p-group, metabelian, nilpotent (class 2), monomial
Aliases: C42.178D4, C24.344C23, C23.480C24, C22.2632+ (1+4), (C2×D4)⋊19Q8, C42⋊8C4⋊48C2, C23.28(C2×Q8), C23⋊Q8⋊23C2, C2.40(D4⋊3Q8), C4.103(C22⋊Q8), C23.8Q8⋊73C2, C23.4Q8⋊24C2, (C22×C4).110C23, (C2×C42).574C22, (C23×C4).125C22, C22.321(C22×D4), C22.115(C22×Q8), (C22×D4).536C22, (C22×Q8).439C22, C23.65C23⋊91C2, C2.65(C22.19C24), C2.27(C22.29C24), C24.3C22.50C2, C2.C42.214C22, C2.22(C22.53C24), (C2×C4×Q8)⋊25C2, (C2×C4×D4).67C2, (C2×C4).312(C2×Q8), C2.38(C2×C22⋊Q8), (C2×C4).1196(C2×D4), (C2×C4).155(C4○D4), (C2×C4⋊C4).326C22, C22.356(C2×C4○D4), (C2×C22⋊C4).195C22, SmallGroup(128,1312)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 532 in 282 conjugacy classes, 112 normal (20 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×16], C22 [×3], C22 [×4], C22 [×20], C2×C4 [×14], C2×C4 [×40], D4 [×8], Q8 [×8], C23, C23 [×4], C23 [×12], C42 [×4], C42 [×4], C22⋊C4 [×16], C4⋊C4 [×16], C22×C4 [×3], C22×C4 [×10], C22×C4 [×8], C2×D4 [×4], C2×D4 [×4], C2×Q8 [×6], C24 [×2], C2.C42 [×8], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×10], C2×C4⋊C4 [×2], C2×C4⋊C4 [×8], C4×D4 [×4], C4×Q8 [×4], C23×C4 [×2], C22×D4, C22×Q8, C42⋊8C4, C23.8Q8 [×4], C23.65C23 [×2], C24.3C22 [×2], C23⋊Q8 [×2], C23.4Q8 [×2], C2×C4×D4, C2×C4×Q8, C42.178D4
Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×6], C24, C22⋊Q8 [×4], C22×D4, C22×Q8, C2×C4○D4 [×3], 2+ (1+4) [×2], C2×C22⋊Q8, C22.19C24, C22.29C24, D4⋊3Q8 [×2], C22.53C24 [×2], C42.178D4
Generators and relations
G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, cbc-1=dbd=b-1, 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 15 21 11)(2 16 22 12)(3 13 23 9)(4 14 24 10)(5 56 38 26)(6 53 39 27)(7 54 40 28)(8 55 37 25)(17 30 50 60)(18 31 51 57)(19 32 52 58)(20 29 49 59)(33 47 61 44)(34 48 62 41)(35 45 63 42)(36 46 64 43)
(1 61 39 49)(2 36 40 19)(3 63 37 51)(4 34 38 17)(5 50 24 62)(6 20 21 33)(7 52 22 64)(8 18 23 35)(9 42 55 57)(10 48 56 30)(11 44 53 59)(12 46 54 32)(13 45 25 31)(14 41 26 60)(15 47 27 29)(16 43 28 58)
(1 55)(2 26)(3 53)(4 28)(5 12)(6 13)(7 10)(8 15)(9 39)(11 37)(14 40)(16 38)(17 60)(18 31)(19 58)(20 29)(21 25)(22 56)(23 27)(24 54)(30 50)(32 52)(33 47)(34 41)(35 45)(36 43)(42 63)(44 61)(46 64)(48 62)(49 59)(51 57)
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,61,39,49)(2,36,40,19)(3,63,37,51)(4,34,38,17)(5,50,24,62)(6,20,21,33)(7,52,22,64)(8,18,23,35)(9,42,55,57)(10,48,56,30)(11,44,53,59)(12,46,54,32)(13,45,25,31)(14,41,26,60)(15,47,27,29)(16,43,28,58), (1,55)(2,26)(3,53)(4,28)(5,12)(6,13)(7,10)(8,15)(9,39)(11,37)(14,40)(16,38)(17,60)(18,31)(19,58)(20,29)(21,25)(22,56)(23,27)(24,54)(30,50)(32,52)(33,47)(34,41)(35,45)(36,43)(42,63)(44,61)(46,64)(48,62)(49,59)(51,57)>;
