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G = C10.402+ 1+4order 320 = 26·5

40th non-split extension by C10 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C10.402+ 1+4, C4⋊C46D10, C5⋊D42D4, (C2×D4)⋊24D10, C4⋊D413D5, C55(D45D4), D108(C4○D4), C22.6(D4×D5), C22⋊C427D10, D10.40(C2×D4), (C22×C4)⋊18D10, C23⋊D1023C2, D10⋊D421C2, D10⋊Q814C2, Dic54D49C2, (D4×C10)⋊30C22, (C2×C20).41C23, Dic5.46(C2×D4), C10.69(C22×D4), Dic5⋊D430C2, (C2×C10).154C24, (C22×C20)⋊40C22, (C4×Dic5)⋊54C22, D10.13D412C2, C23.D552C22, C2.42(D46D10), D10⋊C467C22, Dic5.5D419C2, (C2×Dic10)⋊25C22, (C2×D20).149C22, C23.18D109C2, C10.D416C22, (C22×C10).21C23, (C2×Dic5).74C23, (C23×D5).47C22, C23.182(C22×D5), C22.175(C23×D5), (C22×Dic5)⋊20C22, (C22×D5).198C23, (C2×D4×D5)⋊12C2, C2.42(C2×D4×D5), (C4×C5⋊D4)⋊54C2, (D5×C22⋊C4)⋊5C2, C2.39(D5×C4○D4), (C2×C10).6(C2×D4), (C2×C4×D5)⋊50C22, (C5×C4⋊D4)⋊16C2, (C5×C4⋊C4)⋊12C22, (C2×D42D5)⋊14C2, C10.152(C2×C4○D4), (C2×C5⋊D4)⋊16C22, (C2×D10⋊C4)⋊36C2, (C5×C22⋊C4)⋊14C22, (C2×C4).177(C22×D5), SmallGroup(320,1282)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C10.402+ 1+4
C1C5C10C2×C10C22×D5C23×D5C2×D4×D5 — C10.402+ 1+4
C5C2×C10 — C10.402+ 1+4
C1C22C4⋊D4

Generators and relations for C10.402+ 1+4
 G = < a,b,c,d,e | a10=b4=c2=1, d2=b2, e2=a5, ab=ba, ac=ca, dad-1=a-1, ae=ea, cbc=b-1, dbd-1=ebe-1=a5b, cd=dc, ce=ec, ede-1=a5b2d >

Subgroups: 1414 in 334 conjugacy classes, 105 normal (91 characteristic)
C1, C2 [×3], C2 [×9], C4 [×10], C22, C22 [×2], C22 [×27], C5, C2×C4 [×4], C2×C4 [×15], D4 [×18], Q8 [×2], C23 [×3], C23 [×13], D5 [×5], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4, C4⋊C4 [×3], C22×C4, C22×C4 [×5], C2×D4 [×3], C2×D4 [×10], C2×Q8, C4○D4 [×4], C24 [×2], Dic5 [×2], Dic5 [×4], C20 [×4], D10 [×4], D10 [×15], C2×C10, C2×C10 [×2], C2×C10 [×8], C2×C22⋊C4 [×2], C4×D4 [×2], C22≀C2 [×2], C4⋊D4, C4⋊D4 [×2], C22⋊Q8, C22.D4 [×2], C4.4D4, C22×D4, C2×C4○D4, Dic10 [×2], C4×D5 [×5], D20 [×2], C2×Dic5 [×5], C2×Dic5 [×4], C5⋊D4 [×4], C5⋊D4 [×7], C2×C20 [×4], C2×C20, C5×D4 [×5], C22×D5 [×3], C22×D5 [×10], C22×C10 [×3], D45D4, C4×Dic5, C10.D4 [×3], D10⋊C4 [×7], C23.D5 [×3], C5×C22⋊C4 [×2], C5×C4⋊C4, C2×Dic10, C2×C4×D5 [×3], C2×D20, D4×D5 [×4], D42D5 [×4], C22×Dic5 [×2], C2×C5⋊D4 [×5], C22×C20, D4×C10 [×3], C23×D5 [×2], D5×C22⋊C4, Dic54D4, D10⋊D4, Dic5.5D4, D10.13D4, D10⋊Q8, C2×D10⋊C4, C4×C5⋊D4, C23.18D10, C23⋊D10 [×2], Dic5⋊D4, C5×C4⋊D4, C2×D4×D5, C2×D42D5, C10.402+ 1+4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×2], C24, D10 [×7], C22×D4, C2×C4○D4, 2+ 1+4, C22×D5 [×7], D45D4, D4×D5 [×2], C23×D5, C2×D4×D5, D46D10, D5×C4○D4, C10.402+ 1+4

