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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×Q8).D10
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — Q8×C10 — Dic5⋊Q8 — (C2×Q8).D10
 Lower central C5 — C10 — C2×C10 — C2×C20 — (C2×Q8).D10
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

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, C22, C5, C8, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, M4(2), C2×Q8, C2×Q8, Dic5, C20, C2×C10, C4.10D4, C4.10D4, C4⋊Q8, C52C8, C40, Dic10, C2×Dic5, C2×C20, C5×Q8, C42.3C4, C4.Dic5, C4×Dic5, C10.D4, C5×M4(2), C2×Dic10, Q8×C10, C20.10D4, C5×C4.10D4, Dic5⋊Q8, (C2×Q8).D10
Quotients: C1, C2, C4, C22, C2×C4, D4, 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
(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(49 59)(50 60)
(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 60 79 21 11 50 69 31)(2 49 70 40 12 59 80 30)(3 58 61 39 13 48 71 29)(4 47 72 38 14 57 62 28)(5 56 63 37 15 46 73 27)(6 45 74 36 16 55 64 26)(7 54 65 35 17 44 75 25)(8 43 76 34 18 53 66 24)(9 52 67 33 19 42 77 23)(10 41 78 32 20 51 68 22)
(1 6 11 16)(2 15 12 5)(3 4 13 14)(7 20 17 10)(8 9 18 19)(21 60 31 50)(22 59 32 49)(23 58 33 48)(24 57 34 47)(25 56 35 46)(26 55 36 45)(27 54 37 44)(28 53 38 43)(29 52 39 42)(30 51 40 41)(61 62 71 72)(63 80 73 70)(64 69 74 79)(65 78 75 68)(66 67 76 77)

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

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

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

35 conjugacy classes

 class 1 2A 2B 4A 4B 4C 4D 4E 4F 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 4 4 4 4 4 4 5 5 8 8 8 8 10 10 10 10 20 20 20 20 20 20 20 20 40 ··· 40 size 1 1 2 4 4 4 20 20 40 2 2 8 8 40 40 2 2 4 4 4 4 4 4 8 8 8 8 8 ··· 8

35 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 4 4 4 8 type + + + + + + + + + - - image C1 C2 C2 C2 C4 C4 D4 D5 D10 C4×D5 D20 C5⋊D4 C23⋊C4 C42.3C4 C23.1D10 (C2×Q8).D10 kernel (C2×Q8).D10 C20.10D4 C5×C4.10D4 Dic5⋊Q8 C4×Dic5 C2×Dic10 C2×C20 C4.10D4 C2×Q8 C2×C4 C2×C4 C2×C4 C10 C5 C2 C1 # reps 1 1 1 1 2 2 2 2 2 4 4 4 1 2 4 2

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

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 21 21 0 40
,
 6 40 0 0 0 0 1 0 0 0 0 0 0 0 20 3 0 0 0 0 3 21 0 0 0 0 3 3 17 29 0 0 10 0 31 24
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 20 20 18 2 0 0 3 3 17 29 0 0 40 0 0 0 0 0 6 6 19 18
,
 1 0 0 0 0 0 6 40 0 0 0 0 0 0 20 3 0 0 0 0 3 21 0 0 0 0 20 20 18 2 0 0 0 31 22 23

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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