metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊1Dic5, (C2×C10).3D8, C10.28C4≀C2, (D4×C10)⋊13C4, (C2×D4)⋊1Dic5, C4⋊D4.1D5, (C2×C20).229D4, C22.2(D4⋊D5), (C2×C10).10SD16, (C22×C10).44D4, (C22×C4).59D10, C5⋊5(C22.SD16), C20.55D4⋊28C2, C10.41(C23⋊C4), C2.3(D4⋊Dic5), C23.48(C5⋊D4), C22.2(D4.D5), C2.5(C23⋊Dic5), C10.38(D4⋊C4), C2.4(D4⋊2Dic5), C10.10C42⋊41C2, (C22×C20).371C22, C22.37(C23.D5), (C5×C4⋊C4)⋊8C4, (C2×C4).7(C2×Dic5), (C2×C20).340(C2×C4), (C5×C4⋊D4).10C2, (C2×C4).163(C5⋊D4), (C2×C10).159(C22⋊C4), SmallGroup(320,95)
Series: Derived ►Chief ►Lower central ►Upper central
C1 — C22 — C22×C4 — C4⋊D4 |
Generators and relations for C4⋊C4⋊Dic5
G = < a,b,c,d | a4=b4=c10=1, d2=c5, bab-1=a-1, ac=ca, dad-1=a-1b2, cbc-1=b-1, dbd-1=a-1b, dcd-1=c-1 >
Subgroups: 334 in 90 conjugacy classes, 31 normal (all characteristic)
C1, C2 [×3], C2 [×3], C4 [×5], C22 [×3], C22 [×5], C5, C8, C2×C4 [×2], C2×C4 [×6], D4 [×3], C23, C23, C10 [×3], C10 [×3], C22⋊C4, C4⋊C4, C2×C8, C22×C4, C22×C4, C2×D4, C2×D4, Dic5 [×2], C20 [×3], C2×C10 [×3], C2×C10 [×5], C2.C42, C22⋊C8, C4⋊D4, C5⋊2C8, C2×Dic5 [×4], C2×C20 [×2], C2×C20 [×2], C5×D4 [×3], C22×C10, C22×C10, C22.SD16, C2×C5⋊2C8, C5×C22⋊C4, C5×C4⋊C4, C22×Dic5, C22×C20, D4×C10, D4×C10, C20.55D4, C10.10C42, C5×C4⋊D4, C4⋊C4⋊Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C2×C4, D4 [×2], D5, C22⋊C4, D8, SD16, Dic5 [×2], D10, C23⋊C4, D4⋊C4, C4≀C2, C2×Dic5, C5⋊D4 [×2], C22.SD16, D4⋊D5, D4.D5, C23.D5, D4⋊Dic5, C23⋊Dic5, D4⋊2Dic5, C4⋊C4⋊Dic5
(1 39 9 34)(2 40 10 35)(3 36 6 31)(4 37 7 32)(5 38 8 33)(11 26 16 21)(12 27 17 22)(13 28 18 23)(14 29 19 24)(15 30 20 25)(41 71 54 65)(42 72 55 66)(43 73 56 67)(44 74 57 68)(45 75 58 69)(46 76 59 70)(47 77 60 61)(48 78 51 62)(49 79 52 63)(50 80 53 64)
(1 46 14 41)(2 42 15 47)(3 48 11 43)(4 44 12 49)(5 50 13 45)(6 51 16 56)(7 57 17 52)(8 53 18 58)(9 59 19 54)(10 55 20 60)(21 73 31 78)(22 79 32 74)(23 75 33 80)(24 71 34 76)(25 77 35 72)(26 67 36 62)(27 63 37 68)(28 69 38 64)(29 65 39 70)(30 61 40 66)
(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)
(2 5)(3 4)(6 7)(8 10)(11 12)(13 15)(16 17)(18 20)(21 37)(22 36)(23 40)(24 39)(25 38)(26 32)(27 31)(28 35)(29 34)(30 33)(41 76 46 71)(42 75 47 80)(43 74 48 79)(44 73 49 78)(45 72 50 77)(51 63 56 68)(52 62 57 67)(53 61 58 66)(54 70 59 65)(55 69 60 64)
G:=sub<Sym(80)| (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,71,54,65)(42,72,55,66)(43,73,56,67)(44,74,57,68)(45,75,58,69)(46,76,59,70)(47,77,60,61)(48,78,51,62)(49,79,52,63)(50,80,53,64), (1,46,14,41)(2,42,15,47)(3,48,11,43)(4,44,12,49)(5,50,13,45)(6,51,16,56)(7,57,17,52)(8,53,18,58)(9,59,19,54)(10,55,20,60)(21,73,31,78)(22,79,32,74)(23,75,33,80)(24,71,34,76)(25,77,35,72)(26,67,36,62)(27,63,37,68)(28,69,38,64)(29,65,39,70)(30,61,40,66), (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), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,37)(22,36)(23,40)(24,39)(25,38)(26,32)(27,31)(28,35)(29,34)(30,33)(41,76,46,71)(42,75,47,80)(43,74,48,79)(44,73,49,78)(45,72,50,77)(51,63,56,68)(52,62,57,67)(53,61,58,66)(54,70,59,65)(55,69,60,64)>;
G:=Group( (1,39,9,34)(2,40,10,35)(3,36,6,31)(4,37,7,32)(5,38,8,33)(11,26,16,21)(12,27,17,22)(13,28,18,23)(14,29,19,24)(15,30,20,25)(41,71,54,65)(42,72,55,66)(43,73,56,67)(44,74,57,68)(45,75,58,69)(46,76,59,70)(47,77,60,61)(48,78,51,62)(49,79,52,63)(50,80,53,64), (1,46,14,41)(2,42,15,47)(3,48,11,43)(4,44,12,49)(5,50,13,45)(6,51,16,56)(7,57,17,52)(8,53,18,58)(9,59,19,54)(10,55,20,60)(21,73,31,78)(22,79,32,74)(23,75,33,80)(24,71,34,76)(25,77,35,72)(26,67,36,62)(27,63,37,68)(28,69,38,64)(29,65,39,70)(30,61,40,66), (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), (2,5)(3,4)(6,7)(8,10)(11,12)(13,15)(16,17)(18,20)(21,37)(22,36)(23,40)(24,39)(25,38)(26,32)(27,31)(28,35)(29,34)(30,33)(41,76,46,71)(42,75,47,80)(43,74,48,79)(44,73,49,78)(45,72,50,77)(51,63,56,68)(52,62,57,67)(53,61,58,66)(54,70,59,65)(55,69,60,64) );
