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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — (C2×C4).D20
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — Q8×C10 — C20.23D4 — (C2×C4).D20
 Lower central C5 — C10 — C2×C10 — C2×C20 — (C2×C4).D20
 Upper central C1 — C2 — C22 — C2×Q8 — C4.10D4

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, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, Q8, C23, D5, C10, C10, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, Dic5, C20, D10, C2×C10, C4.10D4, C4.10D4, C4.4D4, C52C8, C40, D20, C2×Dic5, C2×C20, C5×Q8, C22×D5, C42.C4, C4.Dic5, C4×Dic5, D10⋊C4, C5×M4(2), C2×D20, Q8×C10, C20.10D4, C5×C4.10D4, C20.23D4, (C2×C4).D20
Quotients: C1, C2, C4, C22, C2×C4, D4, 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 51 11 71 36 41)(2 62 37 42 12 72 27 52)(3 63 28 53 13 73 38 43)(4 64 39 44 14 74 29 54)(5 65 30 55 15 75 40 45)(6 66 21 46 16 76 31 56)(7 67 32 57 17 77 22 47)(8 68 23 48 18 78 33 58)(9 69 34 59 19 79 24 49)(10 70 25 50 20 80 35 60)

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

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

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

35 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 5A 5B 8A 8B 8C 8D 10A 10B 10C 10D 20A 20B 20C 20D 20E 20F 20G 20H 40A ··· 40H order 1 2 2 2 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 40 4 4 4 20 20 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.C4 C23.1D10 (C2×C4).D20 kernel (C2×C4).D20 C20.10D4 C5×C4.10D4 C20.23D4 C4×Dic5 C2×D20 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×C4).D20 in GL6(𝔽41)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 25 25 0 1
,
 6 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 9 9 1 4 0 0 0 16 20 40
,
 6 40 0 0 0 0 35 35 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 16 16 20 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 0 0 1 0 0 0 32 32 40 37 0 0 32 0 0 0 0 0 2 18 0 9

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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