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## G = C5×C23⋊2Q8order 320 = 26·5

### Direct product of C5 and C23⋊2Q8

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C23⋊2Q8
 Chief series C1 — C2 — C22 — C2×C10 — C2×C20 — C5×C22⋊C4 — C5×C22⋊Q8 — C5×C23⋊2Q8
 Lower central C1 — C22 — C5×C23⋊2Q8
 Upper central C1 — C2×C10 — C5×C23⋊2Q8

Generators and relations for C5×C232Q8
G = < a,b,c,d,e,f | a5=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, ebe-1=fbf-1=bd=db, fcf-1=cd=dc, ce=ec, de=ed, df=fd, fef-1=e-1 >

Subgroups: 386 in 242 conjugacy classes, 162 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×12], C22, C22 [×6], C22 [×10], C5, C2×C4 [×12], C2×C4 [×6], Q8 [×4], C23 [×7], C23 [×2], C10, C10 [×2], C10 [×6], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×6], C2×Q8 [×4], C24, C20 [×12], C2×C10, C2×C10 [×6], C2×C10 [×10], C2×C22⋊C4 [×3], C22⋊Q8 [×12], C2×C20 [×12], C2×C20 [×6], C5×Q8 [×4], C22×C10 [×7], C22×C10 [×2], C232Q8, C5×C22⋊C4 [×12], C5×C4⋊C4 [×12], C22×C20 [×6], Q8×C10 [×4], C23×C10, C10×C22⋊C4 [×3], C5×C22⋊Q8 [×12], C5×C232Q8
Quotients: C1, C2 [×15], C22 [×35], C5, Q8 [×4], C23 [×15], C10 [×15], C2×Q8 [×6], C24, C2×C10 [×35], C22×Q8, 2+ 1+4 [×2], C5×Q8 [×4], C22×C10 [×15], C232Q8, Q8×C10 [×6], C23×C10, Q8×C2×C10, C5×2+ 1+4 [×2], C5×C232Q8

Smallest permutation representation of C5×C232Q8
On 80 points
Generators in S80
(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 30)(2 26)(3 27)(4 28)(5 29)(6 76)(7 77)(8 78)(9 79)(10 80)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)(36 43)(37 44)(38 45)(39 41)(40 42)(46 51)(47 52)(48 53)(49 54)(50 55)(56 63)(57 64)(58 65)(59 61)(60 62)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 76)(17 77)(18 78)(19 79)(20 80)(21 33)(22 34)(23 35)(24 31)(25 32)(36 50)(37 46)(38 47)(39 48)(40 49)(41 53)(42 54)(43 55)(44 51)(45 52)(56 63)(57 64)(58 65)(59 61)(60 62)(66 71)(67 72)(68 73)(69 74)(70 75)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 76)(12 77)(13 78)(14 79)(15 80)(21 28)(22 29)(23 30)(24 26)(25 27)(36 55)(37 51)(38 52)(39 53)(40 54)(41 48)(42 49)(43 50)(44 46)(45 47)(56 75)(57 71)(58 72)(59 73)(60 74)(61 68)(62 69)(63 70)(64 66)(65 67)
(1 50 30 36)(2 46 26 37)(3 47 27 38)(4 48 28 39)(5 49 29 40)(6 56 76 70)(7 57 77 66)(8 58 78 67)(9 59 79 68)(10 60 80 69)(11 63 16 75)(12 64 17 71)(13 65 18 72)(14 61 19 73)(15 62 20 74)(21 53 33 41)(22 54 34 42)(23 55 35 43)(24 51 31 44)(25 52 32 45)
(1 70 30 56)(2 66 26 57)(3 67 27 58)(4 68 28 59)(5 69 29 60)(6 50 76 36)(7 46 77 37)(8 47 78 38)(9 48 79 39)(10 49 80 40)(11 55 16 43)(12 51 17 44)(13 52 18 45)(14 53 19 41)(15 54 20 42)(21 73 33 61)(22 74 34 62)(23 75 35 63)(24 71 31 64)(25 72 32 65)

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

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

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

110 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 4A ··· 4L 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 20A ··· 20AV order 1 2 2 2 2 ··· 2 4 ··· 4 5 5 5 5 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 ··· 4 1 1 1 1 1 ··· 1 2 ··· 2 4 ··· 4

110 irreducible representations

 dim 1 1 1 1 1 1 2 2 4 4 type + + + - + image C1 C2 C2 C5 C10 C10 Q8 C5×Q8 2+ 1+4 C5×2+ 1+4 kernel C5×C23⋊2Q8 C10×C22⋊C4 C5×C22⋊Q8 C23⋊2Q8 C2×C22⋊C4 C22⋊Q8 C22×C10 C23 C10 C2 # reps 1 3 12 4 12 48 4 16 2 8

Matrix representation of C5×C232Q8 in GL6(𝔽41)

 10 0 0 0 0 0 0 10 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1
,
 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 40 0 0 0 0 0 0 40
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 40
,
 0 40 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 35 2 0 0 0 0 2 6 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0

G:=sub<GL(6,GF(41))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[0,1,0,0,0,0,40,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[35,2,0,0,0,0,2,6,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0] >;

C5×C232Q8 in GAP, Magma, Sage, TeX

C_5\times C_2^3\rtimes_2Q_8
% in TeX

G:=Group("C5xC2^3:2Q8");
// GroupNames label

G:=SmallGroup(320,1545);
// by ID

G=gap.SmallGroup(320,1545);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-5,-2,-2,1120,1149,3446,891,856,2467]);
// Polycyclic

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

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