p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.187D4, C24.367C23, C23.528C24, C22.2232- (1+4), C22.3052+ (1+4), (C22×C4)⋊13Q8, C42⋊8C4⋊52C2, C23.67(C2×Q8), C4.86(C22⋊Q8), C23⋊Q8.15C2, (C22×C4).138C23, (C23×C4).430C22, (C2×C42).605C22, C22.353(C22×D4), C23.4Q8.15C2, C23.7Q8.58C2, C22.133(C22×Q8), (C22×Q8).155C22, C23.78C23⋊27C2, C23.83C23⋊60C2, C2.39(C22.29C24), C23.65C23⋊104C2, C2.C42.253C22, C2.47(C22.36C24), C2.39(C23.38C23), C2.20(C23.41C23), (C2×C4⋊Q8)⋊17C2, (C2×C4).387(C2×D4), (C2×C4).130(C2×Q8), C2.43(C2×C22⋊Q8), (C2×C4).660(C4○D4), (C2×C4⋊C4).357C22, C22.400(C2×C4○D4), (C2×C42⋊C2).47C2, (C2×C22⋊C4).217C22, SmallGroup(128,1360)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 452 in 244 conjugacy classes, 108 normal (26 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×12], C2×C4 [×44], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×20], C22×C4 [×6], C22×C4 [×12], C22×C4 [×4], C2×Q8 [×8], C24, C2.C42 [×12], C2×C42 [×2], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×2], C2×C4⋊C4 [×10], C42⋊C2 [×4], C4⋊Q8 [×4], C23×C4, C22×Q8 [×2], C23.7Q8 [×2], C42⋊8C4, C23.65C23 [×2], C23⋊Q8 [×2], C23.78C23 [×2], C23.4Q8 [×2], C23.83C23 [×2], C2×C42⋊C2, C2×C4⋊Q8, C42.187D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×2], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4, 2+ (1+4) [×2], 2- (1+4) [×2], C2×C22⋊Q8, C22.29C24, C23.38C23, C22.36C24 [×2], C23.41C23 [×2], C42.187D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=b-1, 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 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 13 17 45)(6 14 18 46)(7 15 19 47)(8 16 20 48)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 36 57 61)(30 33 58 62)(31 34 59 63)(32 35 60 64)
(1 13 49 59)(2 48 50 30)(3 15 51 57)(4 46 52 32)(5 25 34 43)(6 56 35 10)(7 27 36 41)(8 54 33 12)(9 19 55 61)(11 17 53 63)(14 24 60 38)(16 22 58 40)(18 28 64 42)(20 26 62 44)(21 31 39 45)(23 29 37 47)
(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)
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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,13,49,59)(2,48,50,30)(3,15,51,57)(4,46,52,32)(5,25,34,43)(6,56,35,10)(7,27,36,41)(8,54,33,12)(9,19,55,61)(11,17,53,63)(14,24,60,38)(16,22,58,40)(18,28,64,42)(20,26,62,44)(21,31,39,45)(23,29,37,47), (2,40)(4,38)(5,36)(6,62)(7,34)(8,64)(10,42)(12,44)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(22,50)(24,52)(26,54)(28,56)(29,45)(31,47)(46,58)(48,60)>;
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), (1,43,39,11)(2,44,40,12)(3,41,37,9)(4,42,38,10)(5,13,17,45)(6,14,18,46)(7,15,19,47)(8,16,20,48)(21,53,49,25)(22,54,50,26)(23,55,51,27)(24,56,52,28)(29,36,57,61)(30,33,58,62)(31,34,59,63)(32,35,60,64), (1,13,49,59)(2,48,50,30)(3,15,51,57)(4,46,52,32)(5,25,34,43)(6,56,35,10)(7,27,36,41)(8,54,33,12)(9,19,55,61)(11,17,53,63)(14,24,60,38)(16,22,58,40)(18,28,64,42)(20,26,62,44)(21,31,39,45)(23,29,37,47), (2,40)(4,38)(5,36)(6,62)(7,34)(8,64)(10,42)(12,44)(13,57)(14,30)(15,59)(16,32)(17,61)(18,33)(19,63)(20,35)(22,50)(24,52)(26,54)(28,56)(29,45)(31,47)(46,58)(48,60) );
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)], [(1,43,39,11),(2,44,40,12),(3,41,37,9),(4,42,38,10),(5,13,17,45),(6,14,18,46),(7,15,19,47),(8,16,20,48),(21,53,49,25),(22,54,50,26),(23,55,51,27),(24,56,52,28),(29,36,57,61),(30,33,58,62),(31,34,59,63),(32,35,60,64)], [(1,13,49,59),(2,48,50,30),(3,15,51,57),(4,46,52,32),(5,25,34,43),(6,56,35,10),(7,27,36,41),(8,54,33,12),(9,19,55,61),(11,17,53,63),(14,24,60,38),(16,22,58,40),(18,28,64,42),(20,26,62,44),(21,31,39,45),(23,29,37,47)], [(2,40),(4,38),(5,36),(6,62),(7,34),(8,64),(10,42),(12,44),(13,57),(14,30),(15,59),(16,32),(17,61),(18,33),(19,63),(20,35),(22,50),(24,52),(26,54),(28,56),(29,45),(31,47),(46,58),(48,60)])
Matrix representation ►G ⊆ GL8(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 1 | 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 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 | 4 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3],[0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C42.187D4 | C23.7Q8 | C42⋊8C4 | C23.65C23 | C23⋊Q8 | C23.78C23 | C23.4Q8 | C23.83C23 | C2×C42⋊C2 | C2×C4⋊Q8 | C42 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 4 | 2 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{187}D_4 % in TeX
G:=Group("C4^2.187D4"); // GroupNames label
G:=SmallGroup(128,1360);
// by ID
G=gap.SmallGroup(128,1360);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,185,80]);
// 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*b^2,c*b*c^-1=b^-1,b*d=d*b,d*c*d=a^2*c^-1>;
// generators/relations