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## G = (C2×Q8)⋊7F5order 320 = 26·5

### 5th semidirect product of C2×Q8 and 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 — D10 — C22×D5 — C22⋊F5 — D10.C23 — (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=e4=1, c2=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: 746 in 158 conjugacy classes, 48 normal (18 characteristic)
C1, C2, C2 [×5], C4 [×2], C4 [×8], C22, C22 [×7], C5, C2×C4, C2×C4 [×2], C2×C4 [×13], D4 [×6], Q8 [×2], C23 [×3], D5 [×4], C10, C10, C42 [×2], C22⋊C4 [×4], C4⋊C4 [×2], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×4], Dic5 [×2], C20 [×2], C20 [×2], F5 [×4], D10 [×2], D10 [×5], C2×C10, C23⋊C4 [×4], C42⋊C2 [×2], C2×C4○D4, C4×D5 [×4], C4×D5 [×4], D20 [×6], C2×Dic5, C2×C20, C2×C20 [×2], C5×Q8 [×2], C2×F5 [×4], C22×D5, C22×D5 [×2], C23.C23, C4×F5 [×2], C4⋊F5 [×2], C22⋊F5 [×4], C2×C4×D5, C2×C4×D5 [×2], C2×D20, C2×D20 [×2], Q82D5 [×4], Q8×C10, D10.D4 [×4], D10.C23 [×2], C2×Q82D5, (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, C2×C22⋊C4, C2×F5 [×3], C23.C23, 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 16)(2 17)(3 18)(4 19)(5 20)(6 11)(7 12)(8 13)(9 14)(10 15)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)(41 56)(42 57)(43 58)(44 59)(45 60)(46 51)(47 52)(48 53)(49 54)(50 55)(61 76)(62 77)(63 78)(64 79)(65 80)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 36 6 31)(2 37 7 32)(3 38 8 33)(4 39 9 34)(5 40 10 35)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 46 76)(42 72 47 77)(43 73 48 78)(44 74 49 79)(45 75 50 80)(51 61 56 66)(52 62 57 67)(53 63 58 68)(54 64 59 69)(55 65 60 70)
(1 56 6 51)(2 57 7 52)(3 58 8 53)(4 59 9 54)(5 60 10 55)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)
(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 61 16 71)(2 63 20 74)(3 65 19 72)(4 62 18 75)(5 64 17 73)(6 66 11 76)(7 68 15 79)(8 70 14 77)(9 67 13 80)(10 69 12 78)(21 51 31 46)(22 53 35 49)(23 55 34 47)(24 52 33 50)(25 54 32 48)(26 56 36 41)(27 58 40 44)(28 60 39 42)(29 57 38 45)(30 59 37 43)

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

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

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

32 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 4E 4F 4G 4H ··· 4O 5 10A 10B 10C 20A ··· 20F order 1 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 5 10 10 10 20 ··· 20 size 1 1 2 10 10 20 20 2 2 4 4 5 5 10 20 ··· 20 4 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 C2×F5 C23.C23 C22⋊F5 (C2×Q8)⋊7F5 kernel (C2×Q8)⋊7F5 D10.D4 D10.C23 C2×Q8⋊2D5 C2×C4×D5 C2×D20 Q8×C10 C4×D5 C2×Q8 C2×C4 C5 C4 C1 # reps 1 4 2 1 4 2 2 4 1 3 2 4 2

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

 28 36 0 0 0 0 0 0 9 13 0 0 0 0 0 0 0 1 0 40 0 0 0 0 9 13 40 0 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 13 0 0 5 0 0 0 0 32 0 1 28 0 0 0 0 3 40 0 40 0 0 0 0 7 0 0 28 0 0 0 0 0 0 0 0 19 3 0 38 0 0 0 0 0 22 3 38 0 0 0 0 38 3 22 0 0 0 0 0 38 0 3 19
,
 32 0 0 0 0 0 0 0 0 32 0 0 0 0 0 0 20 0 9 0 0 0 0 0 14 0 0 9 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 40
,
 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 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
,
 9 0 4 0 0 0 0 0 27 0 6 32 0 0 0 0 38 9 32 0 0 0 0 0 24 0 6 0 0 0 0 0 0 0 0 0 38 3 22 0 0 0 0 0 19 3 0 38 0 0 0 0 38 0 3 19 0 0 0 0 0 22 3 38

G:=sub<GL(8,GF(41))| [28,9,0,9,0,0,0,0,36,13,1,13,0,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[13,32,3,7,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,5,28,40,28,0,0,0,0,0,0,0,0,19,0,38,38,0,0,0,0,3,22,3,0,0,0,0,0,0,3,22,3,0,0,0,0,38,38,0,19],[32,0,20,14,0,0,0,0,0,32,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,40,40,40],[9,27,38,24,0,0,0,0,0,0,9,0,0,0,0,0,4,6,32,6,0,0,0,0,0,32,0,0,0,0,0,0,0,0,0,0,38,19,38,0,0,0,0,0,3,3,0,22,0,0,0,0,22,0,3,3,0,0,0,0,0,38,19,38] >;

(C2×Q8)⋊7F5 in GAP, Magma, Sage, TeX

(C_2\times Q_8)\rtimes_7F_5
% in TeX

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

G:=SmallGroup(320,1123);
// by ID

G=gap.SmallGroup(320,1123);
# by ID

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

G:=Group<a,b,c,d,e|a^2=b^4=d^5=e^4=1,c^2=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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