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), (1,15,21,11)(2,16,22,12)(3,13,23,9)(4,14,24,10)(5,56,38,26)(6,53,39,27)(7,54,40,28)(8,55,37,25)(17,30,50,60)(18,31,51,57)(19,32,52,58)(20,29,49,59)(33,47,61,44)(34,48,62,41)(35,45,63,42)(36,46,64,43), (1,61,39,49)(2,36,40,19)(3,63,37,51)(4,34,38,17)(5,50,24,62)(6,20,21,33)(7,52,22,64)(8,18,23,35)(9,42,55,57)(10,48,56,30)(11,44,53,59)(12,46,54,32)(13,45,25,31)(14,41,26,60)(15,47,27,29)(16,43,28,58), (1,55)(2,26)(3,53)(4,28)(5,12)(6,13)(7,10)(8,15)(9,39)(11,37)(14,40)(16,38)(17,60)(18,31)(19,58)(20,29)(21,25)(22,56)(23,27)(24,54)(30,50)(32,52)(33,47)(34,41)(35,45)(36,43)(42,63)(44,61)(46,64)(48,62)(49,59)(51,57) );
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)], [(1,15,21,11),(2,16,22,12),(3,13,23,9),(4,14,24,10),(5,56,38,26),(6,53,39,27),(7,54,40,28),(8,55,37,25),(17,30,50,60),(18,31,51,57),(19,32,52,58),(20,29,49,59),(33,47,61,44),(34,48,62,41),(35,45,63,42),(36,46,64,43)], [(1,61,39,49),(2,36,40,19),(3,63,37,51),(4,34,38,17),(5,50,24,62),(6,20,21,33),(7,52,22,64),(8,18,23,35),(9,42,55,57),(10,48,56,30),(11,44,53,59),(12,46,54,32),(13,45,25,31),(14,41,26,60),(15,47,27,29),(16,43,28,58)], [(1,55),(2,26),(3,53),(4,28),(5,12),(6,13),(7,10),(8,15),(9,39),(11,37),(14,40),(16,38),(17,60),(18,31),(19,58),(20,29),(21,25),(22,56),(23,27),(24,54),(30,50),(32,52),(33,47),(34,41),(35,45),(36,43),(42,63),(44,61),(46,64),(48,62),(49,59),(51,57)])
Matrix representation ►G ⊆ GL6(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 1 | 0 | 0 |
| 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
G:=sub<GL(6,GF(5))| [0,4,0,0,0,0,1,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,3,0,0,0,0,1,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,2,4,0,0,0,0,0,0,1,3,0,0,0,0,0,4] >;
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | ··· | 4H | 4I | ··· | 4V | 4W | 4X | 4Y | 4Z |
| 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 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 | 2+ (1+4) |
| kernel | C42.178D4 | C42⋊8C4 | C23.8Q8 | C23.65C23 | C24.3C22 | C23⋊Q8 | C23.4Q8 | C2×C4×D4 | C2×C4×Q8 | C42 | C2×D4 | C2×C4 | C22 |
| # reps | 1 | 1 | 4 | 2 | 2 | 2 | 2 | 1 | 1 | 4 | 4 | 12 | 2 |
In GAP, Magma, Sage, TeX
C_4^2._{178}D_4 % in TeX
G:=Group("C4^2.178D4"); // GroupNames label
G:=SmallGroup(128,1312);
// by ID
G=gap.SmallGroup(128,1312);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,456,758,723,352,675,80]);
// 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=d*b*d=b^-1,d*c*d=a^2*c^-1>;
// generators/relations