p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.364C24, C24.284C23, C22.1702+ 1+4, C2.25D42, C22⋊C4⋊24D4, C23⋊2D4⋊15C2, C23.173(C2×D4), C2.47(D4⋊5D4), C23.33(C4○D4), (C23×C4).89C22, C23.8Q8⋊51C2, C23.10D4⋊31C2, C23.23D4⋊46C2, (C22×C4).817C23, (C2×C42).507C22, C22.244(C22×D4), C24.C22⋊48C2, C24.3C22⋊42C2, (C22×D4).136C22, C23.81C23⋊16C2, C2.36(C22.19C24), C2.C42.121C22, C2.20(C22.26C24), C2.23(C22.47C24), C2.13(C22.34C24), (C2×C4×D4)⋊39C2, (C2×C4).56(C2×D4), (C2×C4⋊D4)⋊13C2, (C4×C22⋊C4)⋊64C2, (C2×C4).114(C4○D4), (C2×C4⋊C4).850C22, C22.241(C2×C4○D4), (C2×C22.D4)⋊16C2, (C2×C22⋊C4).139C22, SmallGroup(128,1196)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.364C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=a, g2=ba=ab, ac=ca, ede-1=gdg-1=ad=da, ae=ea, af=fa, ag=ga, bc=cb, fdf=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef=ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 756 in 355 conjugacy classes, 108 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C22.D4, C23×C4, C22×D4, C4×C22⋊C4, C23.8Q8, C23.23D4, C24.C22, C24.3C22, C23⋊2D4, C23.10D4, C23.81C23, C2×C4×D4, C2×C4⋊D4, C2×C22.D4, C23.364C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, C22.19C24, C22.26C24, C22.34C24, D42, D4⋊5D4, C22.47C24, C23.364C24
►On 64 points
Generators in S64
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)
(1 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)
(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)
(1 49)(2 52)(3 51)(4 50)(5 54)(6 53)(7 56)(8 55)(9 21)(10 24)(11 23)(12 22)(13 45)(14 48)(15 47)(16 46)(17 41)(18 44)(19 43)(20 42)(25 38)(26 37)(27 40)(28 39)(29 63)(30 62)(31 61)(32 64)(33 60)(34 59)(35 58)(36 57)
(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 45)(2 62)(3 47)(4 64)(5 44)(6 57)(7 42)(8 59)(9 17)(10 33)(11 19)(12 35)(13 21)(14 37)(15 23)(16 39)(18 28)(20 26)(22 32)(24 30)(25 34)(27 36)(29 38)(31 40)(41 49)(43 51)(46 54)(48 56)(50 58)(52 60)(53 61)(55 63)
(1 15 11 41)(2 16 12 42)(3 13 9 43)(4 14 10 44)(5 64 37 33)(6 61 38 34)(7 62 39 35)(8 63 40 36)(17 51 47 21)(18 52 48 22)(19 49 45 23)(20 50 46 24)(25 57 53 29)(26 58 54 30)(27 59 55 31)(28 60 56 32)
G:=sub<Sym(64)| (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,45)(14,48)(15,47)(16,46)(17,41)(18,44)(19,43)(20,42)(25,38)(26,37)(27,40)(28,39)(29,63)(30,62)(31,61)(32,64)(33,60)(34,59)(35,58)(36,57), (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), (1,45)(2,62)(3,47)(4,64)(5,44)(6,57)(7,42)(8,59)(9,17)(10,33)(11,19)(12,35)(13,21)(14,37)(15,23)(16,39)(18,28)(20,26)(22,32)(24,30)(25,34)(27,36)(29,38)(31,40)(41,49)(43,51)(46,54)(48,56)(50,58)(52,60)(53,61)(55,63), (1,15,11,41)(2,16,12,42)(3,13,9,43)(4,14,10,44)(5,64,37,33)(6,61,38,34)(7,62,39,35)(8,63,40,36)(17,51,47,21)(18,52,48,22)(19,49,45,23)(20,50,46,24)(25,57,53,29)(26,58,54,30)(27,59,55,31)(28,60,56,32)>;