G=PermutationGroup([(1,39,9,34),(2,40,10,35),(3,36,6,31),(4,37,7,32),(5,38,8,33),(11,26,16,21),(12,27,17,22),(13,28,18,23),(14,29,19,24),(15,30,20,25),(41,71,54,65),(42,72,55,66),(43,73,56,67),(44,74,57,68),(45,75,58,69),(46,76,59,70),(47,77,60,61),(48,78,51,62),(49,79,52,63),(50,80,53,64)], [(1,46,14,41),(2,42,15,47),(3,48,11,43),(4,44,12,49),(5,50,13,45),(6,51,16,56),(7,57,17,52),(8,53,18,58),(9,59,19,54),(10,55,20,60),(21,73,31,78),(22,79,32,74),(23,75,33,80),(24,71,34,76),(25,77,35,72),(26,67,36,62),(27,63,37,68),(28,69,38,64),(29,65,39,70),(30,61,40,66)], [(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)], [(2,5),(3,4),(6,7),(8,10),(11,12),(13,15),(16,17),(18,20),(21,37),(22,36),(23,40),(24,39),(25,38),(26,32),(27,31),(28,35),(29,34),(30,33),(41,76,46,71),(42,75,47,80),(43,74,48,79),(44,73,49,78),(45,72,50,77),(51,63,56,68),(52,62,57,67),(53,61,58,66),(54,70,59,65),(55,69,60,64)])
47 conjugacy classes
class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 5A | 5B | 8A | 8B | 8C | 8D | 10A | ··· | 10F | 10G | 10H | 10I | 10J | 10K | 10L | 10M | 10N | 20A | ··· | 20H | 20I | 20J | 20K | 20L |
order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 8 | 8 | 8 | 8 | 10 | ··· | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 10 | 20 | ··· | 20 | 20 | 20 | 20 | 20 |
size | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 4 | 8 | 20 | 20 | 20 | 20 | 2 | 2 | 20 | 20 | 20 | 20 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
47 irreducible representations
dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
type | + | + | + | + | + | + | + | + | - | + | - | + | + | - | ||||||||
image | C1 | C2 | C2 | C2 | C4 | C4 | D4 | D4 | D5 | D8 | SD16 | Dic5 | D10 | Dic5 | C4≀C2 | C5⋊D4 | C5⋊D4 | C23⋊C4 | D4⋊D5 | D4.D5 | C23⋊Dic5 | D4⋊2Dic5 |
kernel | C4⋊C4⋊Dic5 | C20.55D4 | C10.10C42 | C5×C4⋊D4 | C5×C4⋊C4 | D4×C10 | C2×C20 | C22×C10 | C4⋊D4 | C2×C10 | C2×C10 | C4⋊C4 | C22×C4 | C2×D4 | C10 | C2×C4 | C23 | C10 | C22 | C22 | C2 | C2 |
# reps | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 4 | 4 |
Matrix representation of C4⋊C4⋊Dic5 ►in GL6(𝔽41)
32 | 0 | 0 | 0 | 0 | 0 |
26 | 9 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 9 | 37 |
0 | 0 | 0 | 0 | 0 | 32 |
3 | 21 | 0 | 0 | 0 | 0 |
21 | 38 | 0 | 0 | 0 | 0 |
0 | 0 | 40 | 0 | 0 | 0 |
0 | 0 | 0 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 3 | 7 |
0 | 0 | 0 | 0 | 34 | 38 |
1 | 0 | 0 | 0 | 0 | 0 |
29 | 40 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 | 0 |
0 | 0 | 40 | 34 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 | 0 |
22 | 32 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 |
0 | 0 | 34 | 40 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 25 | 40 |
G:=sub<GL(6,GF(41))| [32,26,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,37,32],[3,21,0,0,0,0,21,38,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,3,34,0,0,0,0,7,38],[1,29,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,1,34,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,22,0,0,0,0,0,32,0,0,0,0,0,0,1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,25,0,0,0,0,0,40] >;
C4⋊C4⋊Dic5 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes {\rm Dic}_5
% in TeX
G:=Group("C4:C4:Dic5");
// GroupNames label
G:=SmallGroup(320,95);
// by ID
G=gap.SmallGroup(320,95);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,28,141,219,1571,570,136,12550]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^10=1,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,d*a*d^-1=a^-1*b^2,c*b*c^-1=b^-1,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations