metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4⋊C4⋊7D5, (C4×D5)⋊2C4, C4.14(C4×D5), C20.32(C2×C4), C4⋊Dic5⋊12C2, D10.9(C2×C4), (C2×C4).44D10, C5⋊4(C42⋊C2), (C4×Dic5)⋊13C2, D10⋊C4.4C2, C10.26(C4○D4), C2.4(D4⋊2D5), (C2×C20).56C22, (C2×C10).33C23, C10.23(C22×C4), C2.1(Q8⋊2D5), Dic5.21(C2×C4), C22.17(C22×D5), (C2×Dic5).33C22, (C22×D5).22C22, (C5×C4⋊C4)⋊3C2, (C2×C4×D5).2C2, C2.12(C2×C4×D5), SmallGroup(160,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊C4⋊7D5
G = < a,b,c,d | a4=b4=c5=d2=1, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a2b, dcd=c-1 >
Subgroups: 216 in 76 conjugacy classes, 41 normal (19 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C2×C4, C2×C4, C2×C4, C23, D5, C10, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C42⋊C2, C4×D5, C2×Dic5, C2×Dic5, C2×C20, C2×C20, C22×D5, C4×Dic5, C4⋊Dic5, D10⋊C4, C5×C4⋊C4, C2×C4×D5, C4⋊C4⋊7D5
Quotients: C1, C2, C4, C22, C2×C4, C23, D5, C22×C4, C4○D4, D10, C42⋊C2, C4×D5, C22×D5, C2×C4×D5, D4⋊2D5, Q8⋊2D5, C4⋊C4⋊7D5
►On 80 points
(1 59 9 54)(2 60 10 55)(3 56 6 51)(4 57 7 52)(5 58 8 53)(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)
(1 34 14 24)(2 35 15 25)(3 31 11 21)(4 32 12 22)(5 33 13 23)(6 36 16 26)(7 37 17 27)(8 38 18 28)(9 39 19 29)(10 40 20 30)(41 76 51 66)(42 77 52 67)(43 78 53 68)(44 79 54 69)(45 80 55 70)(46 71 56 61)(47 72 57 62)(48 73 58 63)(49 74 59 64)(50 75 60 65)
(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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)(42 45)(43 44)(47 50)(48 49)(52 55)(53 54)(57 60)(58 59)(61 66)(62 70)(63 69)(64 68)(65 67)(71 76)(72 80)(73 79)(74 78)(75 77)
G:=sub<Sym(80)| (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(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), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77)>;
G:=Group( (1,59,9,54)(2,60,10,55)(3,56,6,51)(4,57,7,52)(5,58,8,53)(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), (1,34,14,24)(2,35,15,25)(3,31,11,21)(4,32,12,22)(5,33,13,23)(6,36,16,26)(7,37,17,27)(8,38,18,28)(9,39,19,29)(10,40,20,30)(41,76,51,66)(42,77,52,67)(43,78,53,68)(44,79,54,69)(45,80,55,70)(46,71,56,61)(47,72,57,62)(48,73,58,63)(49,74,59,64)(50,75,60,65), (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), (1,5)(2,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(21,26)(22,30)(23,29)(24,28)(25,27)(31,36)(32,40)(33,39)(34,38)(35,37)(42,45)(43,44)(47,50)(48,49)(52,55)(53,54)(57,60)(58,59)(61,66)(62,70)(63,69)(64,68)(65,67)(71,76)(72,80)(73,79)(74,78)(75,77) );
G=PermutationGroup([[(1,59,9,54),(2,60,10,55),(3,56,6,51),(4,57,7,52),(5,58,8,53),(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)], [(1,34,14,24),(2,35,15,25),(3,31,11,21),(4,32,12,22),(5,33,13,23),(6,36,16,26),(7,37,17,27),(8,38,18,28),(9,39,19,29),(10,40,20,30),(41,76,51,66),(42,77,52,67),(43,78,53,68),(44,79,54,69),(45,80,55,70),(46,71,56,61),(47,72,57,62),(48,73,58,63),(49,74,59,64),(50,75,60,65)], [(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)], [(1,5),(2,4),(7,10),(8,9),(12,15),(13,14),(17,20),(18,19),(21,26),(22,30),(23,29),(24,28),(25,27),(31,36),(32,40),(33,39),(34,38),(35,37),(42,45),(43,44),(47,50),(48,49),(52,55),(53,54),(57,60),(58,59),(61,66),(62,70),(63,69),(64,68),(65,67),(71,76),(72,80),(73,79),(74,78),(75,77)]])
C4⋊C4⋊7D5 is a maximal subgroup of
C20.C42 (D4×D5)⋊C4 D4⋊2D5⋊C4 C40⋊6C4⋊C2 C40⋊5C4⋊C2 (Q8×D5)⋊C4 Q8⋊2D5⋊C4 (C2×C8).D10 D10⋊1C8.C2 (C8×D5)⋊C4 C8⋊(C4×D5) C4.Q8⋊D5 C20.(C4○D4) C8.27(C4×D5) C40⋊20(C2×C4) C2.D8⋊D5 C2.D8⋊7D5 D10.C42 C4⋊C4.7F5 C4⋊C4.9F5 C10.82+ 1+4 C10.52- 1+4 C10.112+ 1+4 D5×C42⋊C2 C42.91D10 C42.97D10 C42.98D10 C4×D4⋊2D5 C42⋊11D10 C42.113D10 C42.117D10 C42.125D10 C4×Q8⋊2D5 C42.132D10 C42.135D10 C4⋊C4⋊21D10 C10.382+ 1+4 C10.452+ 1+4 C10.1152+ 1+4 C22⋊Q8⋊25D5 C4⋊C4⋊26D10 C10.162- 1+4 C10.532+ 1+4 C10.212- 1+4 C10.222- 1+4 C10.232- 1+4 C10.772- 1+4 C4⋊C4.197D10 C10.1222+ 1+4 C10.622+ 1+4 C10.662+ 1+4 C42.236D10 C42.148D10 C42.237D10 C42.151D10 C42.152D10 C42.153D10 C42.154D10 C42⋊24D10 C42.189D10 C42.162D10 C42.164D10 C42.241D10 C42.174D10 C42.177D10 C42.179D10 (C4×D15)⋊8C4 (C4×D5)⋊Dic3 D10.19(C4×S3) D30.C2⋊C4 C4⋊C4⋊7D15
C4⋊C4⋊7D5 is a maximal quotient of
Dic5.15C42 C5⋊2(C42⋊5C4) C10.52(C4×D4) C22.58(D4×D5) D10⋊2C42 C10.55(C4×D4) C42.200D10 C42.202D10 C42.31D10 C4⋊C4×Dic5 C20.48(C4⋊C4) C10.97(C4×D4) D10⋊4(C4⋊C4) C10.90(C4×D4) (C4×D15)⋊8C4 (C4×D5)⋊Dic3 D10.19(C4×S3) D30.C2⋊C4 C4⋊C4⋊7D15
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 4M | 4N | 5A | 5B | 10A | ··· | 10F | 20A | ··· | 20L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 5 | 5 | 10 | ··· | 10 | 20 | ··· | 20 |
| size | 1 | 1 | 1 | 1 | 10 | 10 | 2 | ··· | 2 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D5 | C4○D4 | D10 | C4×D5 | D4⋊2D5 | Q8⋊2D5 |
| kernel | C4⋊C4⋊7D5 | C4×Dic5 | C4⋊Dic5 | D10⋊C4 | C5×C4⋊C4 | C2×C4×D5 | C4×D5 | C4⋊C4 | C10 | C2×C4 | C4 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 8 | 2 | 4 | 6 | 8 | 2 | 2 |
Matrix representation of C4⋊C4⋊7D5 ►in GL5(𝔽41)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 32 | 0 | 0 | 0 |
| 0 | 0 | 9 | 0 | 0 |
| 0 | 0 | 0 | 40 | 0 |
| 0 | 0 | 0 | 0 | 40 |
| 32 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 40 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 6 | 1 |
| 0 | 0 | 0 | 40 | 0 |
| 40 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 40 | 0 | 0 |
| 0 | 0 | 0 | 1 | 6 |
| 0 | 0 | 0 | 0 | 40 |
G:=sub<GL(5,GF(41))| [1,0,0,0,0,0,32,0,0,0,0,0,9,0,0,0,0,0,40,0,0,0,0,0,40],[32,0,0,0,0,0,0,40,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,6,40,0,0,0,1,0],[40,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,6,40] >;
C4⋊C4⋊7D5 in GAP, Magma, Sage, TeX
C_4\rtimes C_4\rtimes_7D_5
% in TeX
G:=Group("C4:C4:7D5"); // GroupNames label
G:=SmallGroup(160,113);
// by ID
G=gap.SmallGroup(160,113);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,103,362,188,50,4613]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^5=d^2=1,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^2*b,d*c*d=c^-1>;
// generators/relations