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G = C24.30D10order 320 = 26·5

30th non-split extension by C24 of D10 acting via D10/C5=C22

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C24.30D10, C10.52+ 1+4, C207D44C2, D10⋊D42C2, (C2×D20)⋊4C22, C242D53C2, C4⋊Dic56C22, C20.48D44C2, (C2×C10).38C24, C22⋊C4.87D10, (C22×C4).45D10, D10.12D42C2, C2.9(D46D10), D10⋊C42C22, (C2×C20).131C23, Dic5.5D42C2, C51(C22.32C24), (C2×Dic10)⋊3C22, (C4×Dic5)⋊48C22, C23.D101C2, C10.D42C22, C23.82(C22×D5), C22.77(C23×D5), C23.D5.2C22, C22.23(C4○D20), (C23×C10).64C22, (C2×Dic5).11C23, (C22×D5).10C23, (C22×C20).355C22, (C22×C10).128C23, (C4×C5⋊D4)⋊34C2, (C2×C4×D5)⋊41C22, (C2×C22⋊C4)⋊17D5, C10.16(C2×C4○D4), C2.18(C2×C4○D20), (C10×C22⋊C4)⋊20C2, (C2×C5⋊D4).7C22, (C2×C4).261(C22×D5), (C2×C10).104(C4○D4), (C5×C22⋊C4).109C22, SmallGroup(320,1166)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C24.30D10
C1C5C10C2×C10C22×D5C2×C5⋊D4C4×C5⋊D4 — C24.30D10
C5C2×C10 — C24.30D10
C1C22C2×C22⋊C4

Generators and relations for C24.30D10
 G = < a,b,c,d,e,f | a2=b2=c2=d2=1, e10=f2=c, ab=ba, ac=ca, faf-1=ad=da, ae=ea, bc=cb, ebe-1=bd=db, fbf-1=bcd, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e9 >

Subgroups: 926 in 250 conjugacy classes, 95 normal (31 characteristic)
C1, C2 [×3], C2 [×6], C4 [×10], C22, C22 [×2], C22 [×18], C5, C2×C4 [×2], C2×C4 [×2], C2×C4 [×10], D4 [×9], Q8, C23, C23 [×2], C23 [×6], D5 [×2], C10 [×3], C10 [×4], C42 [×2], C22⋊C4 [×4], C22⋊C4 [×10], C4⋊C4 [×6], C22×C4 [×2], C22×C4 [×2], C2×D4 [×7], C2×Q8, C24, Dic5 [×6], C20 [×4], D10 [×6], C2×C10, C2×C10 [×2], C2×C10 [×12], C2×C22⋊C4, C4×D4 [×2], C22≀C2 [×2], C4⋊D4 [×3], C22⋊Q8, C22.D4 [×2], C4.4D4 [×2], C422C2 [×2], Dic10, C4×D5 [×2], D20, C2×Dic5 [×6], C5⋊D4 [×8], C2×C20 [×2], C2×C20 [×2], C2×C20 [×2], C22×D5 [×2], C22×C10, C22×C10 [×2], C22×C10 [×4], C22.32C24, C4×Dic5 [×2], C10.D4 [×4], C4⋊Dic5 [×2], D10⋊C4 [×4], C23.D5 [×6], C5×C22⋊C4 [×4], C2×Dic10, C2×C4×D5 [×2], C2×D20, C2×C5⋊D4 [×6], C22×C20 [×2], C23×C10, C23.D10 [×2], D10.12D4 [×2], D10⋊D4 [×2], Dic5.5D4 [×2], C20.48D4, C4×C5⋊D4 [×2], C207D4, C242D5 [×2], C10×C22⋊C4, C24.30D10
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, C4○D20 [×2], C23×D5, C2×C4○D20, D46D10 [×2], C24.30D10

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

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

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

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

62 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G···4L5A5B10A···10N10O···10V20A···20P
order12222222224444444···45510···1010···1020···20
size11112244202022224420···20222···24···44···4

62 irreducible representations

dim111111111122222244
type+++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2D5C4○D4D10D10D10C4○D202+ 1+4D46D10
kernelC24.30D10C23.D10D10.12D4D10⋊D4Dic5.5D4C20.48D4C4×C5⋊D4C207D4C242D5C10×C22⋊C4C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C10C2
# reps1222212121248421628

Matrix representation of C24.30D10 in GL6(𝔽41)

4000000
0400000
0040000
0004000
000010
000001
,
4000000
1810000
001000
0004000
000010
0000040
,
4000000
0400000
001000
000100
000010
000001
,
100000
010000
0040000
0004000
0000400
0000040
,
900000
090000
0001600
0016000
0000018
0000180
,
24300000
4170000
0000018
0000180
0001600
0016000

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,18,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[9,0,0,0,0,0,0,9,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,0,0,0,0,18,0,0,0,0,18,0],[24,4,0,0,0,0,30,17,0,0,0,0,0,0,0,0,0,16,0,0,0,0,16,0,0,0,0,18,0,0,0,0,18,0,0,0] >;

C24.30D10 in GAP, Magma, Sage, TeX

C_2^4._{30}D_{10}
% in TeX

G:=Group("C2^4.30D10");
// GroupNames label

G:=SmallGroup(320,1166);
// by ID

G=gap.SmallGroup(320,1166);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,477,100,675,297,12550]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=1,e^10=f^2=c,a*b=b*a,a*c=c*a,f*a*f^-1=a*d=d*a,a*e=e*a,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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