direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: D4×C4⋊C4, C23.230C24, C24.555C23, C22.662+ (1+4), C22.482- (1+4), C4⋊5(C4×D4), C2.2(D42), C2.2(D4×Q8), (C4×D4)⋊21C4, C42⋊23(C2×C4), (C2×D4).35Q8, (C2×D4).341D4, C42⋊9C4⋊12C2, C2.3(D4⋊6D4), C2.2(D4⋊3Q8), C23.415(C2×D4), C23.113(C2×Q8), C23.8Q8⋊13C2, C23.7Q8⋊23C2, C22.38(C22×Q8), C23.129(C22×C4), (C23×C4).304C22, (C2×C42).429C22, C22.121(C23×C4), C22.105(C22×D4), (C22×C4).1245C23, (C22×D4).609C22, C23.65C23⋊22C2, C2.C42.58C22, C2.27(C23.33C23), C4⋊1(C2×C4⋊C4), (C4×C4⋊C4)⋊36C2, C2.31(C2×C4×D4), C4⋊C4⋊43(C2×C4), C22⋊1(C2×C4⋊C4), (C2×C4×D4).33C2, (C22×C4⋊C4)⋊9C2, C22⋊C4⋊40(C2×C4), (C22×C4)⋊31(C2×C4), C2.14(C22×C4⋊C4), (C2×C4).299(C2×Q8), (C2×D4).243(C2×C4), (C2×C4).1069(C2×D4), (C2×C4).793(C4○D4), (C2×C4⋊C4).820C22, (C2×C4).566(C22×C4), C22.115(C2×C4○D4), (C2×C22⋊C4).438C22, (C2×D4)○(C2×C4⋊C4), (C2×C4⋊C4)○(C22×D4), SmallGroup(128,1080)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 636 in 388 conjugacy classes, 196 normal (28 characteristic)
C1, C2 [×7], C2 [×8], C4 [×8], C4 [×16], C22 [×7], C22 [×8], C22 [×24], C2×C4 [×22], C2×C4 [×60], D4 [×16], C23, C23 [×12], C23 [×8], C42 [×4], C42 [×4], C22⋊C4 [×8], C22⋊C4 [×8], C4⋊C4 [×8], C4⋊C4 [×20], C22×C4 [×3], C22×C4 [×18], C22×C4 [×28], C2×D4 [×12], C24 [×2], C2.C42 [×6], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×6], C2×C4⋊C4 [×3], C2×C4⋊C4 [×10], C2×C4⋊C4 [×8], C4×D4 [×8], C4×D4 [×8], C23×C4 [×6], C22×D4, C4×C4⋊C4, C23.7Q8 [×2], C42⋊9C4, C23.8Q8 [×4], C23.65C23 [×2], C22×C4⋊C4 [×2], C2×C4×D4, C2×C4×D4 [×2], D4×C4⋊C4
Quotients:
C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], D4 [×8], Q8 [×4], C23 [×15], C4⋊C4 [×16], C22×C4 [×14], C2×D4 [×12], C2×Q8 [×6], C4○D4 [×2], C24, C2×C4⋊C4 [×12], C4×D4 [×4], C23×C4, C22×D4 [×2], C22×Q8, C2×C4○D4, 2+ (1+4), 2- (1+4), C22×C4⋊C4, C2×C4×D4, C23.33C23, D42, D4⋊6D4, D4×Q8, D4⋊3Q8, D4×C4⋊C4
Generators and relations
G = < a,b,c,d | a4=b2=c4=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >
(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)
(1 37)(2 40)(3 39)(4 38)(5 21)(6 24)(7 23)(8 22)(9 26)(10 25)(11 28)(12 27)(13 30)(14 29)(15 32)(16 31)(17 64)(18 63)(19 62)(20 61)(33 49)(34 52)(35 51)(36 50)(41 59)(42 58)(43 57)(44 60)(45 55)(46 54)(47 53)(48 56)
(1 31 38 13)(2 32 39 14)(3 29 40 15)(4 30 37 16)(5 46 24 53)(6 47 21 54)(7 48 22 55)(8 45 23 56)(9 49 27 34)(10 50 28 35)(11 51 25 36)(12 52 26 33)(17 43 61 58)(18 44 62 59)(19 41 63 60)(20 42 64 57)
(1 63 27 56)(2 64 28 53)(3 61 25 54)(4 62 26 55)(5 14 57 50)(6 15 58 51)(7 16 59 52)(8 13 60 49)(9 45 38 19)(10 46 39 20)(11 47 40 17)(12 48 37 18)(21 29 43 36)(22 30 44 33)(23 31 41 34)(24 32 42 35)
G:=sub<Sym(64)| (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), (1,37)(2,40)(3,39)(4,38)(5,21)(6,24)(7,23)(8,22)(9,26)(10,25)(11,28)(12,27)(13,30)(14,29)(15,32)(16,31)(17,64)(18,63)(19,62)(20,61)(33,49)(34,52)(35,51)(36,50)(41,59)(42,58)(43,57)(44,60)(45,55)(46,54)(47,53)(48,56), (1,31,38,13)(2,32,39,14)(3,29,40,15)(4,30,37,16)(5,46,24,53)(6,47,21,54)(7,48,22,55)(8,45,23,56)(9,49,27,34)(10,50,28,35)(11,51,25,36)(12,52,26,33)(17,43,61,58)(18,44,62,59)(19,41,63,60)(20,42,64,57), (1,63,27,56)(2,64,28,53)(3,61,25,54)(4,62,26,55)(5,14,57,50)(6,15,58,51)(7,16,59,52)(8,13,60,49)(9,45,38,19)(10,46,39,20)(11,47,40,17)(12,48,37,18)(21,29,43,36)(22,30,44,33)(23,31,41,34)(24,32,42,35)>;
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), (1,37)(2,40)(3,39)(4,38)(5,21)(6,24)(7,23)(8,22)(9,26)(10,25)(11,28)(12,27)(13,30)(14,29)(15,32)(16,31)(17,64)(18,63)(19,62)(20,61)(33,49)(34,52)(35,51)(36,50)(41,59)(42,58)(43,57)(44,60)(45,55)(46,54)(47,53)(48,56), (1,31,38,13)(2,32,39,14)(3,29,40,15)(4,30,37,16)(5,46,24,53)(6,47,21,54)(7,48,22,55)(8,45,23,56)(9,49,27,34)(10,50,28,35)(11,51,25,36)(12,52,26,33)(17,43,61,58)(18,44,62,59)(19,41,63,60)(20,42,64,57), (1,63,27,56)(2,64,28,53)(3,61,25,54)(4,62,26,55)(5,14,57,50)(6,15,58,51)(7,16,59,52)(8,13,60,49)(9,45,38,19)(10,46,39,20)(11,47,40,17)(12,48,37,18)(21,29,43,36)(22,30,44,33)(23,31,41,34)(24,32,42,35) );
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)], [(1,37),(2,40),(3,39),(4,38),(5,21),(6,24),(7,23),(8,22),(9,26),(10,25),(11,28),(12,27),(13,30),(14,29),(15,32),(16,31),(17,64),(18,63),(19,62),(20,61),(33,49),(34,52),(35,51),(36,50),(41,59),(42,58),(43,57),(44,60),(45,55),(46,54),(47,53),(48,56)], [(1,31,38,13),(2,32,39,14),(3,29,40,15),(4,30,37,16),(5,46,24,53),(6,47,21,54),(7,48,22,55),(8,45,23,56),(9,49,27,34),(10,50,28,35),(11,51,25,36),(12,52,26,33),(17,43,61,58),(18,44,62,59),(19,41,63,60),(20,42,64,57)], [(1,63,27,56),(2,64,28,53),(3,61,25,54),(4,62,26,55),(5,14,57,50),(6,15,58,51),(7,16,59,52),(8,13,60,49),(9,45,38,19),(10,46,39,20),(11,47,40,17),(12,48,37,18),(21,29,43,36),(22,30,44,33),(23,31,41,34),(24,32,42,35)])
Matrix representation ►G ⊆ GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 4 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,4,0,0,0,2,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,2,1],[4,0,0,0,0,0,3,0,0,0,0,0,2,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,0,1,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,4] >;
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 4A | ··· | 4P | 4Q | ··· | 4AH |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | D4×C4⋊C4 | C4×C4⋊C4 | C23.7Q8 | C42⋊9C4 | C23.8Q8 | C23.65C23 | C22×C4⋊C4 | C2×C4×D4 | C4×D4 | C4⋊C4 | C2×D4 | C2×D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 3 | 16 | 4 | 4 | 4 | 4 | 1 | 1 |
In GAP, Magma, Sage, TeX
D_4\times C_4\rtimes C_4
% in TeX
G:=Group("D4xC4:C4"); // GroupNames label
G:=SmallGroup(128,1080);
// by ID
G=gap.SmallGroup(128,1080);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,758,184,346]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^4=d^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations