p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.190D4, C23.533C24, C24.371C23, C22.3102+ (1+4), C22.2272- (1+4), C42⋊5C4⋊26C2, C23.66(C4○D4), C23.8Q8⋊86C2, C23.4Q8⋊30C2, C23.11D4⋊61C2, (C23×C4).138C22, (C22×C4).143C23, (C2×C42).610C22, C22.358(C22×D4), C23.10D4.32C2, (C22×D4).542C22, C23.81C23⋊63C2, C2.84(C22.19C24), C2.42(C22.29C24), C2.C42.258C22, C2.41(C23.38C23), C2.40(C22.33C24), (C2×C4×D4).68C2, (C2×C4).392(C2×D4), (C2×C42.C2)⋊17C2, (C2×C4⋊C4).360C22, C22.405(C2×C4○D4), (C2×C22⋊C4).222C22, SmallGroup(128,1365)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 500 in 256 conjugacy classes, 96 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×16], C22, C22 [×6], C22 [×20], C2×C4 [×6], C2×C4 [×44], D4 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×18], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×6], C24 [×2], C2.C42 [×10], C2×C42, C2×C22⋊C4 [×10], C2×C4⋊C4 [×3], C2×C4⋊C4 [×8], C4×D4 [×4], C42.C2 [×4], C23×C4 [×2], C22×D4, C42⋊5C4, C23.8Q8 [×4], C23.10D4 [×2], C23.11D4 [×2], C23.81C23 [×2], C23.4Q8 [×2], C2×C4×D4, C2×C42.C2, C42.190D4
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, C22.33C24 [×4], C42.190D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=a-1, cbc-1=a2b, bd=db, dcd=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 9 57 29)(2 10 58 30)(3 11 59 31)(4 12 60 32)(5 38 49 23)(6 39 50 24)(7 40 51 21)(8 37 52 22)(13 53 25 44)(14 54 26 41)(15 55 27 42)(16 56 28 43)(17 62 36 47)(18 63 33 48)(19 64 34 45)(20 61 35 46)
(1 49 53 64)(2 8 54 48)(3 51 55 62)(4 6 56 46)(5 44 45 57)(7 42 47 59)(9 21 25 36)(10 39 26 20)(11 23 27 34)(12 37 28 18)(13 17 29 40)(14 35 30 24)(15 19 31 38)(16 33 32 22)(41 63 58 52)(43 61 60 50)
(1 56)(2 55)(3 54)(4 53)(5 52)(6 51)(7 50)(8 49)(9 28)(10 27)(11 26)(12 25)(13 32)(14 31)(15 30)(16 29)(17 35)(18 34)(19 33)(20 36)(21 39)(22 38)(23 37)(24 40)(41 59)(42 58)(43 57)(44 60)(45 63)(46 62)(47 61)(48 64)
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), (1,9,57,29)(2,10,58,30)(3,11,59,31)(4,12,60,32)(5,38,49,23)(6,39,50,24)(7,40,51,21)(8,37,52,22)(13,53,25,44)(14,54,26,41)(15,55,27,42)(16,56,28,43)(17,62,36,47)(18,63,33,48)(19,64,34,45)(20,61,35,46), (1,49,53,64)(2,8,54,48)(3,51,55,62)(4,6,56,46)(5,44,45,57)(7,42,47,59)(9,21,25,36)(10,39,26,20)(11,23,27,34)(12,37,28,18)(13,17,29,40)(14,35,30,24)(15,19,31,38)(16,33,32,22)(41,63,58,52)(43,61,60,50), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)(17,35)(18,34)(19,33)(20,36)(21,39)(22,38)(23,37)(24,40)(41,59)(42,58)(43,57)(44,60)(45,63)(46,62)(47,61)(48,64)>;
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), (1,9,57,29)(2,10,58,30)(3,11,59,31)(4,12,60,32)(5,38,49,23)(6,39,50,24)(7,40,51,21)(8,37,52,22)(13,53,25,44)(14,54,26,41)(15,55,27,42)(16,56,28,43)(17,62,36,47)(18,63,33,48)(19,64,34,45)(20,61,35,46), (1,49,53,64)(2,8,54,48)(3,51,55,62)(4,6,56,46)(5,44,45,57)(7,42,47,59)(9,21,25,36)(10,39,26,20)(11,23,27,34)(12,37,28,18)(13,17,29,40)(14,35,30,24)(15,19,31,38)(16,33,32,22)(41,63,58,52)(43,61,60,50), (1,56)(2,55)(3,54)(4,53)(5,52)(6,51)(7,50)(8,49)(9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)(17,35)(18,34)(19,33)(20,36)(21,39)(22,38)(23,37)(24,40)(41,59)(42,58)(43,57)(44,60)(45,63)(46,62)(47,61)(48,64) );
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)], [(1,9,57,29),(2,10,58,30),(3,11,59,31),(4,12,60,32),(5,38,49,23),(6,39,50,24),(7,40,51,21),(8,37,52,22),(13,53,25,44),(14,54,26,41),(15,55,27,42),(16,56,28,43),(17,62,36,47),(18,63,33,48),(19,64,34,45),(20,61,35,46)], [(1,49,53,64),(2,8,54,48),(3,51,55,62),(4,6,56,46),(5,44,45,57),(7,42,47,59),(9,21,25,36),(10,39,26,20),(11,23,27,34),(12,37,28,18),(13,17,29,40),(14,35,30,24),(15,19,31,38),(16,33,32,22),(41,63,58,52),(43,61,60,50)], [(1,56),(2,55),(3,54),(4,53),(5,52),(6,51),(7,50),(8,49),(9,28),(10,27),(11,26),(12,25),(13,32),(14,31),(15,30),(16,29),(17,35),(18,34),(19,33),(20,36),(21,39),(22,38),(23,37),(24,40),(41,59),(42,58),(43,57),(44,60),(45,63),(46,62),(47,61),(48,64)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 2 |
| 2 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 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 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 2 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 1 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3,0,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,4,2],[2,0,0,0,0,0,0,0,3,3,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1],[1,2,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0,0,0,1,0,0,0,0,0,0,0,3,4,0,0] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 4M | ··· | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 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 | C42.190D4 | C42⋊5C4 | C23.8Q8 | C23.10D4 | C23.11D4 | C23.81C23 | C23.4Q8 | C2×C4×D4 | C2×C42.C2 | C42 | C23 | C22 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 8 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{190}D_4 % in TeX
G:=Group("C4^2.190D4"); // GroupNames label
G:=SmallGroup(128,1365);
// by ID
G=gap.SmallGroup(128,1365);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,758,723,100,185,136]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^4=d^2=1,a*b=b*a,c*a*c^-1=a^-1*b^2,d*a*d=a^-1,c*b*c^-1=a^2*b,b*d=d*b,d*c*d=a^2*c^-1>;
// generators/relations