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G = C5×C22⋊Q8order 160 = 25·5

Direct product of C5 and C22⋊Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C5×C22⋊Q8, C20.62D4, C4⋊C43C10, C22⋊(C5×Q8), (C2×C10)⋊2Q8, (C2×Q8)⋊1C10, (Q8×C10)⋊8C2, C2.6(D4×C10), C4.13(C5×D4), C2.3(Q8×C10), C10.69(C2×D4), C10.20(C2×Q8), C22⋊C4.1C10, (C22×C4).5C10, C23.9(C2×C10), C10.42(C4○D4), (C22×C20).15C2, (C2×C10).77C23, (C2×C20).124C22, C22.12(C22×C10), (C22×C10).28C22, (C5×C4⋊C4)⋊12C2, C2.5(C5×C4○D4), (C2×C4).4(C2×C10), (C5×C22⋊C4).4C2, SmallGroup(160,183)

Series: Derived Chief Lower central Upper central

C1C22 — C5×C22⋊Q8
C1C2C22C2×C10C2×C20Q8×C10 — C5×C22⋊Q8
C1C22 — C5×C22⋊Q8
C1C2×C10 — C5×C22⋊Q8

Generators and relations for C5×C22⋊Q8
 G = < a,b,c,d,e | a5=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, ebe-1=bc=cb, bd=db, cd=dc, ce=ec, ede-1=d-1 >

Subgroups: 100 in 74 conjugacy classes, 48 normal (24 characteristic)
C1, C2 [×3], C2 [×2], C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C5, C2×C4 [×2], C2×C4 [×4], C2×C4 [×2], Q8 [×2], C23, C10 [×3], C10 [×2], C22⋊C4 [×2], C4⋊C4, C4⋊C4 [×2], C22×C4, C2×Q8, C20 [×2], C20 [×5], C2×C10, C2×C10 [×2], C2×C10 [×2], C22⋊Q8, C2×C20 [×2], C2×C20 [×4], C2×C20 [×2], C5×Q8 [×2], C22×C10, C5×C22⋊C4 [×2], C5×C4⋊C4, C5×C4⋊C4 [×2], C22×C20, Q8×C10, C5×C22⋊Q8
Quotients: C1, C2 [×7], C22 [×7], C5, D4 [×2], Q8 [×2], C23, C10 [×7], C2×D4, C2×Q8, C4○D4, C2×C10 [×7], C22⋊Q8, C5×D4 [×2], C5×Q8 [×2], C22×C10, D4×C10, Q8×C10, C5×C4○D4, C5×C22⋊Q8

Smallest permutation representation of C5×C22⋊Q8
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)
(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 77)(12 78)(13 79)(14 80)(15 76)(26 34)(27 35)(28 31)(29 32)(30 33)(36 41)(37 42)(38 43)(39 44)(40 45)(46 54)(47 55)(48 51)(49 52)(50 53)(56 61)(57 62)(58 63)(59 64)(60 65)(66 74)(67 75)(68 71)(69 72)(70 73)
(1 55 35 36)(2 51 31 37)(3 52 32 38)(4 53 33 39)(5 54 34 40)(6 60 80 74)(7 56 76 75)(8 57 77 71)(9 58 78 72)(10 59 79 73)(11 68 17 62)(12 69 18 63)(13 70 19 64)(14 66 20 65)(15 67 16 61)(21 47 27 41)(22 48 28 42)(23 49 29 43)(24 50 30 44)(25 46 26 45)
(1 75 35 56)(2 71 31 57)(3 72 32 58)(4 73 33 59)(5 74 34 60)(6 54 80 40)(7 55 76 36)(8 51 77 37)(9 52 78 38)(10 53 79 39)(11 42 17 48)(12 43 18 49)(13 44 19 50)(14 45 20 46)(15 41 16 47)(21 67 27 61)(22 68 28 62)(23 69 29 63)(24 70 30 64)(25 66 26 65)

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

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

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

C5×C22⋊Q8 is a maximal subgroup of
C10.29C4≀C2  C22⋊Q8.D5  (C2×C10).Q16  C10.(C4○D8)  D20.36D4  D20.37D4  C52C824D4  C22⋊Q8⋊D5  Dic10.37D4  (C2×C10)⋊Q16  C5⋊(C8.D4)  (Q8×Dic5)⋊C2  C10.502+ 1+4  C22⋊Q825D5  C10.152- 1+4  C4⋊C426D10  C10.162- 1+4  C10.172- 1+4  D2021D4  D2022D4  Dic1021D4  Dic1022D4  C10.512+ 1+4  C10.1182+ 1+4  C10.522+ 1+4  C10.532+ 1+4  C10.202- 1+4  C10.212- 1+4  C10.222- 1+4  C10.232- 1+4  C10.772- 1+4  C10.242- 1+4  C10.562+ 1+4  C10.572+ 1+4  C10.582+ 1+4  C10.262- 1+4  C5×D4×Q8

70 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G4H5A5B5C5D10A···10L10M···10T20A···20P20Q···20AF
order12222244444444555510···1010···1020···2020···20
size1111222222444411111···12···22···24···4

70 irreducible representations

dim1111111111222222
type++++++-
imageC1C2C2C2C2C5C10C10C10C10D4Q8C4○D4C5×D4C5×Q8C5×C4○D4
kernelC5×C22⋊Q8C5×C22⋊C4C5×C4⋊C4C22×C20Q8×C10C22⋊Q8C22⋊C4C4⋊C4C22×C4C2×Q8C20C2×C10C10C4C22C2
# reps12311481244222888

Matrix representation of C5×C22⋊Q8 in GL4(𝔽41) generated by

16000
01600
00370
00037
,
11600
04000
0010
00140
,
40000
04000
00400
00040
,
322000
0900
0010
0001
,
251500
21600
00139
00040
G:=sub<GL(4,GF(41))| [16,0,0,0,0,16,0,0,0,0,37,0,0,0,0,37],[1,0,0,0,16,40,0,0,0,0,1,1,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[32,0,0,0,20,9,0,0,0,0,1,0,0,0,0,1],[25,2,0,0,15,16,0,0,0,0,1,0,0,0,39,40] >;

C5×C22⋊Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes Q_8
% in TeX

G:=Group("C5xC2^2:Q8");
// GroupNames label

G:=SmallGroup(160,183);
// by ID

G=gap.SmallGroup(160,183);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,240,505,247,1514]);
// Polycyclic

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

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