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G = C4228D10order 320 = 26·5

28th semidirect product of C42 and D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C4228D10, C10.802+ 1+4, C41D49D5, (C2×D4)⋊13D10, (C4×C20)⋊37C22, C23⋊D1028C2, (D4×C10)⋊34C22, C422D519C2, Dic5⋊D439C2, (C2×C10).264C24, (C2×C20).638C23, C2.84(D46D10), C23.D538C22, C23.70(C22×D5), C55(C22.54C24), C10.D437C22, (C22×C10).78C23, (C23×D5).73C22, C22.285(C23×D5), D10⋊C4.75C22, C23.18D1028C2, (C2×Dic5).138C23, (C22×Dic5)⋊30C22, (C22×D5).118C23, (C5×C41D4)⋊15C2, (C2×C4).216(C22×D5), (C2×C5⋊D4).80C22, SmallGroup(320,1392)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C4228D10
C1C5C10C2×C10C22×D5C23×D5C23⋊D10 — C4228D10
C5C2×C10 — C4228D10
C1C22C41D4

Generators and relations for C4228D10
 G = < a,b,c,d | a4=b4=c10=d2=1, ab=ba, cac-1=a-1, dad=a-1b2, cbc-1=b-1, dbd=a2b, dcd=c-1 >

Subgroups: 1062 in 252 conjugacy classes, 91 normal (12 characteristic)
C1, C2 [×3], C2 [×6], C4 [×9], C22, C22 [×22], C5, C2×C4 [×3], C2×C4 [×9], D4 [×12], C23, C23 [×3], C23 [×5], D5 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×12], C4⋊C4 [×6], C22×C4 [×3], C2×D4 [×6], C2×D4 [×6], C24, Dic5 [×6], C20 [×3], D10 [×10], C2×C10, C2×C10 [×12], C22≀C2 [×3], C4⋊D4 [×6], C22.D4 [×3], C422C2 [×2], C41D4, C2×Dic5 [×6], C2×Dic5 [×3], C5⋊D4 [×6], C2×C20 [×3], C5×D4 [×6], C22×D5 [×2], C22×D5 [×3], C22×C10, C22×C10 [×3], C22.54C24, C10.D4 [×6], D10⋊C4 [×6], C23.D5 [×6], C4×C20, C22×Dic5 [×3], C2×C5⋊D4 [×6], D4×C10 [×6], C23×D5, C422D5 [×2], C23.18D10 [×3], C23⋊D10 [×3], Dic5⋊D4 [×6], C5×C41D4, C4228D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C24, D10 [×7], 2+ 1+4 [×3], C22×D5 [×7], C22.54C24, C23×D5, D46D10 [×3], C4228D10

Smallest permutation representation of C4228D10
On 80 points
Generators in S80
(1 66 18 61)(2 62 19 67)(3 68 20 63)(4 64 16 69)(5 70 17 65)(6 54 13 59)(7 60 14 55)(8 56 15 51)(9 52 11 57)(10 58 12 53)(21 49 75 38)(22 39 76 50)(23 41 77 40)(24 31 78 42)(25 43 79 32)(26 33 80 44)(27 45 71 34)(28 35 72 46)(29 47 73 36)(30 37 74 48)
(1 23 8 28)(2 29 9 24)(3 25 10 30)(4 21 6 26)(5 27 7 22)(11 78 19 73)(12 74 20 79)(13 80 16 75)(14 76 17 71)(15 72 18 77)(31 62 47 52)(32 53 48 63)(33 64 49 54)(34 55 50 65)(35 66 41 56)(36 57 42 67)(37 68 43 58)(38 59 44 69)(39 70 45 60)(40 51 46 61)
(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 12)(2 11)(3 15)(4 14)(5 13)(6 17)(7 16)(8 20)(9 19)(10 18)(21 22)(23 30)(24 29)(25 28)(26 27)(31 42)(32 41)(33 50)(34 49)(35 48)(36 47)(37 46)(38 45)(39 44)(40 43)(51 53)(54 60)(55 59)(56 58)(61 63)(64 70)(65 69)(66 68)(71 80)(72 79)(73 78)(74 77)(75 76)

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

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

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

47 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D···4I5A5B10A···10F10G···10N20A···20L
order12222222224444···45510···1010···1020···20
size11114444202044420···20222···28···84···4

47 irreducible representations

dim11111122244
type++++++++++
imageC1C2C2C2C2C2D5D10D102+ 1+4D46D10
kernelC4228D10C422D5C23.18D10C23⋊D10Dic5⋊D4C5×C41D4C41D4C42C2×D4C10C2
# reps1233612212312

Matrix representation of C4228D10 in GL8(𝔽41)

186000000
3523000000
001860000
0035230000
000026077
000039303612
00001312132
00002529213
,
00100000
00010000
400000000
040000000
00002434383
0000617380
0000028186
000013283523
,
16000000
356000000
0040350000
006350000
0000035316
00007351628
000000407
000000347
,
61000000
635000000
00610000
006350000
0000635140
0000403510
000000740
000000734

G:=sub<GL(8,GF(41))| [18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,18,35,0,0,0,0,0,0,6,23,0,0,0,0,0,0,0,0,26,39,13,25,0,0,0,0,0,30,12,29,0,0,0,0,7,36,13,2,0,0,0,0,7,12,2,13],[0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,24,6,0,13,0,0,0,0,34,17,28,28,0,0,0,0,38,38,18,35,0,0,0,0,3,0,6,23],[1,35,0,0,0,0,0,0,6,6,0,0,0,0,0,0,0,0,40,6,0,0,0,0,0,0,35,35,0,0,0,0,0,0,0,0,0,7,0,0,0,0,0,0,35,35,0,0,0,0,0,0,3,16,40,34,0,0,0,0,16,28,7,7],[6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,6,0,0,0,0,0,0,1,35,0,0,0,0,0,0,0,0,6,40,0,0,0,0,0,0,35,35,0,0,0,0,0,0,1,1,7,7,0,0,0,0,40,0,40,34] >;

C4228D10 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{28}D_{10}
% in TeX

G:=Group("C4^2:28D10");
// GroupNames label

G:=SmallGroup(320,1392);
// by ID

G=gap.SmallGroup(320,1392);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,758,219,1571,570,297,136,12550]);
// Polycyclic

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

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