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## G = C42.Dic5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — Q8×C10 — C20.10D4 — C42.Dic5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42.Dic5
 Upper central C1 — C2 — C22 — C2×Q8 — C4.4D4

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, C4, C22, C22, C5, C8, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C10, C10, C42, C22⋊C4, M4(2), C2×D4, C2×Q8, C20, C2×C10, C2×C10, C4.10D4, C4.4D4, C52C8, C2×C20, C2×C20, C2×C20, C5×D4, C5×Q8, C22×C10, C42.C4, C4.Dic5, C4×C20, C5×C22⋊C4, D4×C10, Q8×C10, C20.10D4, C5×C4.4D4, C42.Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, 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 54)(22 45)(23 56)(24 47)(25 58)(26 49)(27 60)(28 51)(29 42)(30 53)(31 44)(32 55)(33 46)(34 57)(35 48)(36 59)(37 50)(38 41)(39 52)(40 43)(61 76 71 66)(62 77 72 67)(63 78 73 68)(64 79 74 69)(65 80 75 70)
(1 64 11 74)(2 75 12 65)(3 66 13 76)(4 77 14 67)(5 68 15 78)(6 79 16 69)(7 70 17 80)(8 61 18 71)(9 72 19 62)(10 63 20 73)(21 49 31 59)(22 60 32 50)(23 51 33 41)(24 42 34 52)(25 53 35 43)(26 44 36 54)(27 55 37 45)(28 46 38 56)(29 57 39 47)(30 48 40 58)
(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 44 6 49 11 54 16 59)(2 53 7 58 12 43 17 48)(3 42 8 47 13 52 18 57)(4 51 9 56 14 41 19 46)(5 60 10 45 15 50 20 55)(21 74 26 79 31 64 36 69)(22 63 27 68 32 73 37 78)(23 72 28 77 33 62 38 67)(24 61 29 66 34 71 39 76)(25 70 30 75 35 80 40 65)

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

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

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

41 conjugacy classes

 class 1 2A 2B 2C 4A ··· 4E 5A 5B 8A 8B 8C 8D 10A ··· 10F 10G 10H 10I 10J 20A ··· 20L 20M 20N 20O 20P order 1 2 2 2 4 ··· 4 5 5 8 8 8 8 10 ··· 10 10 10 10 10 20 ··· 20 20 20 20 20 size 1 1 2 8 4 ··· 4 2 2 40 40 40 40 2 ··· 2 8 8 8 8 4 ··· 4 8 8 8 8

41 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 4 4 4 4 type + + + + + - - + + image C1 C2 C2 C4 C4 D4 D5 Dic5 Dic5 D10 C5⋊D4 C23⋊C4 C42.C4 C23⋊Dic5 C42.Dic5 kernel C42.Dic5 C20.10D4 C5×C4.4D4 C4×C20 D4×C10 C2×C20 C4.4D4 C42 C2×D4 C2×Q8 C2×C4 C10 C5 C2 C1 # reps 1 2 1 2 2 2 2 2 2 2 8 1 2 4 8

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

 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 27 0 0 0 36 0 0 0 0 8 35 0 0 0 33
,
 0 0 1 0 0 0 0 1 9 24 0 0 0 32 0 0
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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