p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.235C24, C24.557C23, C22.702+ 1+4, (C4×D4)⋊24C4, C42⋊26(C2×C4), C42⋊5C4⋊7C2, C42⋊4C4⋊15C2, C23.34D4⋊2C2, C23.91(C22×C4), (C2×C42).19C22, C23.7Q8⋊26C2, C23.291(C4○D4), C22.126(C23×C4), (C23×C4).306C22, (C22×C4).759C23, C23.23D4.8C2, C24.C22⋊14C2, (C22×D4).482C22, C23.63C23⋊16C2, C2.3(C22.45C24), C2.27(C22.11C24), C22.17(C42⋊C2), C2.C42.477C22, C2.4(C22.47C24), C4⋊C4⋊46(C2×C4), (C2×C4×D4).36C2, C2.30(C4×C4○D4), (C4×C22⋊C4)⋊10C2, C22⋊C4⋊43(C2×C4), (C22×C4)⋊13(C2×C4), (C2×D4).215(C2×C4), (C2×C4).721(C4○D4), (C2×C4⋊C4).188C22, (C2×C4).493(C22×C4), C2.32(C2×C42⋊C2), C22.120(C2×C4○D4), (C2×C2.C42)⋊20C2, (C2×C22⋊C4).440C22, SmallGroup(128,1085)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.235C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=d, f2=c, g2=cb=bc, faf-1=ab=ba, ac=ca, ad=da, ae=ea, ag=ga, bd=db, fef-1=be=eb, bf=fb, bg=gb, cd=dc, geg-1=ce=ec, cf=fc, cg=gc, de=ed, df=fd, dg=gd, fg=gf >
Subgroups: 524 in 294 conjugacy classes, 144 normal (34 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C2.C42, C2.C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C23×C4, C23×C4, C22×D4, C2×C2.C42, C42⋊4C4, C4×C22⋊C4, C4×C22⋊C4, C23.7Q8, C23.34D4, C42⋊5C4, C23.23D4, C23.63C23, C24.C22, C2×C4×D4, C23.235C24
Quotients: C1, C2, C4, C22, C2×C4, C23, C22×C4, C4○D4, C24, C42⋊C2, C23×C4, C2×C4○D4, 2+ 1+4, C2×C42⋊C2, C4×C4○D4, C22.11C24, C22.45C24, C22.47C24, C23.235C24
►On 64 points
Generators in S64
(1 43)(2 44)(3 41)(4 42)(5 18)(6 19)(7 20)(8 17)(9 13)(10 14)(11 15)(12 16)(21 25)(22 26)(23 27)(24 28)(29 35)(30 36)(31 33)(32 34)(37 45)(38 46)(39 47)(40 48)(49 53)(50 54)(51 55)(52 56)(57 61)(58 62)(59 63)(60 64)
(1 11)(2 12)(3 9)(4 10)(5 38)(6 39)(7 40)(8 37)(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 53)(26 54)(27 55)(28 56)(29 57)(30 58)(31 59)(32 60)(33 63)(34 64)(35 61)(36 62)
(1 51)(2 52)(3 49)(4 50)(5 36)(6 33)(7 34)(8 35)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(37 61)(38 62)(39 63)(40 64)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 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 39 51 63)(2 7 52 34)(3 37 49 61)(4 5 50 36)(6 23 33 11)(8 21 35 9)(10 38 22 62)(12 40 24 64)(13 45 25 57)(14 18 26 30)(15 47 27 59)(16 20 28 32)(17 53 29 41)(19 55 31 43)(42 46 54 58)(44 48 56 60)
(1 31 23 47)(2 20 24 60)(3 29 21 45)(4 18 22 58)(5 26 62 42)(6 15 63 55)(7 28 64 44)(8 13 61 53)(9 57 49 17)(10 46 50 30)(11 59 51 19)(12 48 52 32)(14 38 54 36)(16 40 56 34)(25 37 41 35)(27 39 43 33)
G:=sub<Sym(64)| (1,43)(2,44)(3,41)(4,42)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,2,3,4)(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), (1,39,51,63)(2,7,52,34)(3,37,49,61)(4,5,50,36)(6,23,33,11)(8,21,35,9)(10,38,22,62)(12,40,24,64)(13,45,25,57)(14,18,26,30)(15,47,27,59)(16,20,28,32)(17,53,29,41)(19,55,31,43)(42,46,54,58)(44,48,56,60), (1,31,23,47)(2,20,24,60)(3,29,21,45)(4,18,22,58)(5,26,62,42)(6,15,63,55)(7,28,64,44)(8,13,61,53)(9,57,49,17)(10,46,50,30)(11,59,51,19)(12,48,52,32)(14,38,54,36)(16,40,56,34)(25,37,41,35)(27,39,43,33)>;
G:=Group( (1,43)(2,44)(3,41)(4,42)(5,18)(6,19)(7,20)(8,17)(9,13)(10,14)(11,15)(12,16)(21,25)(22,26)(23,27)(24,28)(29,35)(30,36)(31,33)(32,34)(37,45)(38,46)(39,47)(40,48)(49,53)(50,54)(51,55)(52,56)(57,61)(58,62)(59,63)(60,64), (1,11)(2,12)(3,9)(4,10)(5,38)(6,39)(7,40)(8,37)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(27,55)(28,56)(29,57)(30,58)(31,59)(32,60)(33,63)(34,64)(35,61)(36,62), (1,51)(2,52)(3,49)(4,50)(5,36)(6,33)(7,34)(8,35)(9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(37,61)(38,62)(39,63)(40,64)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,2,3,4)(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), (1,39,51,63)(2,7,52,34)(3,37,49,61)(4,5,50,36)(6,23,33,11)(8,21,35,9)(10,38,22,62)(12,40,24,64)(13,45,25,57)(14,18,26,30)(15,47,27,59)(16,20,28,32)(17,53,29,41)(19,55,31,43)(42,46,54,58)(44,48,56,60), (1,31,23,47)(2,20,24,60)(3,29,21,45)(4,18,22,58)(5,26,62,42)(6,15,63,55)(7,28,64,44)(8,13,61,53)(9,57,49,17)(10,46,50,30)(11,59,51,19)(12,48,52,32)(14,38,54,36)(16,40,56,34)(25,37,41,35)(27,39,43,33) );
G=PermutationGroup([[(1,43),(2,44),(3,41),(4,42),(5,18),(6,19),(7,20),(8,17),(9,13),(10,14),(11,15),(12,16),(21,25),(22,26),(23,27),(24,28),(29,35),(30,36),(31,33),(32,34),(37,45),(38,46),(39,47),(40,48),(49,53),(50,54),(51,55),(52,56),(57,61),(58,62),(59,63),(60,64)], [(1,11),(2,12),(3,9),(4,10),(5,38),(6,39),(7,40),(8,37),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,53),(26,54),(27,55),(28,56),(29,57),(30,58),(31,59),(32,60),(33,63),(34,64),(35,61),(36,62)], [(1,51),(2,52),(3,49),(4,50),(5,36),(6,33),(7,34),(8,35),(9,21),(10,22),(11,23),(12,24),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(37,61),(38,62),(39,63),(40,64),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,2,3,4),(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)], [(1,39,51,63),(2,7,52,34),(3,37,49,61),(4,5,50,36),(6,23,33,11),(8,21,35,9),(10,38,22,62),(12,40,24,64),(13,45,25,57),(14,18,26,30),(15,47,27,59),(16,20,28,32),(17,53,29,41),(19,55,31,43),(42,46,54,58),(44,48,56,60)], [(1,31,23,47),(2,20,24,60),(3,29,21,45),(4,18,22,58),(5,26,62,42),(6,15,63,55),(7,28,64,44),(8,13,61,53),(9,57,49,17),(10,46,50,30),(11,59,51,19),(12,48,52,32),(14,38,54,36),(16,40,56,34),(25,37,41,35),(27,39,43,33)]])
50 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 4A | ··· | 4T | 4U | ··· | 4AJ |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
50 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4○D4 | C4○D4 | 2+ 1+4 |
| kernel | C23.235C24 | C2×C2.C42 | C42⋊4C4 | C4×C22⋊C4 | C23.7Q8 | C23.34D4 | C42⋊5C4 | C23.23D4 | C23.63C23 | C24.C22 | C2×C4×D4 | C4×D4 | C2×C4 | C23 | C22 |
| # reps | 1 | 1 | 1 | 3 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 16 | 8 | 8 | 2 |
Matrix representation of C23.235C24 ►in GL5(𝔽5)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 | 0 |
| 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
| 4 | 0 | 0 | 0 | 0 |
| 0 | 3 | 1 | 0 | 0 |
| 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 0 | 2 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,4,0,0,0,0,3,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[3,0,0,0,0,0,4,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,1,0],[4,0,0,0,0,0,3,2,0,0,0,1,2,0,0,0,0,0,2,0,0,0,0,0,2],[1,0,0,0,0,0,2,0,0,0,0,0,2,0,0,0,0,0,3,0,0,0,0,0,2] >;
C23.235C24 in GAP, Magma, Sage, TeX
C_2^3._{235}C_2^4 % in TeX
G:=Group("C2^3.235C2^4"); // GroupNames label
G:=SmallGroup(128,1085);
// by ID
G=gap.SmallGroup(128,1085);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,232,758,268,346]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=d,f^2=c,g^2=c*b=b*c,f*a*f^-1=a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*g=g*a,b*d=d*b,f*e*f^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*e*g^-1=c*e=e*c,c*f=f*c,c*g=g*c,d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
// generators/relations