p-group, metabelian, nilpotent (class 2), monomial
Aliases: C23.524C24, C24.365C23, C22.3012+ 1+4, C22.2202- 1+4, C23.195(C2×D4), (C22×C4).402D4, C23.4Q8⋊27C2, C23.7Q8⋊78C2, C23.10D4⋊59C2, C23.11D4⋊58C2, (C22×C4).134C23, (C23×C4).426C22, (C2×C42).603C22, C22.349(C22×D4), C24.3C22⋊66C2, C4.96(C22.D4), (C22×D4).195C22, C2.37(C22.29C24), C23.65C23⋊102C2, C2.C42.250C22, C2.45(C22.36C24), C2.25(C22.31C24), C2.25(C22.34C24), (C2×C4).383(C2×D4), (C2×C4⋊D4).40C2, (C2×C42⋊C2)⋊38C2, (C2×C4).658(C4○D4), (C2×C4⋊C4).355C22, C22.396(C2×C4○D4), C2.42(C2×C22.D4), (C2×C22⋊C4).215C22, SmallGroup(128,1356)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C23.524C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=f2=1, e2=c, g2=b, eae-1=ab=ba, faf=ac=ca, ad=da, ag=ga, bc=cb, bd=db, be=eb, gfg-1=bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef=de=ed, df=fd, dg=gd, eg=ge >
Subgroups: 580 in 274 conjugacy classes, 100 normal (20 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C23×C4, C22×D4, C22×D4, C23.7Q8, C23.65C23, C24.3C22, C23.10D4, C23.11D4, C23.4Q8, C2×C42⋊C2, C2×C4⋊D4, C23.524C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22.D4, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C2×C22.D4, C22.29C24, C22.31C24, C22.34C24, C22.36C24, C23.524C24
►On 64 points
Generators in S64
(1 47)(2 35)(3 45)(4 33)(5 53)(6 51)(7 55)(8 49)(9 50)(10 54)(11 52)(12 56)(13 57)(14 42)(15 59)(16 44)(17 38)(18 31)(19 40)(20 29)(21 36)(22 46)(23 34)(24 48)(25 41)(26 58)(27 43)(28 60)(30 62)(32 64)(37 61)(39 63)
(1 23)(2 24)(3 21)(4 22)(5 9)(6 10)(7 11)(8 12)(13 25)(14 26)(15 27)(16 28)(17 62)(18 63)(19 64)(20 61)(29 37)(30 38)(31 39)(32 40)(33 46)(34 47)(35 48)(36 45)(41 57)(42 58)(43 59)(44 60)(49 56)(50 53)(51 54)(52 55)
(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 27)(2 28)(3 25)(4 26)(5 37)(6 38)(7 39)(8 40)(9 29)(10 30)(11 31)(12 32)(13 21)(14 22)(15 23)(16 24)(17 51)(18 52)(19 49)(20 50)(33 58)(34 59)(35 60)(36 57)(41 45)(42 46)(43 47)(44 48)(53 61)(54 62)(55 63)(56 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)
(2 28)(4 26)(5 9)(6 30)(7 11)(8 32)(10 38)(12 40)(14 22)(16 24)(17 56)(18 61)(19 54)(20 63)(29 37)(31 39)(33 60)(34 36)(35 58)(41 43)(42 48)(44 46)(45 47)(49 62)(50 55)(51 64)(52 53)(57 59)
(1 7 23 11)(2 8 24 12)(3 5 21 9)(4 6 22 10)(13 29 25 37)(14 30 26 38)(15 31 27 39)(16 32 28 40)(17 42 62 58)(18 43 63 59)(19 44 64 60)(20 41 61 57)(33 51 46 54)(34 52 47 55)(35 49 48 56)(36 50 45 53)
G:=sub<Sym(64)| (1,47)(2,35)(3,45)(4,33)(5,53)(6,51)(7,55)(8,49)(9,50)(10,54)(11,52)(12,56)(13,57)(14,42)(15,59)(16,44)(17,38)(18,31)(19,40)(20,29)(21,36)(22,46)(23,34)(24,48)(25,41)(26,58)(27,43)(28,60)(30,62)(32,64)(37,61)(39,63), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,62)(18,63)(19,64)(20,61)(29,37)(30,38)(31,39)(32,40)(33,46)(34,47)(35,48)(36,45)(41,57)(42,58)(43,59)(44,60)(49,56)(50,53)(51,54)(52,55), (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), (1,27)(2,28)(3,25)(4,26)(5,37)(6,38)(7,39)(8,40)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