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G = Q8×C11⋊C5order 440 = 23·5·11

Direct product of Q8 and C11⋊C5

direct product, metacyclic, supersoluble, monomial

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

C1C22 — Q8×C11⋊C5
C1C11C22C2×C11⋊C5C4×C11⋊C5 — Q8×C11⋊C5
C11C22 — Q8×C11⋊C5
C1C2Q8

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 >

11C5
11C10
11C20
11C20
11C20
11C5×Q8

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 4A4B4C5A5B5C5D10A10B10C10D11A11B20A···20L22A22B44A···44F
order12444555510101010111120···20222244···44
size1122211111111111111115522···225510···10

35 irreducible representations

dim1111102255
type++-
imageC1C2C5C10Q8×C11⋊C5Q8C5×Q8C11⋊C5C2×C11⋊C5
kernelQ8×C11⋊C5C4×C11⋊C5Q8×C11C44C1C11⋊C5C11Q8C4
# reps1341221426

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

66057200000
104100000
0010000
0001000
0000100
0000010
0000001
,
11736600000
2454400000
006600000
000660000
000066000
000006600
000000660
,
1000000
0100000
00441660451
0010000
0001000
0000100
0000010
,
197000000
019700000
0010000
0000010
00616659161543
0061543165944
0001000

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

Export

Subgroup lattice of Q8×C11⋊C5 in TeX

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