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## G = C2×C52⋊2D4order 400 = 24·52

### Direct product of C2 and C52⋊2D4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C2×C52⋊2D4
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C52⋊2D4 — C2×C52⋊2D4
 Lower central C52 — C5×C10 — C2×C52⋊2D4
 Upper central C1 — C22

Generators and relations for C2×C522D4
G = < a,b,c,d,e | a2=b5=c5=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=b-1, be=eb, dcd-1=ece=c-1, ede=d-1 >

Subgroups: 652 in 124 conjugacy classes, 40 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, C23, D5, C10, C10, C2×D4, Dic5, D10, D10, C2×C10, C2×C10, C52, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C5×D5, C5×C10, C5×C10, C2×C5⋊D4, C526C4, D5×C10, D5×C10, C102, C522D4, C2×C526C4, D5×C2×C10, C2×C522D4
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, C5⋊D4, C22×D5, C2×C5⋊D4, D52, C522D4, C2×D52, C2×C522D4

Smallest permutation representation of C2×C522D4
On 80 points
Generators in S80
(1 11)(2 12)(3 13)(4 14)(5 15)(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 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 25 24 23 22)(26 30 29 28 27)(31 35 34 33 32)(36 40 39 38 37)(41 45 44 43 42)(46 50 49 48 47)(51 55 54 53 52)(56 60 59 58 57)(61 62 63 64 65)(66 67 68 69 70)(71 72 73 74 75)(76 77 78 79 80)
(1 69 6 64)(2 68 7 63)(3 67 8 62)(4 66 9 61)(5 70 10 65)(11 79 16 74)(12 78 17 73)(13 77 18 72)(14 76 19 71)(15 80 20 75)(21 49 26 44)(22 48 27 43)(23 47 28 42)(24 46 29 41)(25 50 30 45)(31 59 36 54)(32 58 37 53)(33 57 38 52)(34 56 39 51)(35 60 40 55)
(1 51)(2 52)(3 53)(4 54)(5 55)(6 56)(7 57)(8 58)(9 59)(10 60)(11 41)(12 42)(13 43)(14 44)(15 45)(16 46)(17 47)(18 48)(19 49)(20 50)(21 71)(22 72)(23 73)(24 74)(25 75)(26 76)(27 77)(28 78)(29 79)(30 80)(31 61)(32 62)(33 63)(34 64)(35 65)(36 66)(37 67)(38 68)(39 69)(40 70)

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 10Y ··· 10AN order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 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 D4 D5 D10 D10 C5⋊D4 D52 C52⋊2D4 C2×D52 kernel C2×C52⋊2D4 C52⋊2D4 C2×C52⋊6C4 D5×C2×C10 C5×C10 C22×D5 D10 C2×C10 C10 C22 C2 C2 # reps 1 4 1 2 2 4 8 4 16 4 8 4

Matrix representation of C2×C522D4 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 35 6 0 0 35 40 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 40 6
,
 4 10 0 0 27 37 0 0 0 0 1 0 0 0 6 40
,
 23 35 0 0 6 18 0 0 0 0 1 0 0 0 6 40
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[35,35,0,0,6,40,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,40,0,0,1,6],[4,27,0,0,10,37,0,0,0,0,1,6,0,0,0,40],[23,6,0,0,35,18,0,0,0,0,1,6,0,0,0,40] >;

C2×C522D4 in GAP, Magma, Sage, TeX

C_2\times C_5^2\rtimes_2D_4
% in TeX

G:=Group("C2xC5^2:2D4");
// GroupNames label

G:=SmallGroup(400,176);
// by ID

G=gap.SmallGroup(400,176);
# by ID

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

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

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