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## G = (C2×Q8)⋊4F5order 320 = 26·5

### 2nd semidirect product of C2×Q8 and F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — (C2×Q8)⋊4F5
 Chief series C1 — C5 — C10 — D10 — C4×D5 — C4⋊F5 — D10.C23 — (C2×Q8)⋊4F5
 Lower central C5 — C10 — C20 — (C2×Q8)⋊4F5
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for (C2×Q8)⋊4F5
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 >

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, (C2×Q8)⋊4F5
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, (C2×Q8)⋊4F5

Smallest permutation representation of (C2×Q8)⋊4F5
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 44 19 62)(2 45 20 63)(3 46 11 64)(4 47 12 65)(5 48 13 66)(6 49 14 67)(7 50 15 68)(8 41 16 69)(9 42 17 70)(10 43 18 61)(21 34 60 76)(22 35 51 77)(23 36 52 78)(24 37 53 79)(25 38 54 80)(26 39 55 71)(27 40 56 72)(28 31 57 73)(29 32 58 74)(30 33 59 75)
(1 22 19 51)(2 23 20 52)(3 24 11 53)(4 25 12 54)(5 26 13 55)(6 27 14 56)(7 28 15 57)(8 29 16 58)(9 30 17 59)(10 21 18 60)(31 50 73 68)(32 41 74 69)(33 42 75 70)(34 43 76 61)(35 44 77 62)(36 45 78 63)(37 46 79 64)(38 47 80 65)(39 48 71 66)(40 49 72 67)
(2 16 10 12)(3 5 9 7)(4 20 8 18)(6 14)(11 13 17 15)(21 75 23 79)(22 40)(24 34 30 36)(25 73 29 71)(26 38 28 32)(27 77)(31 58 39 54)(33 52 37 60)(35 56)(41 61 47 63)(42 50 46 48)(43 65 45 69)(49 67)(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), (1,44,19,62)(2,45,20,63)(3,46,11,64)(4,47,12,65)(5,48,13,66)(6,49,14,67)(7,50,15,68)(8,41,16,69)(9,42,17,70)(10,43,18,61)(21,34,60,76)(22,35,51,77)(23,36,52,78)(24,37,53,79)(25,38,54,80)(26,39,55,71)(27,40,56,72)(28,31,57,73)(29,32,58,74)(30,33,59,75), (1,22,19,51)(2,23,20,52)(3,24,11,53)(4,25,12,54)(5,26,13,55)(6,27,14,56)(7,28,15,57)(8,29,16,58)(9,30,17,59)(10,21,18,60)(31,50,73,68)(32,41,74,69)(33,42,75,70)(34,43,76,61)(35,44,77,62)(36,45,78,63)(37,46,79,64)(38,47,80,65)(39,48,71,66)(40,49,72,67), (2,16,10,12)(3,5,9,7)(4,20,8,18)(6,14)(11,13,17,15)(21,75,23,79)(22,40)(24,34,30,36)(25,73,29,71)(26,38,28,32)(27,77)(31,58,39,54)(33,52,37,60)(35,56)(41,61,47,63)(42,50,46,48)(43,65,45,69)(49,67)(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), (1,44,19,62)(2,45,20,63)(3,46,11,64)(4,47,12,65)(5,48,13,66)(6,49,14,67)(7,50,15,68)(8,41,16,69)(9,42,17,70)(10,43,18,61)(21,34,60,76)(22,35,51,77)(23,36,52,78)(24,37,53,79)(25,38,54,80)(26,39,55,71)(27,40,56,72)(28,31,57,73)(29,32,58,74)(30,33,59,75), (1,22,19,51)(2,23,20,52)(3,24,11,53)(4,25,12,54)(5,26,13,55)(6,27,14,56)(7,28,15,57)(8,29,16,58)(9,30,17,59)(10,21,18,60)(31,50,73,68)(32,41,74,69)(33,42,75,70)(34,43,76,61)(35,44,77,62)(36,45,78,63)(37,46,79,64)(38,47,80,65)(39,48,71,66)(40,49,72,67), (2,16,10,12)(3,5,9,7)(4,20,8,18)(6,14)(11,13,17,15)(21,75,23,79)(22,40)(24,34,30,36)(25,73,29,71)(26,38,28,32)(27,77)(31,58,39,54)(33,52,37,60)(35,56)(41,61,47,63)(42,50,46,48)(43,65,45,69)(49,67)(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)], [(1,44,19,62),(2,45,20,63),(3,46,11,64),(4,47,12,65),(5,48,13,66),(6,49,14,67),(7,50,15,68),(8,41,16,69),(9,42,17,70),(10,43,18,61),(21,34,60,76),(22,35,51,77),(23,36,52,78),(24,37,53,79),(25,38,54,80),(26,39,55,71),(27,40,56,72),(28,31,57,73),(29,32,58,74),(30,33,59,75)], [(1,22,19,51),(2,23,20,52),(3,24,11,53),(4,25,12,54),(5,26,13,55),(6,27,14,56),(7,28,15,57),(8,29,16,58),(9,30,17,59),(10,21,18,60),(31,50,73,68),(32,41,74,69),(33,42,75,70),(34,43,76,61),(35,44,77,62),(36,45,78,63),(37,46,79,64),(38,47,80,65),(39,48,71,66),(40,49,72,67)], [(2,16,10,12),(3,5,9,7),(4,20,8,18),(6,14),(11,13,17,15),(21,75,23,79),(22,40),(24,34,30,36),(25,73,29,71),(26,38,28,32),(27,77),(31,58,39,54),(33,52,37,60),(35,56),(41,61,47,63),(42,50,46,48),(43,65,45,69),(49,67),(51,72),(53,76,59,78),(55,80,57,74),(64,66,70,68)])

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 4A 4B 4C 4D 4E 4F 4G ··· 4L 5 8A 8B 8C 8D 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 4 4 4 4 4 4 4 ··· 4 5 8 8 8 8 10 10 10 20 ··· 20 size 1 1 2 5 5 10 2 2 4 4 10 10 20 ··· 20 4 20 20 20 20 4 4 4 8 ··· 8

32 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 4 4 4 4 4 4 8 type + + + + + + + + + - + + + + - image C1 C2 C2 C2 C2 C4 C4 C4 D4 D4 D4 F5 C8.C22 C2×F5 C2×F5 C22⋊F5 C22⋊F5 (C2×Q8)⋊4F5 kernel (C2×Q8)⋊4F5 Q8⋊F5 D5⋊M4(2) D10.C23 C2×Q8×D5 C2×Dic10 Q8×D5 Q8×C10 C4×D5 C2×Dic5 C22×D5 C2×Q8 D5 C2×C4 Q8 C4 C22 C1 # reps 1 4 1 1 1 2 4 2 2 1 1 1 2 1 2 2 2 2

Matrix representation of (C2×Q8)⋊4F5 in GL8(𝔽41)

 0 40 1 0 0 0 0 0 0 40 0 1 0 0 0 0 0 40 0 0 0 0 0 0 1 40 0 0 0 0 0 0 0 0 0 0 0 34 4 37 0 0 0 0 7 0 37 37 0 0 0 0 37 4 0 34 0 0 0 0 4 4 7 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 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 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

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] >;

(C2×Q8)⋊4F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_4F_5
% in TeX

G:=Group("(C2xQ8):4F5");
// 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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