Copied to
clipboard

## G = C5×C22⋊Q8order 160 = 25·5

### Direct product of C5 and C22⋊Q8

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22⋊Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — Q8×C10 — C5×C22⋊Q8
 Lower central C1 — C22 — C5×C22⋊Q8
 Upper central C1 — C2×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, C2, C4, C4, C22, C22, C22, C5, C2×C4, C2×C4, C2×C4, Q8, C23, C10, C10, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C2×Q8, C20, C20, C2×C10, C2×C10, C2×C10, C22⋊Q8, C2×C20, C2×C20, C2×C20, C5×Q8, C22×C10, C5×C22⋊C4, C5×C4⋊C4, C5×C4⋊C4, C22×C20, Q8×C10, C5×C22⋊Q8
Quotients: C1, C2, C22, C5, D4, Q8, C23, C10, C2×D4, C2×Q8, C4○D4, C2×C10, C22⋊Q8, C5×D4, C5×Q8, 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 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)(36 41)(37 42)(38 43)(39 44)(40 45)(46 51)(47 52)(48 53)(49 54)(50 55)(56 61)(57 62)(58 63)(59 64)(60 65)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 46 26 36)(2 47 27 37)(3 48 28 38)(4 49 29 39)(5 50 30 40)(6 71 16 61)(7 72 17 62)(8 73 18 63)(9 74 19 64)(10 75 20 65)(11 56 76 66)(12 57 77 67)(13 58 78 68)(14 59 79 69)(15 60 80 70)(21 51 31 41)(22 52 32 42)(23 53 33 43)(24 54 34 44)(25 55 35 45)
(1 66 26 56)(2 67 27 57)(3 68 28 58)(4 69 29 59)(5 70 30 60)(6 41 16 51)(7 42 17 52)(8 43 18 53)(9 44 19 54)(10 45 20 55)(11 46 76 36)(12 47 77 37)(13 48 78 38)(14 49 79 39)(15 50 80 40)(21 71 31 61)(22 72 32 62)(23 73 33 63)(24 74 34 64)(25 75 35 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), (6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,21)(2,22)(3,23)(4,24)(5,25)(6,76)(7,77)(8,78)(9,79)(10,80)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35)(36,41)(37,42)(38,43)(39,44)(40,45)(46,51)(47,52)(48,53)(49,54)(50,55)(56,61)(57,62)(58,63)(59,64)(60,65)(66,71)(67,72)(68,73)(69,74)(70,75), (1,46,26,36)(2,47,27,37)(3,48,28,38)(4,49,29,39)(5,50,30,40)(6,71,16,61)(7,72,17,62)(8,73,18,63)(9,74,19,64)(10,75,20,65)(11,56,76,66)(12,57,77,67)(13,58,78,68)(14,59,79,69)(15,60,80,70)(21,51,31,41)(22,52,32,42)(23,53,33,43)(24,54,34,44)(25,55,35,45), (1,66,26,56)(2,67,27,57)(3,68,28,58)(4,69,29,59)(5,70,30,60)(6,41,16,51)(7,42,17,52)(8,43,18,53)(9,44,19,54)(10,45,20,55)(11,46,76,36)(12,47,77,37)(13,48,78,38)(14,49,79,39)(15,50,80,40)(21,71,31,61)(22,72,32,62)(23,73,33,63)(24,74,34,64)(25,75,35,65)>;

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

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

70 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 5C 5D 10A ··· 10L 10M ··· 10T 20A ··· 20P 20Q ··· 20AF order 1 2 2 2 2 2 4 4 4 4 4 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 2 2 4 4 4 4 1 1 1 1 1 ··· 1 2 ··· 2 2 ··· 2 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 type + + + + + + - image C1 C2 C2 C2 C2 C5 C10 C10 C10 C10 D4 Q8 C4○D4 C5×D4 C5×Q8 C5×C4○D4 kernel C5×C22⋊Q8 C5×C22⋊C4 C5×C4⋊C4 C22×C20 Q8×C10 C22⋊Q8 C22⋊C4 C4⋊C4 C22×C4 C2×Q8 C20 C2×C10 C10 C4 C22 C2 # reps 1 2 3 1 1 4 8 12 4 4 2 2 2 8 8 8

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

 16 0 0 0 0 16 0 0 0 0 37 0 0 0 0 37
,
 1 16 0 0 0 40 0 0 0 0 1 0 0 0 1 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 32 20 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 25 15 0 0 2 16 0 0 0 0 1 39 0 0 0 40
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

׿
×
𝔽