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G = (C2×Q8)⋊7F5order 320 = 26·5

5th semidirect product of C2×Q8 and F5 acting via F5/D5=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×Q8)⋊7F5, (Q8×C10)⋊7C4, (C2×D20)⋊13C4, D10.9(C2×D4), (C4×D5).44D4, D10.D44C2, C4.20(C22⋊F5), C20.20(C22⋊C4), C22⋊F5.3C22, C22.15(C22×F5), (C2×D20).140C22, C53(C23.C23), D10.C235C2, Dic5.48(C22⋊C4), (C22×D5).149C23, (C2×C4×D5)⋊5C4, (C2×C4).6(C2×F5), (C2×C20).63(C2×C4), C2.30(C2×C22⋊F5), C10.29(C2×C22⋊C4), (C2×C4×D5).208C22, (C2×C10).84(C22×C4), (C2×Q82D5).11C2, (C22×D5).10(C2×C4), (C2×Dic5).194(C2×C4), SmallGroup(320,1123)

Series: Derived Chief Lower central Upper central

C1C2×C10 — (C2×Q8)⋊7F5
C1C5C10D10C22×D5C22⋊F5D10.C23 — (C2×Q8)⋊7F5
C5C10C2×C10 — (C2×Q8)⋊7F5
C1C2C2×C4C2×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 2A2B2C2D2E2F4A4B4C4D4E4F4G4H···4O 5 10A10B10C20A···20F
order122222244444444···4510101020···20
size112101020202244551020···2044448···8

32 irreducible representations

dim1111111244448
type+++++++++
imageC1C2C2C2C4C4C4D4F5C2×F5C23.C23C22⋊F5(C2×Q8)⋊7F5
kernel(C2×Q8)⋊7F5D10.D4D10.C23C2×Q82D5C2×C4×D5C2×D20Q8×C10C4×D5C2×Q8C2×C4C5C4C1
# reps1421422413242

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

2836000000
913000000
010400000
9134000000
000040000
000004000
000000400
000000040
,
130050000
3201280000
3400400000
700280000
0000193038
0000022338
0000383220
0000380319
,
320000000
032000000
200900000
140090000
000040000
000004000
000000400
000000040
,
10000000
01000000
00100000
00010000
000000040
000010040
000001040
000000140
,
90400000
2706320000
3893200000
240600000
0000383220
0000193038
0000380319
0000022338

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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