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G = C42.3Dic5order 320 = 26·5

3rd non-split extension by C42 of Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.3Dic5, C4⋊Q8.3D5, (C2×C20).6D4, (C4×C20).20C4, (C2×Q8).4D10, (Q8×C10).13C4, (C2×Q8).3Dic5, C54(C42.3C4), (Q8×C10).4C22, C10.47(C23⋊C4), C20.10D4.2C2, C2.11(C23⋊Dic5), C22.17(C23.D5), (C5×C4⋊Q8).3C2, (C2×C4).8(C5⋊D4), (C2×C4).4(C2×Dic5), (C2×C20).184(C2×C4), (C2×C10).170(C22⋊C4), SmallGroup(320,106)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.3Dic5
C1C5C10C2×C10C2×C20Q8×C10C20.10D4 — C42.3Dic5
C5C10C2×C10C2×C20 — C42.3Dic5
C1C2C22C2×Q8C4⋊Q8

Generators and relations for C42.3Dic5
 G = < a,b,c,d | a4=b4=1, c10=b2, d2=c5, ab=ba, cac-1=a-1, dad-1=a-1b, cbc-1=b-1, dbd-1=a2b, dcd-1=c9 >

Subgroups: 174 in 60 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2, C4 [×6], C22, C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], Q8 [×2], C10, C10, C42, C4⋊C4 [×2], M4(2) [×2], C2×Q8 [×2], C20 [×6], C2×C10, C4.10D4 [×2], C4⋊Q8, C52C8 [×2], C2×C20, C2×C20 [×2], C2×C20 [×2], C5×Q8 [×2], C42.3C4, C4.Dic5 [×2], C4×C20, C5×C4⋊C4 [×2], Q8×C10 [×2], C20.10D4 [×2], C5×C4⋊Q8, C42.3Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, Dic5 [×2], D10, C23⋊C4, C2×Dic5, C5⋊D4 [×2], C42.3C4, C23.D5, C23⋊Dic5, C42.3Dic5

Smallest permutation representation of C42.3Dic5
On 80 points
Generators in S80
(21 46 31 56)(22 57 32 47)(23 48 33 58)(24 59 34 49)(25 50 35 60)(26 41 36 51)(27 52 37 42)(28 43 38 53)(29 54 39 44)(30 45 40 55)
(1 77 11 67)(2 68 12 78)(3 79 13 69)(4 70 14 80)(5 61 15 71)(6 72 16 62)(7 63 17 73)(8 74 18 64)(9 65 19 75)(10 76 20 66)(21 46 31 56)(22 57 32 47)(23 48 33 58)(24 59 34 49)(25 50 35 60)(26 41 36 51)(27 52 37 42)(28 43 38 53)(29 54 39 44)(30 45 40 55)
(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 36 6 21 11 26 16 31)(2 25 7 30 12 35 17 40)(3 34 8 39 13 24 18 29)(4 23 9 28 14 33 19 38)(5 32 10 37 15 22 20 27)(41 72 46 77 51 62 56 67)(42 61 47 66 52 71 57 76)(43 70 48 75 53 80 58 65)(44 79 49 64 54 69 59 74)(45 68 50 73 55 78 60 63)

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

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

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

41 conjugacy classes

class 1 2A2B4A···4E4F5A5B8A8B8C8D10A···10F20A···20L20M···20T
order1224···4455888810···1020···2020···20
size1124···4822404040402···24···48···8

41 irreducible representations

dim111112222224444
type+++++--++-
imageC1C2C2C4C4D4D5Dic5Dic5D10C5⋊D4C23⋊C4C42.3C4C23⋊Dic5C42.3Dic5
kernelC42.3Dic5C20.10D4C5×C4⋊Q8C4×C20Q8×C10C2×C20C4⋊Q8C42C2×Q8C2×Q8C2×C4C10C5C2C1
# reps121222222281248

Matrix representation of C42.3Dic5 in GL6(𝔽41)

4000000
0400000
001000
000100
0000040
000010
,
100000
010000
000100
0040000
0000040
000010
,
2300000
16250000
00261500
00151500
00002615
00001515
,
25390000
26160000
000010
000001
00261500
00151500

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[23,16,0,0,0,0,0,25,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,0,0,0,0,26,15,0,0,0,0,15,15],[25,26,0,0,0,0,39,16,0,0,0,0,0,0,0,0,26,15,0,0,0,0,15,15,0,0,1,0,0,0,0,0,0,1,0,0] >;

C42.3Dic5 in GAP, Magma, Sage, TeX

C_4^2._3{\rm Dic}_5
% in TeX

G:=Group("C4^2.3Dic5");
// GroupNames label

G:=SmallGroup(320,106);
// by ID

G=gap.SmallGroup(320,106);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,232,219,184,1571,570,297,136,1684,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a^-1*b,c*b*c^-1=b^-1,d*b*d^-1=a^2*b,d*c*d^-1=c^9>;
// generators/relations

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