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G = D106(C4⋊C4)  order 320 = 26·5

5th semidirect product of D10 and C4⋊C4 acting via C4⋊C4/C2×C4=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D106(C4⋊C4), (C22×C4)⋊8F5, C221(C4⋊F5), C42(C22⋊F5), C202(C22⋊C4), (C22×C20)⋊12C4, C5⋊(C23.7Q8), D10.33(C2×D4), (C4×D5).113D4, D10.17(C2×Q8), C23.50(C2×F5), D5.3(C4⋊D4), D10.3Q85C2, D5.4(C22⋊Q8), D10.27(C4○D4), (C22×D5).22Q8, Dic56(C22⋊C4), (C22×Dic5)⋊19C4, (C22×D5).100D4, (C22×F5).7C22, C22.90(C22×F5), C10.19(C42⋊C2), (C23×D5).135C22, (C22×D5).277C23, C2.19(D10.C23), (C2×C4×D5)⋊20C4, (C2×C4⋊F5)⋊8C2, (C2×C10)⋊2(C4⋊C4), C2.23(C2×C4⋊F5), C10.23(C2×C4⋊C4), (C2×C4).145(C2×F5), (D5×C22×C4).29C2, (C2×C20).113(C2×C4), (C2×C22⋊F5).7C2, C2.13(C2×C22⋊F5), C10.12(C2×C22⋊C4), (C2×C4×D5).368C22, (C22×C10).72(C2×C4), (C2×C10).72(C22×C4), (C2×Dic5).191(C2×C4), (C22×D5).128(C2×C4), SmallGroup(320,1103)

Series: Derived Chief Lower central Upper central

C1C2×C10 — D106(C4⋊C4)
C1C5D5D10C22×D5C22×F5C2×C4⋊F5 — D106(C4⋊C4)
C5C2×C10 — D106(C4⋊C4)
C1C22C22×C4

Generators and relations for D106(C4⋊C4)
 G = < a,b,c,d | a10=b2=c4=d4=1, bab=cac-1=a-1, dad-1=a3, cbc-1=a8b, dbd-1=a7b, dcd-1=c-1 >

Subgroups: 1002 in 234 conjugacy classes, 70 normal (32 characteristic)
C1, C2 [×3], C2 [×8], C4 [×2], C4 [×8], C22, C22 [×2], C22 [×20], C5, C2×C4 [×2], C2×C4 [×28], C23, C23 [×10], D5 [×4], D5 [×2], C10 [×3], C10 [×2], C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C22×C4 [×13], C24, Dic5 [×2], Dic5, C20 [×2], C20, F5 [×4], D10 [×2], D10 [×6], D10 [×10], C2×C10, C2×C10 [×2], C2×C10 [×2], C2.C42 [×2], C2×C22⋊C4 [×2], C2×C4⋊C4 [×2], C23×C4, C4×D5 [×4], C4×D5 [×6], C2×Dic5 [×2], C2×Dic5 [×2], C2×C20 [×2], C2×C20 [×2], C2×F5 [×12], C22×D5 [×2], C22×D5 [×4], C22×D5 [×4], C22×C10, C23.7Q8, C4⋊F5 [×4], C22⋊F5 [×4], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C22×C20, C22×F5 [×4], C23×D5, D10.3Q8 [×2], C2×C4⋊F5 [×2], C2×C22⋊F5 [×2], D5×C22×C4, D106(C4⋊C4)
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], F5, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C2×F5 [×3], C23.7Q8, C4⋊F5 [×2], C22⋊F5 [×2], C22×F5, C2×C4⋊F5, D10.C23, C2×C22⋊F5, D106(C4⋊C4)

Smallest permutation representation of D106(C4⋊C4)
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 34)(2 33)(3 32)(4 31)(5 40)(6 39)(7 38)(8 37)(9 36)(10 35)(11 69)(12 68)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 70)(21 43)(22 42)(23 41)(24 50)(25 49)(26 48)(27 47)(28 46)(29 45)(30 44)(51 73)(52 72)(53 71)(54 80)(55 79)(56 78)(57 77)(58 76)(59 75)(60 74)
(1 65 30 55)(2 64 21 54)(3 63 22 53)(4 62 23 52)(5 61 24 51)(6 70 25 60)(7 69 26 59)(8 68 27 58)(9 67 28 57)(10 66 29 56)(11 50 75 40)(12 49 76 39)(13 48 77 38)(14 47 78 37)(15 46 79 36)(16 45 80 35)(17 44 71 34)(18 43 72 33)(19 42 73 32)(20 41 74 31)
(1 40 6 35)(2 37 5 38)(3 34 4 31)(7 32 10 33)(8 39 9 36)(11 60 16 55)(12 57 15 58)(13 54 14 51)(17 52 20 53)(18 59 19 56)(21 47 24 48)(22 44 23 41)(25 45 30 50)(26 42 29 43)(27 49 28 46)(61 77 64 78)(62 74 63 71)(65 75 70 80)(66 72 69 73)(67 79 68 76)

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

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

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

44 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H4I···4P 5 10A···10G20A···20H
order122222222222444444444···4510···1020···20
size1111225555101022221010101020···2044···44···4

44 irreducible representations

dim111111112222444444
type+++++++-++++
imageC1C2C2C2C2C4C4C4D4D4Q8C4○D4F5C2×F5C2×F5C22⋊F5C4⋊F5D10.C23
kernelD106(C4⋊C4)D10.3Q8C2×C4⋊F5C2×C22⋊F5D5×C22×C4C2×C4×D5C22×Dic5C22×C20C4×D5C22×D5C22×D5D10C22×C4C2×C4C23C4C22C2
# reps122214224224121444

Matrix representation of D106(C4⋊C4) in GL6(𝔽41)

100000
010000
0000140
000010
0040010
0004010
,
100000
010000
003822193
00190223
00383220
00031922
,
3220000
090000
002701434
00271470
00071427
003414027
,
3200000
190000
00383220
00193038
00380319
00022338

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,1,1,1,1,0,0,40,0,0,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,38,19,38,0,0,0,22,0,3,3,0,0,19,22,22,19,0,0,3,3,0,22],[32,0,0,0,0,0,2,9,0,0,0,0,0,0,27,27,0,34,0,0,0,14,7,14,0,0,14,7,14,0,0,0,34,0,27,27],[32,1,0,0,0,0,0,9,0,0,0,0,0,0,38,19,38,0,0,0,3,3,0,22,0,0,22,0,3,3,0,0,0,38,19,38] >;

D106(C4⋊C4) in GAP, Magma, Sage, TeX

D_{10}\rtimes_6(C_4\rtimes C_4)
% in TeX

G:=Group("D10:6(C4:C4)");
// GroupNames label

G:=SmallGroup(320,1103);
// by ID

G=gap.SmallGroup(320,1103);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,477,232,422,6278,1595]);
// Polycyclic

G:=Group<a,b,c,d|a^10=b^2=c^4=d^4=1,b*a*b=c*a*c^-1=a^-1,d*a*d^-1=a^3,c*b*c^-1=a^8*b,d*b*d^-1=a^7*b,d*c*d^-1=c^-1>;
// generators/relations

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