p-group, metabelian, nilpotent (class 2), monomial
Aliases: C24.96C23, C23.700C24, C22.4732+ (1+4), C22.3622- (1+4), C23.Q8⋊89C2, (C2×C42).114C22, (C22×C4).609C23, C23.11D4⋊123C2, C23.10D4.68C2, (C22×D4).286C22, C24.C22⋊174C2, C24.3C22.77C2, C23.65C23⋊159C2, C23.63C23⋊194C2, C2.107(C22.32C24), C2.C42.404C22, C2.44(C22.56C24), C2.66(C22.34C24), C2.75(C22.50C24), C2.122(C22.36C24), C2.120(C22.47C24), (C2×C4).241(C4○D4), (C2×C4⋊C4).510C22, C22.561(C2×C4○D4), (C2×C22⋊C4).328C22, SmallGroup(128,1532)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Subgroups: 436 in 210 conjugacy classes, 88 normal (82 characteristic)
C1, C2 [×7], C2 [×2], C4 [×16], C22 [×7], C22 [×14], C2×C4 [×6], C2×C4 [×36], D4 [×4], C23, C23 [×14], C42 [×3], C22⋊C4 [×16], C4⋊C4 [×13], C22×C4 [×13], C2×D4 [×4], C24 [×2], C2.C42 [×10], C2×C42 [×3], C2×C22⋊C4 [×12], C2×C4⋊C4 [×9], C22×D4, C23.63C23 [×2], C24.C22 [×4], C23.65C23 [×2], C24.3C22, C23.10D4 [×2], C23.Q8 [×2], C23.11D4 [×2], C23.700C24
Quotients:
C1, C2 [×15], C22 [×35], C23 [×15], C4○D4 [×6], C24, C2×C4○D4 [×3], 2+ (1+4) [×3], 2- (1+4), C22.32C24, C22.34C24, C22.36C24, C22.47C24 [×2], C22.50C24, C22.56C24, C23.700C24
Generators and relations
G = < a,b,c,d,e,f,g | a2=b2=c2=1, d2=cb=bc, e2=a, f2=ca=ac, g2=b, ab=ba, ede-1=ad=da, geg-1=ae=ea, af=fa, ag=ga, fdf-1=bd=db, be=eb, bf=fb, bg=gb, cd=dc, fef-1=ce=ec, cf=fc, cg=gc, gdg-1=abd, fg=gf >
(1 42)(2 43)(3 44)(4 41)(5 24)(6 21)(7 22)(8 23)(9 18)(10 19)(11 20)(12 17)(13 32)(14 29)(15 30)(16 31)(25 64)(26 61)(27 62)(28 63)(33 60)(34 57)(35 58)(36 59)(37 53)(38 54)(39 55)(40 56)(45 52)(46 49)(47 50)(48 51)
(1 14)(2 15)(3 16)(4 13)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(25 49)(26 50)(27 51)(28 52)(29 42)(30 43)(31 44)(32 41)(33 38)(34 39)(35 40)(36 37)(45 63)(46 64)(47 61)(48 62)(53 59)(54 60)(55 57)(56 58)
(1 16)(2 13)(3 14)(4 15)(5 19)(6 20)(7 17)(8 18)(9 23)(10 24)(11 21)(12 22)(25 51)(26 52)(27 49)(28 50)(29 44)(30 41)(31 42)(32 43)(33 40)(34 37)(35 38)(36 39)(45 61)(46 62)(47 63)(48 64)(53 57)(54 58)(55 59)(56 60)
(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 61 42 26)(2 27 43 62)(3 63 44 28)(4 25 41 64)(5 60 24 33)(6 34 21 57)(7 58 22 35)(8 36 23 59)(9 55 18 39)(10 40 19 56)(11 53 20 37)(12 38 17 54)(13 49 32 46)(14 47 29 50)(15 51 30 48)(16 45 31 52)
(1 17 31 22)(2 6 32 11)(3 19 29 24)(4 8 30 9)(5 44 10 14)(7 42 12 16)(13 20 43 21)(15 18 41 23)(25 39 48 59)(26 35 45 54)(27 37 46 57)(28 33 47 56)(34 62 53 49)(36 64 55 51)(38 61 58 52)(40 63 60 50)
(1 18 14 6)(2 22 15 10)(3 20 16 8)(4 24 13 12)(5 32 17 41)(7 30 19 43)(9 29 21 42)(11 31 23 44)(25 60 49 54)(26 39 50 34)(27 58 51 56)(28 37 52 36)(33 46 38 64)(35 48 40 62)(45 59 63 53)(47 57 61 55)
