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## G = (C2×C20)⋊15D4order 320 = 26·5

### 11st semidirect product of C2×C20 and D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — (C2×C20)⋊15D4
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C23×D5 — D5×C22×C4 — (C2×C20)⋊15D4
 Lower central C5 — C2×C10 — (C2×C20)⋊15D4
 Upper central C1 — C2×C4 — C2×C4○D4

Generators and relations for (C2×C20)⋊15D4
G = < a,b,c,d | a2=b20=c4=d2=1, ab=ba, cac-1=ab10, ad=da, cbc-1=dbd=b9, dcd=c-1 >

Subgroups: 1118 in 330 conjugacy classes, 115 normal (25 characteristic)
C1, C2, C2 [×2], C2 [×8], C4 [×4], C4 [×8], C22, C22 [×2], C22 [×24], C5, C2×C4 [×2], C2×C4 [×6], C2×C4 [×20], D4 [×14], Q8 [×2], C23, C23 [×2], C23 [×8], D5 [×4], C10, C10 [×2], C10 [×4], C42 [×2], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C22×C4 [×9], C2×D4, C2×D4 [×2], C2×D4 [×4], C2×Q8, C4○D4 [×4], C24, Dic5 [×6], C20 [×4], C20 [×2], D10 [×4], D10 [×12], C2×C10, C2×C10 [×2], C2×C10 [×8], C42⋊C2, C4×D4 [×4], C22≀C2 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C22.D4 [×2], C23×C4, C2×C4○D4, C4×D5 [×8], C2×Dic5 [×6], C2×Dic5 [×2], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×6], C2×C20 [×4], C5×D4 [×6], C5×Q8 [×2], C22×D5 [×2], C22×D5 [×6], C22×C10, C22×C10 [×2], C22.19C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×6], C2×C4×D5 [×4], C2×C4×D5 [×4], C22×Dic5, C2×C5⋊D4 [×4], C22×C20, C22×C20 [×2], D4×C10, D4×C10 [×2], Q8×C10, C5×C4○D4 [×4], C23×D5, C23.21D10, C4×C5⋊D4 [×4], C23.18D10 [×2], C23⋊D10 [×2], C202D4 [×2], D103Q8 [×2], D5×C22×C4, C10×C4○D4, (C2×C20)⋊15D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], D5, C2×D4 [×6], C4○D4 [×4], C24, D10 [×7], C22×D4, C2×C4○D4 [×2], C5⋊D4 [×4], C22×D5 [×7], C22.19C24, C2×C5⋊D4 [×6], C23×D5, D5×C4○D4 [×2], C22×C5⋊D4, (C2×C20)⋊15D4

Smallest permutation representation of (C2×C20)⋊15D4
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 33 64 45)(2 22 65 54)(3 31 66 43)(4 40 67 52)(5 29 68 41)(6 38 69 50)(7 27 70 59)(8 36 71 48)(9 25 72 57)(10 34 73 46)(11 23 74 55)(12 32 75 44)(13 21 76 53)(14 30 77 42)(15 39 78 51)(16 28 79 60)(17 37 80 49)(18 26 61 58)(19 35 62 47)(20 24 63 56)
(1 11)(2 20)(3 9)(4 18)(5 7)(6 16)(8 14)(10 12)(13 19)(15 17)(21 47)(22 56)(23 45)(24 54)(25 43)(26 52)(27 41)(28 50)(29 59)(30 48)(31 57)(32 46)(33 55)(34 44)(35 53)(36 42)(37 51)(38 60)(39 49)(40 58)(61 67)(62 76)(63 65)(64 74)(66 72)(68 70)(69 79)(71 77)(73 75)(78 80)

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

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

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

68 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L 4M 4N 4O 4P 5A 5B 10A ··· 10F 10G ··· 10R 20A ··· 20H 20I ··· 20T order 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 5 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 4 4 10 10 10 10 1 1 1 1 2 2 4 4 10 10 10 10 20 20 20 20 2 2 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

68 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 type + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 C2 C2 D4 D5 C4○D4 D10 D10 D10 C5⋊D4 D5×C4○D4 kernel (C2×C20)⋊15D4 C23.21D10 C4×C5⋊D4 C23.18D10 C23⋊D10 C20⋊2D4 D10⋊3Q8 D5×C22×C4 C10×C4○D4 C2×C20 C2×C4○D4 D10 C22×C4 C2×D4 C2×Q8 C2×C4 C2 # reps 1 1 4 2 2 2 2 1 1 4 2 8 6 6 2 16 8

Matrix representation of (C2×C20)⋊15D4 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 1 0 0 0 40 40
,
 1 1 0 0 5 6 0 0 0 0 32 0 0 0 0 32
,
 20 3 0 0 3 21 0 0 0 0 40 39 0 0 1 1
,
 6 7 0 0 36 35 0 0 0 0 40 0 0 0 1 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,1,40,0,0,0,40],[1,5,0,0,1,6,0,0,0,0,32,0,0,0,0,32],[20,3,0,0,3,21,0,0,0,0,40,1,0,0,39,1],[6,36,0,0,7,35,0,0,0,0,40,1,0,0,0,1] >;

(C2×C20)⋊15D4 in GAP, Magma, Sage, TeX

(C_2\times C_{20})\rtimes_{15}D_4
% in TeX

G:=Group("(C2xC20):15D4");
// GroupNames label

G:=SmallGroup(320,1500);
// by ID

G=gap.SmallGroup(320,1500);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,184,675,570,12550]);
// Polycyclic

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

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