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## G = Q8×C11⋊C5order 440 = 23·5·11

### Direct product of Q8 and C11⋊C5

Aliases: Q8×C11⋊C5, C44.3C10, (Q8×C11)⋊C5, C112(C5×Q8), C22.8(C2×C10), C4.(C2×C11⋊C5), (C4×C11⋊C5).3C2, C2.3(C22×C11⋊C5), (C2×C11⋊C5).8C22, SmallGroup(440,14)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — Q8×C11⋊C5
 Chief series C1 — C11 — C22 — C2×C11⋊C5 — C4×C11⋊C5 — Q8×C11⋊C5
 Lower central C11 — C22 — Q8×C11⋊C5
 Upper central C1 — C2 — Q8

Generators and relations for Q8×C11⋊C5
G = < a,b,c,d | a4=c11=d5=1, b2=a2, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

Smallest permutation representation of Q8×C11⋊C5
On 88 points
Generators in S88
(1 34 12 23)(2 35 13 24)(3 36 14 25)(4 37 15 26)(5 38 16 27)(6 39 17 28)(7 40 18 29)(8 41 19 30)(9 42 20 31)(10 43 21 32)(11 44 22 33)(45 67 56 78)(46 68 57 79)(47 69 58 80)(48 70 59 81)(49 71 60 82)(50 72 61 83)(51 73 62 84)(52 74 63 85)(53 75 64 86)(54 76 65 87)(55 77 66 88)
(1 56 12 45)(2 57 13 46)(3 58 14 47)(4 59 15 48)(5 60 16 49)(6 61 17 50)(7 62 18 51)(8 63 19 52)(9 64 20 53)(10 65 21 54)(11 66 22 55)(23 78 34 67)(24 79 35 68)(25 80 36 69)(26 81 37 70)(27 82 38 71)(28 83 39 72)(29 84 40 73)(30 85 41 74)(31 86 42 75)(32 87 43 76)(33 88 44 77)
(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 81 82 83 84 85 86 87 88)
(2 5 6 10 4)(3 9 11 8 7)(13 16 17 21 15)(14 20 22 19 18)(24 27 28 32 26)(25 31 33 30 29)(35 38 39 43 37)(36 42 44 41 40)(46 49 50 54 48)(47 53 55 52 51)(57 60 61 65 59)(58 64 66 63 62)(68 71 72 76 70)(69 75 77 74 73)(79 82 83 87 81)(80 86 88 85 84)

G:=sub<Sym(88)| (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88), (1,56,12,45)(2,57,13,46)(3,58,14,47)(4,59,15,48)(5,60,16,49)(6,61,17,50)(7,62,18,51)(8,63,19,52)(9,64,20,53)(10,65,21,54)(11,66,22,55)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77), (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)(78,79,80,81,82,83,84,85,86,87,88), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84)>;

G:=Group( (1,34,12,23)(2,35,13,24)(3,36,14,25)(4,37,15,26)(5,38,16,27)(6,39,17,28)(7,40,18,29)(8,41,19,30)(9,42,20,31)(10,43,21,32)(11,44,22,33)(45,67,56,78)(46,68,57,79)(47,69,58,80)(48,70,59,81)(49,71,60,82)(50,72,61,83)(51,73,62,84)(52,74,63,85)(53,75,64,86)(54,76,65,87)(55,77,66,88), (1,56,12,45)(2,57,13,46)(3,58,14,47)(4,59,15,48)(5,60,16,49)(6,61,17,50)(7,62,18,51)(8,63,19,52)(9,64,20,53)(10,65,21,54)(11,66,22,55)(23,78,34,67)(24,79,35,68)(25,80,36,69)(26,81,37,70)(27,82,38,71)(28,83,39,72)(29,84,40,73)(30,85,41,74)(31,86,42,75)(32,87,43,76)(33,88,44,77), (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)(78,79,80,81,82,83,84,85,86,87,88), (2,5,6,10,4)(3,9,11,8,7)(13,16,17,21,15)(14,20,22,19,18)(24,27,28,32,26)(25,31,33,30,29)(35,38,39,43,37)(36,42,44,41,40)(46,49,50,54,48)(47,53,55,52,51)(57,60,61,65,59)(58,64,66,63,62)(68,71,72,76,70)(69,75,77,74,73)(79,82,83,87,81)(80,86,88,85,84) );

