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G = (C2×Q8).D10order 320 = 26·5

2nd non-split extension by C2×Q8 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C4).4D20, (C2×C20).16D4, C4.10D4.D5, (C2×Q8).2D10, (C4×Dic5).2C4, C53(C42.3C4), (C2×Dic10).4C4, (Q8×C10).2C22, C10.35(C23⋊C4), Dic5⋊Q8.1C2, C20.10D4.1C2, C22.15(D10⋊C4), C2.15(C23.1D10), (C2×C4).4(C4×D5), (C2×C20).4(C2×C4), (C2×C4).4(C5⋊D4), (C5×C4.10D4).1C2, (C2×C10).72(C22⋊C4), SmallGroup(320,36)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×Q8).D10
C1C5C10C2×C10C2×C20Q8×C10Dic5⋊Q8 — (C2×Q8).D10
C5C10C2×C10C2×C20 — (C2×Q8).D10
C1C2C22C2×Q8C4.10D4

Generators and relations for (C2×Q8).D10
 G = < a,b,c,d | a2=b20=1, c4=d2=b10, ab=ba, cac-1=ab10, ad=da, cbc-1=dbd-1=ab9, dcd-1=b5c3 >

Subgroups: 238 in 60 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2, C4 [×6], C22, C5, C8 [×2], C2×C4 [×3], C2×C4 [×2], Q8 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], C2×Q8, C2×Q8, Dic5 [×3], C20 [×3], C2×C10, C4.10D4, C4.10D4, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5 [×2], C2×C20 [×3], C5×Q8, C42.3C4, C4.Dic5, C4×Dic5, C10.D4 [×2], C5×M4(2), C2×Dic10, Q8×C10, C20.10D4, C5×C4.10D4, Dic5⋊Q8, (C2×Q8).D10
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C42.3C4, D10⋊C4, C23.1D10, (C2×Q8).D10

Smallest permutation representation of (C2×Q8).D10
On 80 points
Generators in S80
(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)(61 71)(62 72)(63 73)(64 74)(65 75)(66 76)(67 77)(68 78)(69 79)(70 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 71 37 44 11 61 27 54)(2 80 28 43 12 70 38 53)(3 69 39 42 13 79 29 52)(4 78 30 41 14 68 40 51)(5 67 21 60 15 77 31 50)(6 76 32 59 16 66 22 49)(7 65 23 58 17 75 33 48)(8 74 34 57 18 64 24 47)(9 63 25 56 19 73 35 46)(10 72 36 55 20 62 26 45)
(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(21 38 31 28)(22 27 32 37)(23 36 33 26)(24 25 34 35)(29 30 39 40)(41 74 51 64)(42 73 52 63)(43 72 53 62)(44 71 54 61)(45 70 55 80)(46 69 56 79)(47 68 57 78)(48 67 58 77)(49 66 59 76)(50 65 60 75)

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

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

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

35 conjugacy classes

class 1 2A2B4A4B4C4D4E4F5A5B8A8B8C8D10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12244444455888810101010202020202020202040···40
size112444202040228840402244444488888···8

35 irreducible representations

dim1111112222224448
type+++++++++--
imageC1C2C2C2C4C4D4D5D10C4×D5D20C5⋊D4C23⋊C4C42.3C4C23.1D10(C2×Q8).D10
kernel(C2×Q8).D10C20.10D4C5×C4.10D4Dic5⋊Q8C4×Dic5C2×Dic10C2×C20C4.10D4C2×Q8C2×C4C2×C4C2×C4C10C5C2C1
# reps1111222224441242

Matrix representation of (C2×Q8).D10 in GL6(𝔽41)

100000
010000
001000
000100
0000400
002121040
,
6400000
100000
0020300
0032100
00331729
001003124
,
100000
6400000
002020182
00331729
0040000
00661918
,
100000
6400000
0020300
0032100
002020182
000312223

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,21,0,0,0,1,0,21,0,0,0,0,40,0,0,0,0,0,0,40],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,20,3,3,10,0,0,3,21,3,0,0,0,0,0,17,31,0,0,0,0,29,24],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,20,3,40,6,0,0,20,3,0,6,0,0,18,17,0,19,0,0,2,29,0,18],[1,6,0,0,0,0,0,40,0,0,0,0,0,0,20,3,20,0,0,0,3,21,20,31,0,0,0,0,18,22,0,0,0,0,2,23] >;

(C2×Q8).D10 in GAP, Magma, Sage, TeX

(C_2\times Q_8).D_{10}
% in TeX

G:=Group("(C2xQ8).D10");
// GroupNames label

G:=SmallGroup(320,36);
// by ID

G=gap.SmallGroup(320,36);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,224,141,36,422,184,1123,794,297,136,851,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^20=1,c^4=d^2=b^10,a*b=b*a,c*a*c^-1=a*b^10,a*d=d*a,c*b*c^-1=d*b*d^-1=a*b^9,d*c*d^-1=b^5*c^3>;
// generators/relations

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