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G = C24.32D10order 320 = 26·5

32nd non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.32D10, C10.242+ 1+4, (D4×Dic5)⋊11C2, C22≀C2.2D5, C22⋊C4.1D10, (C2×D4).149D10, (C2×C20).26C23, C4⋊Dic524C22, C20.17D411C2, (C2×C10).131C24, (C4×Dic5)⋊13C22, C10.D47C22, C23.D512C22, C2.26(D46D10), C54(C22.45C24), (C2×Dic10)⋊19C22, (D4×C10).110C22, C23.D1011C2, C23.18D103C2, C23.11D102C2, (C23×C10).67C22, C23.176(C22×D5), C22.152(C23×D5), Dic5.14D412C2, C22.17(D42D5), (C22×C10).180C23, (C2×Dic5).230C23, (C22×Dic5)⋊11C22, C10.76(C2×C4○D4), (C5×C22≀C2).2C2, C2.27(C2×D42D5), (C2×C23.D5)⋊19C2, (C2×C4).26(C22×D5), (C2×C10).43(C4○D4), (C5×C22⋊C4).2C22, SmallGroup(320,1259)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.32D10
C1C5C10C2×C10C2×Dic5C22×Dic5D4×Dic5 — C24.32D10
C5C2×C10 — C24.32D10
C1C22C22≀C2

Generators and relations for C24.32D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e10=1, f2=d, ab=ba, eae-1=faf-1=ac=ca, ad=da, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >

Subgroups: 758 in 248 conjugacy classes, 99 normal (27 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×11], C22, C22 [×4], C22 [×14], C5, C2×C4, C2×C4 [×2], C2×C4 [×15], D4 [×5], Q8, C23 [×2], C23 [×2], C23 [×5], C10, C10 [×2], C10 [×6], C42 [×3], C22⋊C4, C22⋊C4 [×2], C22⋊C4 [×11], C4⋊C4 [×8], C22×C4 [×5], C2×D4, C2×D4 [×2], C2×Q8, C24, Dic5 [×8], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×14], C2×C22⋊C4 [×2], C42⋊C2 [×2], C4×D4 [×2], C22≀C2, C22⋊Q8 [×2], C22.D4 [×3], C4.4D4, C422C2 [×2], Dic10, C2×Dic5 [×8], C2×Dic5 [×7], C2×C20, C2×C20 [×2], C5×D4 [×5], C22×C10 [×2], C22×C10 [×2], C22×C10 [×5], C22.45C24, C4×Dic5, C4×Dic5 [×2], C10.D4 [×6], C4⋊Dic5 [×2], C23.D5, C23.D5 [×10], C5×C22⋊C4, C5×C22⋊C4 [×2], C2×Dic10, C22×Dic5, C22×Dic5 [×4], D4×C10, D4×C10 [×2], C23×C10, C23.11D10 [×2], Dic5.14D4 [×2], C23.D10 [×2], D4×Dic5 [×2], C23.18D10, C23.18D10 [×2], C20.17D4, C2×C23.D5 [×2], C5×C22≀C2, C24.32D10
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, D42D5 [×4], C23×D5, C2×D42D5 [×2], D46D10, C24.32D10

Smallest permutation representation of C24.32D10
On 80 points
Generators in S80
(2 20)(4 12)(6 14)(8 16)(10 18)(21 78)(23 80)(25 72)(27 74)(29 76)(32 50)(34 42)(36 44)(38 46)(40 48)(52 69)(54 61)(56 63)(58 65)(60 67)
(1 33)(3 35)(5 37)(7 39)(9 31)(11 43)(13 45)(15 47)(17 49)(19 41)(21 60)(22 79)(23 52)(24 71)(25 54)(26 73)(27 56)(28 75)(29 58)(30 77)(51 68)(53 70)(55 62)(57 64)(59 66)(61 72)(63 74)(65 76)(67 78)(69 80)
(1 19)(2 20)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(9 17)(10 18)(21 78)(22 79)(23 80)(24 71)(25 72)(26 73)(27 74)(28 75)(29 76)(30 77)(31 49)(32 50)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)(51 68)(52 69)(53 70)(54 61)(55 62)(56 63)(57 64)(58 65)(59 66)(60 67)
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 31)(10 32)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 49)(18 50)(19 41)(20 42)(21 67)(22 68)(23 69)(24 70)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(51 79)(52 80)(53 71)(54 72)(55 73)(56 74)(57 75)(58 76)(59 77)(60 78)
(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 76 33 58)(2 75 34 57)(3 74 35 56)(4 73 36 55)(5 72 37 54)(6 71 38 53)(7 80 39 52)(8 79 40 51)(9 78 31 60)(10 77 32 59)(11 27 43 63)(12 26 44 62)(13 25 45 61)(14 24 46 70)(15 23 47 69)(16 22 48 68)(17 21 49 67)(18 30 50 66)(19 29 41 65)(20 28 42 64)

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

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

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

53 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D···4K4L4M4N4O5A5B10A···10F10G···10R10S10T20A···20F
order12222222224444···444445510···1010···10101020···20
size111122224444410···1020202020222···24···4888···8

53 irreducible representations

dim11111111122222444
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2D5C4○D4D10D10D102+ 1+4D42D5D46D10
kernelC24.32D10C23.11D10Dic5.14D4C23.D10D4×Dic5C23.18D10C20.17D4C2×C23.D5C5×C22≀C2C22≀C2C2×C10C22⋊C4C2×D4C24C10C22C2
# reps12222312128662184

Matrix representation of C24.32D10 in GL6(𝔽41)

100000
010000
001000
000100
000010
0000040
,
4000000
1810000
0040000
0004000
0000400
000001
,
100000
010000
001000
000100
0000400
0000040
,
4000000
0400000
001000
000100
0000400
0000040
,
4090000
010000
007700
00344000
000001
000010
,
3200000
0320000
00271100
00271400
000009
000090

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,9,1,0,0,0,0,0,0,7,34,0,0,0,0,7,40,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[32,0,0,0,0,0,0,32,0,0,0,0,0,0,27,27,0,0,0,0,11,14,0,0,0,0,0,0,0,9,0,0,0,0,9,0] >;

C24.32D10 in GAP, Magma, Sage, TeX

C_2^4._{32}D_{10}
% in TeX

G:=Group("C2^4.32D10");
// GroupNames label

G:=SmallGroup(320,1259);
// by ID

G=gap.SmallGroup(320,1259);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,219,1571,297,12550]);
// Polycyclic

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

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