Smallest permutation representation of C10.402+ 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 48 13 58)(2 49 14 59)(3 50 15 60)(4 41 16 51)(5 42 17 52)(6 43 18 53)(7 44 19 54)(8 45 20 55)(9 46 11 56)(10 47 12 57)(21 66 31 76)(22 67 32 77)(23 68 33 78)(24 69 34 79)(25 70 35 80)(26 61 36 71)(27 62 37 72)(28 63 38 73)(29 64 39 74)(30 65 40 75)
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 80)
(1 38 13 28)(2 37 14 27)(3 36 15 26)(4 35 16 25)(5 34 17 24)(6 33 18 23)(7 32 19 22)(8 31 20 21)(9 40 11 30)(10 39 12 29)(41 75 51 65)(42 74 52 64)(43 73 53 63)(44 72 54 62)(45 71 55 61)(46 80 56 70)(47 79 57 69)(48 78 58 68)(49 77 59 67)(50 76 60 66)
(1 28 6 23)(2 29 7 24)(3 30 8 25)(4 21 9 26)(5 22 10 27)(11 36 16 31)(12 37 17 32)(13 38 18 33)(14 39 19 34)(15 40 20 35)(41 61 46 66)(42 62 47 67)(43 63 48 68)(44 64 49 69)(45 65 50 70)(51 71 56 76)(52 72 57 77)(53 73 58 78)(54 74 59 79)(55 75 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), (1,48,13,58)(2,49,14,59)(3,50,15,60)(4,41,16,51)(5,42,17,52)(6,43,18,53)(7,44,19,54)(8,45,20,55)(9,46,11,56)(10,47,12,57)(21,66,31,76)(22,67,32,77)(23,68,33,78)(24,69,34,79)(25,70,35,80)(26,61,36,71)(27,62,37,72)(28,63,38,73)(29,64,39,74)(30,65,40,75), (41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(49,59)(50,60)(61,71)(62,72)(63,73)(64,74)(65,75)(66,76)(67,77)(68,78)(69,79)(70,80), (1,38,13,28)(2,37,14,27)(3,36,15,26)(4,35,16,25)(5,34,17,24)(6,33,18,23)(7,32,19,22)(8,31,20,21)(9,40,11,30)(10,39,12,29)(41,75,51,65)(42,74,52,64)(43,73,53,63)(44,72,54,62)(45,71,55,61)(46,80,56,70)(47,79,57,69)(48,78,58,68)(49,77,59,67)(50,76,60,66), (1,28,6,23)(2,29,7,24)(3,30,8,25)(4,21,9,26)(5,22,10,27)(11,36,16,31)(12,37,17,32)(13,38,18,33)(14,39,19,34)(15,40,20,35)(41,61,46,66)(42,62,47,67)(43,63,48,68)(44,64,49,69)(45,65,50,70)(51,71,56,76)(52,72,57,77)(53,73,58,78)(54,74,59,79)(55,75,60,80)>;

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L4A4B4C4D4E4F4G4H4I4J4K4L5A5B10A···10F10G10H10I10J10K10L10M10N20A···20H20I20J20K20L
order12222222222224444444444445510···10101010101010101020···2020202020
size1111224410101010202244410101010202020222···2444488884···48888

53 irreducible representations

dim11111111111111122222224444
type+++++++++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2C2D4D5C4○D4D10D10D10D102+ 1+4D4×D5D46D10D5×C4○D4
kernelC10.402+ 1+4D5×C22⋊C4Dic54D4D10⋊D4Dic5.5D4D10.13D4D10⋊Q8C2×D10⋊C4C4×C5⋊D4C23.18D10C23⋊D10Dic5⋊D4C5×C4⋊D4C2×D4×D5C2×D42D5C5⋊D4C4⋊D4D10C22⋊C4C4⋊C4C22×C4C2×D4C10C22C2C2
# reps11111111112111142442261444

Matrix representation of C10.402+ 1+4 in GL6(𝔽41)

770000
34400000
0040000
0004000
0000400
0000040
,
100000
010000
001000
0014000
00004018
000091
,
100000
010000
001000
000100
000010
00003240
,
100000
34400000
0040200
000100
000090
0000132
,
100000
010000
0013900
0014000
0000320
0000409

G:=sub<GL(6,GF(41))| [7,34,0,0,0,0,7,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,40,0,0,0,0,0,0,40,9,0,0,0,0,18,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,32,0,0,0,0,0,40],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,2,1,0,0,0,0,0,0,9,1,0,0,0,0,0,32],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,39,40,0,0,0,0,0,0,32,40,0,0,0,0,0,9] >;

C10.402+ 1+4 in GAP, Magma, Sage, TeX

C_{10}._{40}2_+^{1+4}
% in TeX

G:=Group("C10.40ES+(2,2)");
// GroupNames label

G:=SmallGroup(320,1282);
// by ID

G=gap.SmallGroup(320,1282);
# by ID

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

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

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