G=PermutationGroup([[(1,34,12,23),(2,35,13,24),(3,36,14,25),(4,37,15,26),(5,38,16,27),(6,39,17,28),(7,40,18,29),(8,41,19,30),(9,42,20,31),(10,43,21,32),(11,44,22,33),(45,67,56,78),(46,68,57,79),(47,69,58,80),(48,70,59,81),(49,71,60,82),(50,72,61,83),(51,73,62,84),(52,74,63,85),(53,75,64,86),(54,76,65,87),(55,77,66,88)], [(1,56,12,45),(2,57,13,46),(3,58,14,47),(4,59,15,48),(5,60,16,49),(6,61,17,50),(7,62,18,51),(8,63,19,52),(9,64,20,53),(10,65,21,54),(11,66,22,55),(23,78,34,67),(24,79,35,68),(25,80,36,69),(26,81,37,70),(27,82,38,71),(28,83,39,72),(29,84,40,73),(30,85,41,74),(31,86,42,75),(32,87,43,76),(33,88,44,77)], [(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),(78,79,80,81,82,83,84,85,86,87,88)], [(2,5,6,10,4),(3,9,11,8,7),(13,16,17,21,15),(14,20,22,19,18),(24,27,28,32,26),(25,31,33,30,29),(35,38,39,43,37),(36,42,44,41,40),(46,49,50,54,48),(47,53,55,52,51),(57,60,61,65,59),(58,64,66,63,62),(68,71,72,76,70),(69,75,77,74,73),(79,82,83,87,81),(80,86,88,85,84)]])

35 conjugacy classes

 class 1 2 4A 4B 4C 5A 5B 5C 5D 10A 10B 10C 10D 11A 11B 20A ··· 20L 22A 22B 44A ··· 44F order 1 2 4 4 4 5 5 5 5 10 10 10 10 11 11 20 ··· 20 22 22 44 ··· 44 size 1 1 2 2 2 11 11 11 11 11 11 11 11 5 5 22 ··· 22 5 5 10 ··· 10

35 irreducible representations

 dim 1 1 1 1 10 2 2 5 5 type + + - image C1 C2 C5 C10 Q8×C11⋊C5 Q8 C5×Q8 C11⋊C5 C2×C11⋊C5 kernel Q8×C11⋊C5 C4×C11⋊C5 Q8×C11 C44 C1 C11⋊C5 C11 Q8 C4 # reps 1 3 4 12 2 1 4 2 6

Matrix representation of Q8×C11⋊C5 in GL7(𝔽661)

 660 572 0 0 0 0 0 104 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
,
 117 366 0 0 0 0 0 24 544 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 660 0 0 0 0 0 0 0 660
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 44 1 660 45 1 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 197 0 0 0 0 0 0 0 197 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 616 659 1 615 43 0 0 615 43 1 659 44 0 0 0 1 0 0 0

G:=sub<GL(7,GF(661))| [660,104,0,0,0,0,0,572,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[117,24,0,0,0,0,0,366,544,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,660,0,0,0,0,0,0,0,660],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,44,1,0,0,0,0,0,1,0,1,0,0,0,0,660,0,0,1,0,0,0,45,0,0,0,1,0,0,1,0,0,0,0],[197,0,0,0,0,0,0,0,197,0,0,0,0,0,0,0,1,0,616,615,0,0,0,0,0,659,43,1,0,0,0,0,1,1,0,0,0,0,1,615,659,0,0,0,0,0,43,44,0] >;

Q8×C11⋊C5 in GAP, Magma, Sage, TeX

Q_8\times C_{11}\rtimes C_5
% in TeX

G:=Group("Q8xC11:C5");
// GroupNames label

G:=SmallGroup(440,14);
// by ID

G=gap.SmallGroup(440,14);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-11,100,221,106,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^11=d^5=1,b^2=a^2,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations

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