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G = C42.Dic5order 320 = 26·5

2nd non-split extension by C42 of Dic5 acting via Dic5/C5=C4

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C42.2Dic5, (C2×C20).5D4, (C4×C20).19C4, (C2×Q8).3D10, (D4×C10).15C4, (C2×D4).2Dic5, C4.4D4.3D5, C20.10D42C2, C54(C42.C4), (Q8×C10).3C22, C10.45(C23⋊C4), C2.9(C23⋊Dic5), C22.15(C23.D5), (C2×C4).7(C5⋊D4), (C2×C4).2(C2×Dic5), (C2×C20).182(C2×C4), (C5×C4.4D4).1C2, (C2×C10).164(C22⋊C4), SmallGroup(320,100)

Series: Derived Chief Lower central Upper central

C1C2×C20 — C42.Dic5
C1C5C10C2×C10C2×C20Q8×C10C20.10D4 — C42.Dic5
C5C10C2×C10C2×C20 — C42.Dic5
C1C2C22C2×Q8C4.4D4

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

Subgroups: 206 in 64 conjugacy classes, 23 normal (17 characteristic)
C1, C2, C2 [×2], C4 [×4], C22, C22 [×3], C5, C8 [×2], C2×C4, C2×C4 [×2], C2×C4, D4, Q8, C23, C10, C10 [×2], C42, C22⋊C4 [×2], M4(2) [×2], C2×D4, C2×Q8, C20 [×4], C2×C10, C2×C10 [×3], C4.10D4 [×2], C4.4D4, C52C8 [×2], C2×C20, C2×C20 [×2], C2×C20, C5×D4, C5×Q8, C22×C10, C42.C4, C4.Dic5 [×2], C4×C20, C5×C22⋊C4 [×2], D4×C10, Q8×C10, C20.10D4 [×2], C5×C4.4D4, C42.Dic5
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.C4, C23.D5, C23⋊Dic5, C42.Dic5

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

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

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

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

41 conjugacy classes

class 1 2A2B2C4A···4E5A5B8A8B8C8D10A···10F10G10H10I10J20A···20L20M20N20O20P
order12224···455888810···101010101020···2020202020
size11284···422404040402···288884···48888

41 irreducible representations

dim111112222224444
type+++++--++
imageC1C2C2C4C4D4D5Dic5Dic5D10C5⋊D4C23⋊C4C42.C4C23⋊Dic5C42.Dic5
kernelC42.Dic5C20.10D4C5×C4.4D4C4×C20D4×C10C2×C20C4.4D4C42C2×D4C2×Q8C2×C4C10C5C2C1
# reps121222222281248

Matrix representation of C42.Dic5 in GL4(𝔽41) generated by

9000
0900
003217
00179
,
13000
304000
004011
00111
,
52700
03600
00835
00033
,
0010
0001
92400
03200
G:=sub<GL(4,GF(41))| [9,0,0,0,0,9,0,0,0,0,32,17,0,0,17,9],[1,30,0,0,30,40,0,0,0,0,40,11,0,0,11,1],[5,0,0,0,27,36,0,0,0,0,8,0,0,0,35,33],[0,0,9,0,0,0,24,32,1,0,0,0,0,1,0,0] >;

C42.Dic5 in GAP, Magma, Sage, TeX

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

G:=Group("C4^2.Dic5");
// GroupNames label

G:=SmallGroup(320,100);
// by ID

G=gap.SmallGroup(320,100);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,184,1571,570,297,136,1684,12550]);
// 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*b^2,d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,d*b*d^-1=a^2*b^-1,d*c*d^-1=c^9>;
// generators/relations

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