53 68)(49 72 54 67)(50 71 55 66)
(1 23)(2 24)(3 25)(4 26)(5 27)(6 28)(7 29)(8 30)(9 21)(10 22)(11 31)(12 32)(13 33)(14 34)(15 35)(16 36)(17 37)(18 38)(19 39)(20 40)(41 61)(42 62)(43 63)(44 64)(45 65)(46 66)(47 67)(48 68)(49 69)(50 70)(51 71)(52 72)(53 73)(54 74)(55 75)(56 76)(57 77)(58 78)(59 79)(60 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), (1,43,18,58)(2,44,19,59)(3,45,20,60)(4,46,11,51)(5,47,12,52)(6,48,13,53)(7,49,14,54)(8,50,15,55)(9,41,16,56)(10,42,17,57)(21,66,36,71)(22,67,37,72)(23,68,38,73)(24,69,39,74)(25,70,40,75)(26,61,31,76)(27,62,32,77)(28,63,33,78)(29,64,34,79)(30,65,35,80), (21,26)(22,27)(23,28)(24,29)(25,30)(31,36)(32,37)(33,38)(34,39)(35,40)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,76)(62,77)(63,78)(64,79)(65,80)(66,71)(67,72)(68,73)(69,74)(70,75), (1,38,18,23)(2,37,19,22)(3,36,20,21)(4,35,11,30)(5,34,12,29)(6,33,13,28)(7,32,14,27)(8,31,15,26)(9,40,16,25)(10,39,17,24)(41,80,56,65)(42,79,57,64)(43,78,58,63)(44,77,59,62)(45,76,60,61)(46,75,51,70)(47,74,52,69)(48,73,53,68)(49,72,54,67)(50,71,55,66), (1,23)(2,24)(3,25)(4,26)(5,27)(6,28)(7,29)(8,30)(9,21)(10,22)(11,31)(12,32)(13,33)(14,34)(15,35)(16,36)(17,37)(18,38)(19,39)(20,40)(41,61)(42,62)(43,63)(44,64)(45,65)(46,66)(47,67)(48,68)(49,69)(50,70)(51,71)(52,72)(53,73)(54,74)(55,75)(56,76)(57,77)(58,78)(59,79)(60,80)>;

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O5A5B10A···10F10G10H10I10J10K10L20A···20H20I···20N
order12222222224444444444444445510···1010101010101020···2020···20
size1111441010101022224441010101020202020222···24444884···48···8

53 irreducible representations

dim1111111111222222444
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10D102+ 1+4D46D10D5×C4○D4
kernelC10.622+ 1+4C23.D10D5×C22⋊C4D10.12D4C4⋊C47D5D10⋊Q8C4×C5⋊D4C20.17D4C23⋊D10C5×C22.D4C22.D4D10C22⋊C4C4⋊C4C22×C4C2×D4C10C2C2
# reps1222222111286422148

Matrix representation of C10.622+ 1+4 in GL6(𝔽41)

4000000
0400000
00353500
0064000
0000400
0000040
,
3200000
090000
0040000
0004000
0000322
000019
,
100000
0400000
001000
000100
000010
0000940
,
010000
4000000
001000
0064000
0000118
0000040
,
010000
100000
001000
000100
0000118
0000040

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,35,6,0,0,0,0,35,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[32,0,0,0,0,0,0,9,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,32,1,0,0,0,0,2,9],[1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,9,0,0,0,0,0,40],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,18,40],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,18,40] >;

C10.622+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{62}2_+^{1+4}
% in TeX

G:=Group("C10.62ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1331);
// by ID

G=gap.SmallGroup(320,1331);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,219,184,1571,570,12550]);
// Polycyclic

G:=Group<a,b,c,d,e|a^10=b^4=c^2=e^2=1,d^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,c*b*c=a^5*b^-1,d*b*d^-1=e*b*e=a^5*b,d*c*d^-1=e*c*e=a^5*c,e*d*e=b^2*d>;
// generators/relations

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