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G = (C2×C4).D20order 320 = 26·5

3rd non-split extension by C2×C4 of D20 acting via D20/C5=D4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: (C2×C4).3D20, (C2×D20).4C4, (C2×C20).15D4, (C2×Q8).1D10, C4.10D45D5, (C4×Dic5).1C4, C20.10D41C2, C53(C42.C4), (Q8×C10).1C22, C10.34(C23⋊C4), C20.23D4.1C2, C22.14(D10⋊C4), C2.14(C23.1D10), (C2×C4).3(C4×D5), (C2×C20).3(C2×C4), (C2×C4).3(C5⋊D4), (C5×C4.10D4)⋊11C2, (C2×C10).71(C22⋊C4), SmallGroup(320,35)

Series: Derived Chief Lower central Upper central

C1C2×C20 — (C2×C4).D20
C1C5C10C2×C10C2×C20Q8×C10C20.23D4 — (C2×C4).D20
C5C10C2×C10C2×C20 — (C2×C4).D20
C1C2C22C2×Q8C4.10D4

Generators and relations for (C2×C4).D20
 G = < a,b,c,d | a2=b20=c2=1, d4=b10, ab=ba, ac=ca, dad-1=ab10, cbc=b-1, dbd-1=ab11, dcd-1=b15c >

Subgroups: 334 in 64 conjugacy classes, 21 normal (all characteristic)
C1, C2, C2 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4 [×3], C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, Dic5, C20 [×3], D10 [×3], C2×C10, C4.10D4, C4.10D4, C4.4D4, C52C8, C40, D20, C2×Dic5, C2×C20 [×3], C5×Q8, C22×D5, C42.C4, C4.Dic5, C4×Dic5, D10⋊C4 [×2], C5×M4(2), C2×D20, Q8×C10, C20.10D4, C5×C4.10D4, C20.23D4, (C2×C4).D20
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D10, C23⋊C4, C4×D5, D20, C5⋊D4, C42.C4, D10⋊C4, C23.1D10, (C2×C4).D20

Smallest permutation representation of (C2×C4).D20
On 80 points
Generators in S80
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)
(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 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 30)(22 29)(23 28)(24 27)(25 26)(31 40)(32 39)(33 38)(34 37)(35 36)(41 45)(42 44)(46 60)(47 59)(48 58)(49 57)(50 56)(51 55)(52 54)(61 65)(62 64)(66 80)(67 79)(68 78)(69 77)(70 76)(71 75)(72 74)
(1 61 26 41 11 71 36 51)(2 62 37 52 12 72 27 42)(3 63 28 43 13 73 38 53)(4 64 39 54 14 74 29 44)(5 65 30 45 15 75 40 55)(6 66 21 56 16 76 31 46)(7 67 32 47 17 77 22 57)(8 68 23 58 18 78 33 48)(9 69 34 49 19 79 24 59)(10 70 25 60 20 80 35 50)

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

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

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

35 conjugacy classes

class 1 2A2B2C4A4B4C4D4E5A5B8A8B8C8D10A10B10C10D20A20B20C20D20E20F20G20H40A···40H
order12224444455888810101010202020202020202040···40
size112404442020228840402244444488888···8

35 irreducible representations

dim1111112222224448
type++++++++++
imageC1C2C2C2C4C4D4D5D10C4×D5D20C5⋊D4C23⋊C4C42.C4C23.1D10(C2×C4).D20
kernel(C2×C4).D20C20.10D4C5×C4.10D4C20.23D4C4×Dic5C2×D20C2×C20C4.10D4C2×Q8C2×C4C2×C4C2×C4C10C5C2C1
# reps1111222224441242

Matrix representation of (C2×C4).D20 in GL6(𝔽41)

100000
010000
0040000
0004000
000010
00252501
,
6400000
100000
000100
0040000
009914
000162040
,
6400000
35350000
000100
001000
000010
0016162040
,
4000000
0400000
000010
0032324037
0032000
0021809

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,25,0,0,0,40,0,25,0,0,0,0,1,0,0,0,0,0,0,1],[6,1,0,0,0,0,40,0,0,0,0,0,0,0,0,40,9,0,0,0,1,0,9,16,0,0,0,0,1,20,0,0,0,0,4,40],[6,35,0,0,0,0,40,35,0,0,0,0,0,0,0,1,0,16,0,0,1,0,0,16,0,0,0,0,1,20,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,32,32,2,0,0,0,32,0,18,0,0,1,40,0,0,0,0,0,37,0,9] >;

(C2×C4).D20 in GAP, Magma, Sage, TeX

(C_2\times C_4).D_{20}
% in TeX

G:=Group("(C2xC4).D20");
// GroupNames label

G:=SmallGroup(320,35);
// by ID

G=gap.SmallGroup(320,35);
# by ID

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

G:=Group<a,b,c,d|a^2=b^20=c^2=1,d^4=b^10,a*b=b*a,a*c=c*a,d*a*d^-1=a*b^10,c*b*c=b^-1,d*b*d^-1=a*b^11,d*c*d^-1=b^15*c>;
// generators/relations

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