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G = C24.30D10  order 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), C20⋊7D4⋊4C2, D10⋊D4⋊2C2, (C2×D20)⋊4C22, C24⋊2D5⋊3C2, C4⋊Dic5⋊6C22, C20.48D4⋊4C2, (C2×C10).38C24, C22⋊C4.87D10, (C22×C4).45D10, D10.12D4⋊2C2, C2.9(D4⋊6D10), D10⋊C4⋊2C22, (C2×C20).131C23, Dic5.5D4⋊2C2, C5⋊1(C22.32C24), (C4×Dic5)⋊48C22, (C2×Dic10)⋊3C22, C10.D4⋊2C22, C23.D10⋊1C2, 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×C10).128C23, (C22×C20).355C22, (C4×C5⋊D4)⋊34C2, (C2×C4×D5)⋊41C22, (C2×C22⋊C4)⋊17D5, C2.18(C2×C4○D20), C10.16(C2×C4○D4), (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

Derived series
C1 — C2×C10 — C24.30D10
Chief series
C1 — C5 — C10 — C2×C10 — C22×D5 — C2×C5⋊D4 — C4×C5⋊D4 — C24.30D10
Lower central
C5 — C2×C10 — C24.30D10

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], C42⋊2C2 [×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], C20⋊7D4, C24⋊2D5 [×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, D4⋊6D10 [×2], C24.30D10

Generators and relations
 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 >

Smallest permutation representation
►On 80 points
Generators in S80
(1 50)(2 51)(3 52)(4 53)(5 54)(6 55)(7 56)(8 57)(9 58)(10 59)(11 60)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 49)

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

Matrix representation ►G ⊆ 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] >;
 

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+4)D4⋊6D10
kernelC24.30D10C23.D10D10.12D4D10⋊D4Dic5.5D4C20.48D4C4×C5⋊D4C20⋊7D4C24⋊2D5C10×C22⋊C4C2×C22⋊C4C2×C10C22⋊C4C22×C4C24C22C10C2
# reps1222212121248421628

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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