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## G = C2×C52⋊2Q8order 400 = 24·52

### Direct product of C2 and C52⋊2Q8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C2×C52⋊2Q8
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — C52⋊2Q8 — C2×C52⋊2Q8
 Lower central C52 — C5×C10 — C2×C52⋊2Q8
 Upper central C1 — C22

Generators and relations for C2×C522Q8
G = < a,b,c,d,e | a2=b5=c5=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >

Subgroups: 460 in 92 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C4, C22, C5, C5, C2×C4, Q8, C10, C10, C2×Q8, Dic5, Dic5, C20, C2×C10, C2×C10, C52, Dic10, C2×Dic5, C2×Dic5, C2×C20, C5×C10, C5×C10, C2×Dic10, C5×Dic5, C526C4, C102, C522Q8, C10×Dic5, C2×C526C4, C2×C522Q8
Quotients: C1, C2, C22, Q8, C23, D5, C2×Q8, D10, Dic10, C22×D5, C2×Dic10, D52, C522Q8, C2×D52, C2×C522Q8

Smallest permutation representation of C2×C522Q8
On 80 points
Generators in S80
(1 12)(2 13)(3 14)(4 15)(5 11)(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 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 4 2 5 3)(6 9 7 10 8)(11 14 12 15 13)(16 19 17 20 18)(21 23 25 22 24)(26 28 30 27 29)(31 33 35 32 34)(36 38 40 37 39)(41 43 45 42 44)(46 48 50 47 49)(51 53 55 52 54)(56 58 60 57 59)(61 64 62 65 63)(66 69 67 70 68)(71 74 72 75 73)(76 79 77 80 78)
(1 38 7 33)(2 37 8 32)(3 36 9 31)(4 40 10 35)(5 39 6 34)(11 29 16 24)(12 28 17 23)(13 27 18 22)(14 26 19 21)(15 30 20 25)(41 74 46 79)(42 73 47 78)(43 72 48 77)(44 71 49 76)(45 75 50 80)(51 64 56 69)(52 63 57 68)(53 62 58 67)(54 61 59 66)(55 65 60 70)
(1 57 7 52)(2 58 8 53)(3 59 9 54)(4 60 10 55)(5 56 6 51)(11 46 16 41)(12 47 17 42)(13 48 18 43)(14 49 19 44)(15 50 20 45)(21 76 26 71)(22 77 27 72)(23 78 28 73)(24 79 29 74)(25 80 30 75)(31 66 36 61)(32 67 37 62)(33 68 38 63)(34 69 39 64)(35 70 40 65)

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 20A ··· 20P order 1 2 2 2 4 4 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 10 10 50 50 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 4 4 4 type + + + + - + + + - + - + image C1 C2 C2 C2 Q8 D5 D10 D10 Dic10 D52 C52⋊2Q8 C2×D52 kernel C2×C52⋊2Q8 C52⋊2Q8 C10×Dic5 C2×C52⋊6C4 C5×C10 C2×Dic5 Dic5 C2×C10 C10 C22 C2 C2 # reps 1 4 2 1 2 4 8 4 16 4 8 4

Matrix representation of C2×C522Q8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 34 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 25 12 0 0 23 16 0 0 0 0 1 0 0 0 0 1
,
 30 9 0 0 32 11 0 0 0 0 1 0 0 0 6 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[34,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[25,23,0,0,12,16,0,0,0,0,1,0,0,0,0,1],[30,32,0,0,9,11,0,0,0,0,1,6,0,0,0,40] >;

C2×C522Q8 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_2Q_8
% in TeX

G:=Group("C2xC5^2:2Q8");
// GroupNames label

G:=SmallGroup(400,178);
// by ID

G=gap.SmallGroup(400,178);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,121,55,970,11525]);
// Polycyclic

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

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