G:=Group( (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,49)(2,52)(3,51)(4,50)(5,54)(6,53)(7,56)(8,55)(9,21)(10,24)(11,23)(12,22)(13,45)(14,48)(15,47)(16,46)(17,41)(18,44)(19,43)(20,42)(25,38)(26,37)(27,40)(28,39)(29,63)(30,62)(31,61)(32,64)(33,60)(34,59)(35,58)(36,57), (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), (1,45)(2,62)(3,47)(4,64)(5,44)(6,57)(7,42)(8,59)(9,17)(10,33)(11,19)(12,35)(13,21)(14,37)(15,23)(16,39)(18,28)(20,26)(22,32)(24,30)(25,34)(27,36)(29,38)(31,40)(41,49)(43,51)(46,54)(48,56)(50,58)(52,60)(53,61)(55,63), (1,15,11,41)(2,16,12,42)(3,13,9,43)(4,14,10,44)(5,64,37,33)(6,61,38,34)(7,62,39,35)(8,63,40,36)(17,51,47,21)(18,52,48,22)(19,49,45,23)(20,50,46,24)(25,57,53,29)(26,58,54,30)(27,59,55,31)(28,60,56,32) );
G=PermutationGroup([[(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,9),(2,10),(3,11),(4,12),(5,39),(6,40),(7,37),(8,38),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,62),(34,63),(35,64),(36,61)], [(1,53),(2,54),(3,55),(4,56),(5,52),(6,49),(7,50),(8,51),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(1,49),(2,52),(3,51),(4,50),(5,54),(6,53),(7,56),(8,55),(9,21),(10,24),(11,23),(12,22),(13,45),(14,48),(15,47),(16,46),(17,41),(18,44),(19,43),(20,42),(25,38),(26,37),(27,40),(28,39),(29,63),(30,62),(31,61),(32,64),(33,60),(34,59),(35,58),(36,57)], [(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)], [(1,45),(2,62),(3,47),(4,64),(5,44),(6,57),(7,42),(8,59),(9,17),(10,33),(11,19),(12,35),(13,21),(14,37),(15,23),(16,39),(18,28),(20,26),(22,32),(24,30),(25,34),(27,36),(29,38),(31,40),(41,49),(43,51),(46,54),(48,56),(50,58),(52,60),(53,61),(55,63)], [(1,15,11,41),(2,16,12,42),(3,13,9,43),(4,14,10,44),(5,64,37,33),(6,61,38,34),(7,62,39,35),(8,63,40,36),(17,51,47,21),(18,52,48,22),(19,49,45,23),(20,50,46,24),(25,57,53,29),(26,58,54,30),(27,59,55,31),(28,60,56,32)]])
38 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2M | 2N | 4A | ··· | 4H | 4I | ··· | 4T | 4U | 4V | 4W |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 4 | ··· | 4 | 8 | 2 | ··· | 2 | 4 | ··· | 4 | 8 | 8 | 8 |
38 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | C4○D4 | 2+ 1+4 |
| kernel | C23.364C24 | C4×C22⋊C4 | C23.8Q8 | C23.23D4 | C24.C22 | C24.3C22 | C23⋊2D4 | C23.10D4 | C23.81C23 | C2×C4×D4 | C2×C4⋊D4 | C2×C22.D4 | C22⋊C4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 2 | 1 | 1 | 1 | 2 | 1 | 8 | 8 | 4 | 2 |
Matrix representation of C23.364C24 ►in GL6(𝔽5)
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 4 | 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 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 3 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(5))| [4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,3,2,0,0,0,0,1,2,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1],[0,4,0,0,0,0,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
C23.364C24 in GAP, Magma, Sage, TeX
C_2^3._{364}C_2^4 % in TeX
G:=Group("C2^3.364C2^4"); // GroupNames label
G:=SmallGroup(128,1196);
// by ID
G=gap.SmallGroup(128,1196);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,675,192]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=a,g^2=b*a=a*b,a*c=c*a,e*d*e^-1=g*d*g^-1=a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f=c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations