p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.189D4, C24.40C23, C23.532C24, C22.3092+ (1+4), C22.2262- (1+4), C42⋊5C4⋊25C2, C23⋊Q8⋊30C2, C23.4Q8⋊29C2, (C22×C4).142C23, (C2×C42).609C22, C22.357(C22×D4), C23.10D4.31C2, (C22×D4).196C22, (C22×Q8).449C22, C23.78C23⋊28C2, C24.C22⋊105C2, C2.83(C22.19C24), C2.41(C22.29C24), C2.C42.257C22, C2.49(C22.36C24), C2.40(C23.38C23), (C2×C4×Q8)⋊29C2, (C2×C4).391(C2×D4), (C2×C4).168(C4○D4), (C2×C4⋊C4).359C22, (C2×C4.4D4).27C2, C22.404(C2×C4○D4), (C2×C22⋊C4).221C22, SmallGroup(128,1364)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 250 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×18], C22, C22 [×6], C22 [×14], C2×C4 [×10], C2×C4 [×34], D4 [×4], Q8 [×12], C23, C23 [×14], C42 [×4], C42 [×4], C22⋊C4 [×20], C4⋊C4 [×10], C22×C4 [×3], C22×C4 [×10], C2×D4 [×4], C2×Q8 [×10], C24 [×2], C2.C42 [×10], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×12], C2×C4⋊C4, C2×C4⋊C4 [×6], C4×Q8 [×4], C4.4D4 [×4], C22×D4, C22×Q8 [×2], C42⋊5C4, C24.C22 [×4], C23⋊Q8 [×2], C23.10D4 [×2], C23.78C23 [×2], C23.4Q8 [×2], C2×C4×Q8, C2×C4.4D4, C42.189D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C22×D4, C2×C4○D4 [×2], 2+ (1+4) [×2], 2- (1+4) [×2], C22.19C24, C22.29C24, C23.38C23, C22.36C24 [×4], C42.189D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=1, d2=a2, ab=ba, cac-1=a-1b2, dad-1=a-1, cbc-1=a2b, bd=db, dcd-1=a2c-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 21 42 15)(2 22 43 16)(3 23 44 13)(4 24 41 14)(5 30 40 57)(6 31 37 58)(7 32 38 59)(8 29 39 60)(9 20 49 46)(10 17 50 47)(11 18 51 48)(12 19 52 45)(25 33 56 63)(26 34 53 64)(27 35 54 61)(28 36 55 62)
(1 40 46 64)(2 8 47 33)(3 38 48 62)(4 6 45 35)(5 20 34 42)(7 18 36 44)(9 28 21 59)(10 54 22 31)(11 26 23 57)(12 56 24 29)(13 30 51 53)(14 60 52 25)(15 32 49 55)(16 58 50 27)(17 63 43 39)(19 61 41 37)
(1 54 3 56)(2 53 4 55)(5 52 7 50)(6 51 8 49)(9 37 11 39)(10 40 12 38)(13 33 15 35)(14 36 16 34)(17 57 19 59)(18 60 20 58)(21 61 23 63)(22 64 24 62)(25 42 27 44)(26 41 28 43)(29 46 31 48)(30 45 32 47)
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), (1,21,42,15)(2,22,43,16)(3,23,44,13)(4,24,41,14)(5,30,40,57)(6,31,37,58)(7,32,38,59)(8,29,39,60)(9,20,49,46)(10,17,50,47)(11,18,51,48)(12,19,52,45)(25,33,56,63)(26,34,53,64)(27,35,54,61)(28,36,55,62), (1,40,46,64)(2,8,47,33)(3,38,48,62)(4,6,45,35)(5,20,34,42)(7,18,36,44)(9,28,21,59)(10,54,22,31)(11,26,23,57)(12,56,24,29)(13,30,51,53)(14,60,52,25)(15,32,49,55)(16,58,50,27)(17,63,43,39)(19,61,41,37), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,37,11,39)(10,40,12,38)(13,33,15,35)(14,36,16,34)(17,57,19,59)(18,60,20,58)(21,61,23,63)(22,64,24,62)(25,42,27,44)(26,41,28,43)(29,46,31,48)(30,45,32,47)>;
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), (1,21,42,15)(2,22,43,16)(3,23,44,13)(4,24,41,14)(5,30,40,57)(6,31,37,58)(7,32,38,59)(8,29,39,60)(9,20,49,46)(10,17,50,47)(11,18,51,48)(12,19,52,45)(25,33,56,63)(26,34,53,64)(27,35,54,61)(28,36,55,62), (1,40,46,64)(2,8,47,33)(3,38,48,62)(4,6,45,35)(5,20,34,42)(7,18,36,44)(9,28,21,59)(10,54,22,31)(11,26,23,57)(12,56,24,29)(13,30,51,53)(14,60,52,25)(15,32,49,55)(16,58,50,27)(17,63,43,39)(19,61,41,37), (1,54,3,56)(2,53,4,55)(5,52,7,50)(6,51,8,49)(9,37,11,39)(10,40,12,38)(13,33,15,35)(14,36,16,34)(17,57,19,59)(18,60,20,58)(21,61,23,63)(22,64,24,62)(25,42,27,44)(26,41,28,43)(29,46,31,48)(30,45,32,47) );
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)], [(1,21,42,15),(2,22,43,16),(3,23,44,13),(4,24,41,14),(5,30,40,57),(6,31,37,58),(7,32,38,59),(8,29,39,60),(9,20,49,46),(10,17,50,47),(11,18,51,48),(12,19,52,45),(25,33,56,63),(26,34,53,64),(27,35,54,61),(28,36,55,62)], [(1,40,46,64),(2,8,47,33),(3,38,48,62),(4,6,45,35),(5,20,34,42),(7,18,36,44),(9,28,21,59),(10,54,22,31),(11,26,23,57),(12,56,24,29),(13,30,51,53),(14,60,52,25),(15,32,49,55),(16,58,50,27),(17,63,43,39),(19,61,41,37)], [(1,54,3,56),(2,53,4,55),(5,52,7,50),(6,51,8,49),(9,37,11,39),(10,40,12,38),(13,33,15,35),(14,36,16,34),(17,57,19,59),(18,60,20,58),(21,61,23,63),(22,64,24,62),(25,42,27,44),(26,41,28,43),(29,46,31,48),(30,45,32,47)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 4 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 3 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 | 3 | 1 |
| 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 2 | 3 |
| 0 | 0 | 0 | 0 | 2 | 4 | 0 | 3 |
G:=sub<GL(8,GF(5))| [0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,4,3,3,0,0,0,0,3,1,1,0,0,0,0,0,2,0,0,1],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3],[3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,4,0,0,4,0,0,0,0,0,1,2,2,0,0,0,0,3,4,4,3,0,0,0,0,2,0,0,1],[0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,3,1,0,0,0,0,0,0,2,2,0,0,0,0,0,0,0,0,0,1,2,2,0,0,0,0,4,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4P | 4Q | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.189D4 | C42⋊5C4 | C24.C22 | C23⋊Q8 | C23.10D4 | C23.78C23 | C23.4Q8 | C2×C4×Q8 | C2×C4.4D4 | C42 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{189}D_4 % in TeX
G:=Group("C4^2.189D4"); // GroupNames label
G:=SmallGroup(128,1364);
// by ID
G=gap.SmallGroup(128,1364);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,120,758,723,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=1,d^2=a^2,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d^-1=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d^-1=a^2*c^-1>;
// generators/relations