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## G = C24.35D10order 320 = 26·5

### 35th non-split extension by C24 of D10 acting via D10/C5=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C24.35D10
 Chief series C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C4×D5 — D10.12D4 — C24.35D10
 Lower central C5 — C2×C10 — C24.35D10
 Upper central C1 — C22 — C22≀C2

Generators and relations for C24.35D10
G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=d, ab=ba, eae-1=ac=ca, ad=da, faf-1=acd, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 878 in 250 conjugacy classes, 95 normal (91 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×3], C2×C4 [×11], D4 [×9], Q8, C23 [×4], C23 [×5], D5, C10 [×3], C10 [×5], C42 [×2], C22⋊C4 [×3], C22⋊C4 [×11], C4⋊C4 [×6], C22×C4 [×4], C2×D4 [×3], C2×D4 [×4], C2×Q8, C24, Dic5 [×7], C20 [×3], D10 [×3], C2×C10, C2×C10 [×2], C2×C10 [×15], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C22≀C2, C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2 [×2], Dic10, C4×D5, C2×Dic5 [×7], C2×Dic5 [×3], C5⋊D4 [×5], C2×C20 [×3], C5×D4 [×4], C22×D5, C22×C10 [×4], C22×C10 [×4], C22.32C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×2], C23.D5 [×9], C5×C22⋊C4 [×3], C2×Dic10, C2×C4×D5, C22×Dic5 [×3], C2×C5⋊D4 [×4], D4×C10 [×3], C23×C10, Dic5.14D4, C23.D10 [×2], Dic54D4, D10.12D4, Dic5.5D4, D4×Dic5, C23.18D10, C20.17D4, C202D4, Dic5⋊D4 [×2], C2×C23.D5, C242D5, C5×C22≀C2, C24.35D10
Quotients: C1, C2 [×15], C22 [×35], C23 [×15], D5, C4○D4 [×2], C24, D10 [×7], C2×C4○D4, 2+ 1+4 [×2], C22×D5 [×7], C22.32C24, D42D5 [×2], C23×D5, C2×D42D5, D46D10 [×2], C24.35D10

Smallest permutation representation of C24.35D10
On 80 points
Generators in S80
(2 30)(4 32)(6 34)(8 36)(10 38)(12 40)(14 22)(16 24)(18 26)(20 28)(41 73)(42 52)(43 75)(44 54)(45 77)(46 56)(47 79)(48 58)(49 61)(50 60)(51 63)(53 65)(55 67)(57 69)(59 71)(62 72)(64 74)(66 76)(68 78)(70 80)
(1 11)(3 13)(5 15)(7 17)(9 19)(21 31)(23 33)(25 35)(27 37)(29 39)(41 63)(42 74)(43 65)(44 76)(45 67)(46 78)(47 69)(48 80)(49 71)(50 62)(51 73)(52 64)(53 75)(54 66)(55 77)(56 68)(57 79)(58 70)(59 61)(60 72)
(1 29)(2 30)(3 31)(4 32)(5 33)(6 34)(7 35)(8 36)(9 37)(10 38)(11 39)(12 40)(13 21)(14 22)(15 23)(16 24)(17 25)(18 26)(19 27)(20 28)(41 63)(42 64)(43 65)(44 66)(45 67)(46 68)(47 69)(48 70)(49 71)(50 72)(51 73)(52 74)(53 75)(54 76)(55 77)(56 78)(57 79)(58 80)(59 61)(60 62)
(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)(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 41 11 51)(2 50 12 60)(3 59 13 49)(4 48 14 58)(5 57 15 47)(6 46 16 56)(7 55 17 45)(8 44 18 54)(9 53 19 43)(10 42 20 52)(21 71 31 61)(22 80 32 70)(23 69 33 79)(24 78 34 68)(25 67 35 77)(26 76 36 66)(27 65 37 75)(28 74 38 64)(29 63 39 73)(30 72 40 62)

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

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

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

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 4A 4B 4C 4D 4E 4F 4G 4H ··· 4L 5A 5B 10A ··· 10F 10G ··· 10R 10S 10T 20A ··· 20F order 1 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 ··· 4 5 5 10 ··· 10 10 ··· 10 10 10 20 ··· 20 size 1 1 1 1 2 2 4 4 4 20 4 4 4 10 10 10 10 20 ··· 20 2 2 2 ··· 2 4 ··· 4 8 8 8 ··· 8

50 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 C2 D5 C4○D4 D10 D10 D10 2+ 1+4 D4⋊2D5 D4⋊6D10 kernel C24.35D10 Dic5.14D4 C23.D10 Dic5⋊4D4 D10.12D4 Dic5.5D4 D4×Dic5 C23.18D10 C20.17D4 C20⋊2D4 Dic5⋊D4 C2×C23.D5 C24⋊2D5 C5×C22≀C2 C22≀C2 C2×C10 C22⋊C4 C2×D4 C24 C10 C22 C2 # reps 1 1 2 1 1 1 1 1 1 1 2 1 1 1 2 4 6 6 2 2 4 8

Matrix representation of C24.35D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 40
,
 40 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 0 0 0 40
,
 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 40 0 0 0 0 0 0 40 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 1
,
 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 37 0 0 0 0 37 0 0 0 0 0 0 0 0 10 0 0 0 0 10 0
,
 9 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 10 0 0 0 0 10 0 0 0 0 37 0 0 0 0 37 0 0 0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,0,0,0,0,10,0,0,0,0,10,0],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,0,0,37,0,0,0,0,37,0,0,0,0,10,0,0,0,0,10,0,0,0] >;

C24.35D10 in GAP, Magma, Sage, TeX

C_2^4._{35}D_{10}
% in TeX

G:=Group("C2^4.35D10");
// GroupNames label

G:=SmallGroup(320,1265);
// by ID

G=gap.SmallGroup(320,1265);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,232,758,219,675,570,12550]);
// Polycyclic

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

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