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G = (Q8×C10)⋊C4  order 320 = 26·5

4th semidirect product of Q8×C10 and C4 acting faithfully

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (Q8×D5)⋊7C4, (C2×Q8)⋊4F5, (Q8×C10)⋊4C4, Q8.7(C2×F5), Q8⋊F5⋊3C2, (C4×D5).41D4, C4⋊F5.8C22, D10.98(C2×D4), D5⋊C8.7C22, Dic5.8(C2×D4), C5⋊(C23.38D4), C4.18(C22×F5), (C2×Dic10)⋊12C4, C20.18(C22×C4), Dic10.8(C2×C4), D5⋊M4(2).5C2, (C4×D5).40C23, C4.17(C22⋊F5), (Q8×D5).11C22, C20.17(C22⋊C4), (C2×Dic5).122D4, (C22×D5).148D4, D5.4(C8.C22), D10.46(C22⋊C4), C22.28(C22⋊F5), Dic5.13(C22⋊C4), D10.C23.5C2, (C2×Q8×D5).10C2, (C2×C4).39(C2×F5), (C5×Q8).7(C2×C4), (C2×C20).60(C2×C4), (C4×D5).24(C2×C4), C2.27(C2×C22⋊F5), C10.26(C2×C22⋊C4), (C2×C4×D5).205C22, (C2×C10).58(C22⋊C4), SmallGroup(320,1120)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C20 — (Q8×C10)⋊C4
Chief series
C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D10.C23 — (Q8×C10)⋊C4
Lower central
C5 — C10 — C20 — (Q8×C10)⋊C4

Subgroups: 602 in 150 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×8], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4 [×13], Q8 [×2], Q8 [×8], C23, D5 [×2], D5, C10, C10, C42, C22⋊C4, C4⋊C4 [×2], C2×C8 [×2], M4(2) [×2], C22×C4 [×2], C2×Q8, C2×Q8 [×8], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], F5 [×2], D10 [×2], D10 [×2], C2×C10, Q8⋊C4 [×4], C42⋊C2, C2×M4(2), C22×Q8, C5⋊C8 [×2], Dic10 [×2], Dic10 [×5], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5, C2×C20, C2×C20, C5×Q8 [×2], C5×Q8, C2×F5 [×2], C22×D5, C23.38D4, D5⋊C8 [×2], C4.F5, C4×F5, C4⋊F5 [×2], C22.F5, C22⋊F5, C2×Dic10, C2×Dic10, C2×C4×D5, C2×C4×D5, Q8×D5 [×4], Q8×D5 [×2], Q8×C10, Q8⋊F5 [×4], D5⋊M4(2), D10.C23, C2×Q8×D5, (Q8×C10)⋊C4

Quotients:
C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C2×C22⋊C4, C8.C22 [×2], C2×F5 [×3], C23.38D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (Q8×C10)⋊C4

Generators and relations
 G = < a,b,c,d | a10=b4=d4=1, c2=b2, ab=ba, ac=ca, dad-1=a3b2, cbc-1=b-1, bd=db, dcd-1=a5b-1c >

Smallest permutation representation
►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)

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

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

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

Matrix representation ►G ⊆ GL8(𝔽41)

040100000
040010000
040000000
140000000
0000034437
0000703737
0000374034
00004470
,
10000000
01000000
00100000
00010000
000034044
0000034437
00004470
000043707
,
10000000
01000000
00100000
00010000
0000374034
00004470
000007374
000034044
,
00100000
10000000
00010000
01000000
00009000
000003200
00000009
00000090

G:=sub<GL(8,GF(41))| [0,0,0,1,0,0,0,0,40,40,40,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4,0,0,0,0,4,37,0,7,0,0,0,0,37,37,34,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,34,0,4,4,0,0,0,0,0,34,4,37,0,0,0,0,4,4,7,0,0,0,0,0,4,37,0,7],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,37,4,0,34,0,0,0,0,4,4,7,0,0,0,0,0,0,7,37,4,0,0,0,0,34,0,4,4],[0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0] >;
 

32 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E4F4G···4L 5 8A8B8C8D10A10B10C20A···20F
order1222224444444···45888810101020···20
size11255102244101020···204202020204448···8

32 irreducible representations

dim111111112224444448
type+++++++++-++++-
imageC1C2C2C2C2C4C4C4D4D4D4F5C8.C22C2×F5C2×F5C22⋊F5C22⋊F5(Q8×C10)⋊C4
kernel(Q8×C10)⋊C4Q8⋊F5D5⋊M4(2)D10.C23C2×Q8×D5C2×Dic10Q8×D5Q8×C10C4×D5C2×Dic5C22×D5C2×Q8D5C2×C4Q8C4C22C1
# reps141112422111212222

In GAP, Magma, Sage, TeX 

(Q_8\times C_{10})\rtimes C_4
 
% in TeX
 
G:=Group("(Q8xC10):C4");
 
// GroupNames label
 
G:=SmallGroup(320,1120);
 
// by ID
 
G=gap.SmallGroup(320,1120);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,1684,438,102,6278,1595]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=b^4=d^4=1,c^2=b^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^3*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=a^5*b^-1*c>;
 
// generators/relations
 

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