G:=sub<Sym(64)| (1,42)(2,43)(3,44)(4,41)(5,24)(6,21)(7,22)(8,23)(9,18)(10,19)(11,20)(12,17)(13,32)(14,29)(15,30)(16,31)(25,64)(26,61)(27,62)(28,63)(33,60)(34,57)(35,58)(36,59)(37,53)(38,54)(39,55)(40,56)(45,52)(46,49)(47,50)(48,51), (1,14)(2,15)(3,16)(4,13)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,49)(26,50)(27,51)(28,52)(29,42)(30,43)(31,44)(32,41)(33,38)(34,39)(35,40)(36,37)(45,63)(46,64)(47,61)(48,62)(53,59)(54,60)(55,57)(56,58), (1,16)(2,13)(3,14)(4,15)(5,19)(6,20)(7,17)(8,18)(9,23)(10,24)(11,21)(12,22)(25,51)(26,52)(27,49)(28,50)(29,44)(30,41)(31,42)(32,43)(33,40)(34,37)(35,38)(36,39)(45,61)(46,62)(47,63)(48,64)(53,57)(54,58)(55,59)(56,60), (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), (1,61,42,26)(2,27,43,62)(3,63,44,28)(4,25,41,64)(5,60,24,33)(6,34,21,57)(7,58,22,35)(8,36,23,59)(9,55,18,39)(10,40,19,56)(11,53,20,37)(12,38,17,54)(13,49,32,46)(14,47,29,50)(15,51,30,48)(16,45,31,52), (1,17,31,22)(2,6,32,11)(3,19,29,24)(4,8,30,9)(5,44,10,14)(7,42,12,16)(13,20,43,21)(15,18,41,23)(25,39,48,59)(26,35,45,54)(27,37,46,57)(28,33,47,56)(34,62,53,49)(36,64,55,51)(38,61,58,52)(40,63,60,50), (1,18,14,6)(2,22,15,10)(3,20,16,8)(4,24,13,12)(5,32,17,41)(7,30,19,43)(9,29,21,42)(11,31,23,44)(25,60,49,54)(26,39,50,34)(27,58,51,56)(28,37,52,36)(33,46,38,64)(35,48,40,62)(45,59,63,53)(47,57,61,55)>;
G:=Group( (1,42)(2,43)(3,44)(4,41)(5,24)(6,21)(7,22)(8,23)(9,18)(10,19)(11,20)(12,17)(13,32)(14,29)(15,30)(16,31)(25,64)(26,61)(27,62)(28,63)(33,60)(34,57)(35,58)(36,59)(37,53)(38,54)(39,55)(40,56)(45,52)(46,49)(47,50)(48,51), (1,14)(2,15)(3,16)(4,13)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(25,49)(26,50)(27,51)(28,52)(29,42)(30,43)(31,44)(32,41)(33,38)(34,39)(35,40)(36,37)(45,63)(46,64)(47,61)(48,62)(53,59)(54,60)(55,57)(56,58), (1,16)(2,13)(3,14)(4,15)(5,19)(6,20)(7,17)(8,18)(9,23)(10,24)(11,21)(12,22)(25,51)(26,52)(27,49)(28,50)(29,44)(30,41)(31,42)(32,43)(33,40)(34,37)(35,38)(36,39)(45,61)(46,62)(47,63)(48,64)(53,57)(54,58)(55,59)(56,60), (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), (1,61,42,26)(2,27,43,62)(3,63,44,28)(4,25,41,64)(5,60,24,33)(6,34,21,57)(7,58,22,35)(8,36,23,59)(9,55,18,39)(10,40,19,56)(11,53,20,37)(12,38,17,54)(13,49,32,46)(14,47,29,50)(15,51,30,48)(16,45,31,52), (1,17,31,22)(2,6,32,11)(3,19,29,24)(4,8,30,9)(5,44,10,14)(7,42,12,16)(13,20,43,21)(15,18,41,23)(25,39,48,59)(26,35,45,54)(27,37,46,57)(28,33,47,56)(34,62,53,49)(36,64,55,51)(38,61,58,52)(40,63,60,50), (1,18,14,6)(2,22,15,10)(3,20,16,8)(4,24,13,12)(5,32,17,41)(7,30,19,43)(9,29,21,42)(11,31,23,44)(25,60,49,54)(26,39,50,34)(27,58,51,56)(28,37,52,36)(33,46,38,64)(35,48,40,62)(45,59,63,53)(47,57,61,55) );
G=PermutationGroup([(1,42),(2,43),(3,44),(4,41),(5,24),(6,21),(7,22),(8,23),(9,18),(10,19),(11,20),(12,17),(13,32),(14,29),(15,30),(16,31),(25,64),(26,61),(27,62),(28,63),(33,60),(34,57),(35,58),(36,59),(37,53),(38,54),(39,55),(40,56),(45,52),(46,49),(47,50),(48,51)], [(1,14),(2,15),(3,16),(4,13),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(25,49),(26,50),(27,51),(28,52),(29,42),(30,43),(31,44),(32,41),(33,38),(34,39),(35,40),(36,37),(45,63),(46,64),(47,61),(48,62),(53,59),(54,60),(55,57),(56,58)], [(1,16),(2,13),(3,14),(4,15),(5,19),(6,20),(7,17),(8,18),(9,23),(10,24),(11,21),(12,22),(25,51),(26,52),(27,49),(28,50),(29,44),(30,41),(31,42),(32,43),(33,40),(34,37),(35,38),(36,39),(45,61),(46,62),(47,63),(48,64),(53,57),(54,58),(55,59),(56,60)], [(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)], [(1,61,42,26),(2,27,43,62),(3,63,44,28),(4,25,41,64),(5,60,24,33),(6,34,21,57),(7,58,22,35),(8,36,23,59),(9,55,18,39),(10,40,19,56),(11,53,20,37),(12,38,17,54),(13,49,32,46),(14,47,29,50),(15,51,30,48),(16,45,31,52)], [(1,17,31,22),(2,6,32,11),(3,19,29,24),(4,8,30,9),(5,44,10,14),(7,42,12,16),(13,20,43,21),(15,18,41,23),(25,39,48,59),(26,35,45,54),(27,37,46,57),(28,33,47,56),(34,62,53,49),(36,64,55,51),(38,61,58,52),(40,63,60,50)], [(1,18,14,6),(2,22,15,10),(3,20,16,8),(4,24,13,12),(5,32,17,41),(7,30,19,43),(9,29,21,42),(11,31,23,44),(25,60,49,54),(26,39,50,34),(27,58,51,56),(28,37,52,36),(33,46,38,64),(35,48,40,62),(45,59,63,53),(47,57,61,55)])
Matrix representation ►G ⊆ 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 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 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 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 4 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 2 |
| 0 | 0 | 0 | 0 | 3 | 0 |
| 1 | 3 | 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 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,3,0,0,0,0,4,2,0,0,0,0,0,0,3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[3,3,0,0,0,0,0,2,0,0,0,0,0,0,4,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,2,3,0,0,0,0,0,3,0,0,0,0,0,0,0,3,0,0,0,0,2,0],[1,0,0,0,0,0,3,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 4A | ··· | 4R | 4S | 4T | 4U | 4V |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 4 | ··· | 4 | 4 | 4 | 4 | 4 |
| size | 1 | 1 | ··· | 1 | 8 | 8 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4○D4 | 2+ (1+4) | 2- (1+4) |
| kernel | C23.700C24 | C23.63C23 | C24.C22 | C23.65C23 | C24.3C22 | C23.10D4 | C23.Q8 | C23.11D4 | C2×C4 | C22 | C22 |
| # reps | 1 | 2 | 4 | 2 | 1 | 2 | 2 | 2 | 12 | 3 | 1 |
In GAP, Magma, Sage, TeX
C_2^3._{700}C_2^4 % in TeX
G:=Group("C2^3.700C2^4"); // GroupNames label
G:=SmallGroup(128,1532);
// by ID
G=gap.SmallGroup(128,1532);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,672,253,120,758,723,436,1571,346,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=1,d^2=c*b=b*c,e^2=a,f^2=c*a=a*c,g^2=b,a*b=b*a,e*d*e^-1=a*d=d*a,g*e*g^-1=a*e=e*a,a*f=f*a,a*g=g*a,f*d*f^-1=b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,f*e*f^-1=c*e=e*c,c*f=f*c,c*g=g*c,g*d*g^-1=a*b*d,f*g=g*f>;
// generators/relations