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## G = C42.3Dic5order 320 = 26·5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C20 — C42.3Dic5
 Chief series C1 — C5 — C10 — C2×C10 — C2×C20 — Q8×C10 — C20.10D4 — C42.3Dic5
 Lower central C5 — C10 — C2×C10 — C2×C20 — C42.3Dic5
 Upper central C1 — C2 — C22 — C2×Q8 — C4⋊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, C22, C5, C8, C2×C4, C2×C4, C2×C4, Q8, C10, C10, C42, C4⋊C4, M4(2), C2×Q8, C20, C2×C10, C4.10D4, C4⋊Q8, C52C8, C2×C20, C2×C20, C2×C20, C5×Q8, C42.3C4, C4.Dic5, C4×C20, C5×C4⋊C4, Q8×C10, C20.10D4, C5×C4⋊Q8, C42.3Dic5
Quotients: C1, C2, C4, C22, C2×C4, D4, D5, C22⋊C4, Dic5, D10, C23⋊C4, C2×Dic5, C5⋊D4, C42.3C4, C23.D5, C23⋊Dic5, C42.3Dic5

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

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

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

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

41 conjugacy classes

 class 1 2A 2B 4A ··· 4E 4F 5A 5B 8A 8B 8C 8D 10A ··· 10F 20A ··· 20L 20M ··· 20T order 1 2 2 4 ··· 4 4 5 5 8 8 8 8 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 2 4 ··· 4 8 2 2 40 40 40 40 2 ··· 2 4 ··· 4 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 C42.3C4 C23⋊Dic5 C42.3Dic5 kernel C42.3Dic5 C20.10D4 C5×C4⋊Q8 C4×C20 Q8×C10 C2×C20 C4⋊Q8 C42 C2×Q8 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 C42.3Dic5 in GL6(𝔽41)

 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 40 0 0 0 0 1 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 40 0 0 0 0 0 0 0 0 40 0 0 0 0 1 0
,
 23 0 0 0 0 0 16 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 39 0 0 0 0 26 16 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 26 15 0 0 0 0 15 15 0 0

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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