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## G = (C2×Q8).7F5order 320 = 26·5

### 4th non-split extension by C2×Q8 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×Q8).7F5
 Chief series C1 — C5 — C10 — Dic5 — C2×Dic5 — C22.F5 — D5⋊M4(2) — (C2×Q8).7F5
 Lower central C5 — C10 — C2×C10 — (C2×Q8).7F5
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for (C2×Q8).7F5
G = < a,b,c,d,e | a2=b4=d5=1, c2=e4=b2, ab=ba, ac=ca, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, ebe-1=ab-1, cd=dc, ece-1=b2c, ede-1=d3 >

Subgroups: 554 in 146 conjugacy classes, 50 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C5, C8 [×4], C2×C4, C2×C4 [×2], C2×C4 [×11], Q8 [×8], C23, D5 [×2], D5, C10, C10, C2×C8 [×2], M4(2) [×6], C22×C4 [×3], C2×Q8, C2×Q8 [×7], Dic5 [×2], Dic5 [×2], C20 [×2], C20 [×2], D10 [×2], D10 [×2], C2×C10, C4.10D4 [×4], C2×M4(2) [×2], C22×Q8, C5⋊C8 [×4], Dic10 [×6], C4×D5 [×4], C4×D5 [×4], C2×Dic5, C2×Dic5 [×2], C2×C20, C2×C20 [×2], C5×Q8 [×2], C22×D5, C2×C4.10D4, D5⋊C8 [×2], C4.F5 [×2], C22.F5 [×4], C2×Dic10, C2×Dic10 [×2], C2×C4×D5, C2×C4×D5 [×2], Q8×D5 [×4], Q8×C10, Dic5.D4 [×4], D5⋊M4(2) [×2], C2×Q8×D5, (C2×Q8).7F5
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], F5, C4.10D4 [×2], C2×C22⋊C4, C2×F5 [×3], C2×C4.10D4, C22⋊F5 [×2], C22×F5, C2×C22⋊F5, (C2×Q8).7F5

Smallest permutation representation of (C2×Q8).7F5
On 80 points
Generators in S80
(1 5)(3 7)(9 13)(11 15)(17 21)(19 23)(26 30)(28 32)(33 37)(35 39)(41 45)(43 47)(49 53)(51 55)(57 61)(59 63)(66 70)(68 72)(73 77)(75 79)
(1 68 5 72)(2 69 6 65)(3 66 7 70)(4 67 8 71)(9 63 13 59)(10 64 14 60)(11 61 15 57)(12 62 16 58)(17 43 21 47)(18 44 22 48)(19 41 23 45)(20 42 24 46)(25 40 29 36)(26 37 30 33)(27 38 31 34)(28 35 32 39)(49 79 53 75)(50 80 54 76)(51 77 55 73)(52 78 56 74)
(1 3 5 7)(2 8 6 4)(9 55 13 51)(10 52 14 56)(11 49 15 53)(12 54 16 50)(17 39 21 35)(18 36 22 40)(19 33 23 37)(20 38 24 34)(25 44 29 48)(26 41 30 45)(27 46 31 42)(28 43 32 47)(57 79 61 75)(58 76 62 80)(59 73 63 77)(60 78 64 74)(65 71 69 67)(66 68 70 72)
(1 57 37 17 77)(2 18 58 78 38)(3 79 19 39 59)(4 40 80 60 20)(5 61 33 21 73)(6 22 62 74 34)(7 75 23 35 63)(8 36 76 64 24)(9 66 53 41 28)(10 42 67 29 54)(11 30 43 55 68)(12 56 31 69 44)(13 70 49 45 32)(14 46 71 25 50)(15 26 47 51 72)(16 52 27 65 48)
(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)

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

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

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

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 1 1 1 2 4 4 4 4 8 type + + + + + + - + + - image C1 C2 C2 C2 C4 C4 C4 D4 F5 C4.10D4 C2×F5 C22⋊F5 (C2×Q8).7F5 kernel (C2×Q8).7F5 Dic5.D4 D5⋊M4(2) C2×Q8×D5 C2×Dic10 C2×C4×D5 Q8×C10 C4×D5 C2×Q8 D5 C2×C4 C4 C1 # reps 1 4 2 1 2 4 2 4 1 2 3 4 2

Matrix representation of (C2×Q8).7F5 in GL8(𝔽41)

 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 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 23 19 1 0 0 0 0 0 30 18 0 1
,
 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 36 19 0 0 0 0 0 0 31 5 0 0 0 0 0 0 0 34 26 15 0 0 0 0 9 9 15 15
,
 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 15 37 0 0 0 0 0 0 36 26 0 0 0 0 0 0 24 36 0 40 0 0 0 0 16 20 1 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 40 40 40 40 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 1
,
 2 0 38 38 0 0 0 0 38 38 0 2 0 0 0 0 3 5 3 0 0 0 0 0 39 36 36 39 0 0 0 0 0 0 0 0 37 27 5 0 0 0 0 0 38 8 29 9 0 0 0 0 16 35 36 17 0 0 0 0 13 15 33 1

G:=sub<GL(8,GF(41))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,23,30,0,0,0,0,0,40,19,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,36,31,0,9,0,0,0,0,19,5,34,9,0,0,0,0,0,0,26,15,0,0,0,0,0,0,15,15],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,15,36,24,16,0,0,0,0,37,26,36,20,0,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0],[0,0,0,40,0,0,0,0,1,0,0,40,0,0,0,0,0,1,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[2,38,3,39,0,0,0,0,0,38,5,36,0,0,0,0,38,0,3,36,0,0,0,0,38,2,0,39,0,0,0,0,0,0,0,0,37,38,16,13,0,0,0,0,27,8,35,15,0,0,0,0,5,29,36,33,0,0,0,0,0,9,17,1] >;

(C2×Q8).7F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)._7F_5
% in TeX

G:=Group("(C2xQ8).7F5");
// GroupNames label

G:=SmallGroup(320,1127);
// by ID

G=gap.SmallGroup(320,1127);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,232,422,387,184,297,136,1684,6278,1595]);
// Polycyclic

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

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