,51)(18,52)(19,49)(20,50)(33,58)(34,59)(35,60)(36,57)(41,45)(42,46)(43,47)(44,48)(53,61)(54,62)(55,63)(56,64), (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), (2,28)(4,26)(5,9)(6,30)(7,11)(8,32)(10,38)(12,40)(14,22)(16,24)(17,56)(18,61)(19,54)(20,63)(29,37)(31,39)(33,60)(34,36)(35,58)(41,43)(42,48)(44,46)(45,47)(49,62)(50,55)(51,64)(52,53)(57,59), (1,7,23,11)(2,8,24,12)(3,5,21,9)(4,6,22,10)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,42,62,58)(18,43,63,59)(19,44,64,60)(20,41,61,57)(33,51,46,54)(34,52,47,55)(35,49,48,56)(36,50,45,53)>;
G:=Group( (1,47)(2,35)(3,45)(4,33)(5,53)(6,51)(7,55)(8,49)(9,50)(10,54)(11,52)(12,56)(13,57)(14,42)(15,59)(16,44)(17,38)(18,31)(19,40)(20,29)(21,36)(22,46)(23,34)(24,48)(25,41)(26,58)(27,43)(28,60)(30,62)(32,64)(37,61)(39,63), (1,23)(2,24)(3,21)(4,22)(5,9)(6,10)(7,11)(8,12)(13,25)(14,26)(15,27)(16,28)(17,62)(18,63)(19,64)(20,61)(29,37)(30,38)(31,39)(32,40)(33,46)(34,47)(35,48)(36,45)(41,57)(42,58)(43,59)(44,60)(49,56)(50,53)(51,54)(52,55), (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), (1,27)(2,28)(3,25)(4,26)(5,37)(6,38)(7,39)(8,40)(9,29)(10,30)(11,31)(12,32)(13,21)(14,22)(15,23)(16,24)(17,51)(18,52)(19,49)(20,50)(33,58)(34,59)(35,60)(36,57)(41,45)(42,46)(43,47)(44,48)(53,61)(54,62)(55,63)(56,64), (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), (2,28)(4,26)(5,9)(6,30)(7,11)(8,32)(10,38)(12,40)(14,22)(16,24)(17,56)(18,61)(19,54)(20,63)(29,37)(31,39)(33,60)(34,36)(35,58)(41,43)(42,48)(44,46)(45,47)(49,62)(50,55)(51,64)(52,53)(57,59), (1,7,23,11)(2,8,24,12)(3,5,21,9)(4,6,22,10)(13,29,25,37)(14,30,26,38)(15,31,27,39)(16,32,28,40)(17,42,62,58)(18,43,63,59)(19,44,64,60)(20,41,61,57)(33,51,46,54)(34,52,47,55)(35,49,48,56)(36,50,45,53) );
G=PermutationGroup([[(1,47),(2,35),(3,45),(4,33),(5,53),(6,51),(7,55),(8,49),(9,50),(10,54),(11,52),(12,56),(13,57),(14,42),(15,59),(16,44),(17,38),(18,31),(19,40),(20,29),(21,36),(22,46),(23,34),(24,48),(25,41),(26,58),(27,43),(28,60),(30,62),(32,64),(37,61),(39,63)], [(1,23),(2,24),(3,21),(4,22),(5,9),(6,10),(7,11),(8,12),(13,25),(14,26),(15,27),(16,28),(17,62),(18,63),(19,64),(20,61),(29,37),(30,38),(31,39),(32,40),(33,46),(34,47),(35,48),(36,45),(41,57),(42,58),(43,59),(44,60),(49,56),(50,53),(51,54),(52,55)], [(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)], [(1,27),(2,28),(3,25),(4,26),(5,37),(6,38),(7,39),(8,40),(9,29),(10,30),(11,31),(12,32),(13,21),(14,22),(15,23),(16,24),(17,51),(18,52),(19,49),(20,50),(33,58),(34,59),(35,60),(36,57),(41,45),(42,46),(43,47),(44,48),(53,61),(54,62),(55,63),(56,64)], [(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)], [(2,28),(4,26),(5,9),(6,30),(7,11),(8,32),(10,38),(12,40),(14,22),(16,24),(17,56),(18,61),(19,54),(20,63),(29,37),(31,39),(33,60),(34,36),(35,58),(41,43),(42,48),(44,46),(45,47),(49,62),(50,55),(51,64),(52,53),(57,59)], [(1,7,23,11),(2,8,24,12),(3,5,21,9),(4,6,22,10),(13,29,25,37),(14,30,26,38),(15,31,27,39),(16,32,28,40),(17,42,62,58),(18,43,63,59),(19,44,64,60),(20,41,61,57),(33,51,46,54),(34,52,47,55),(35,49,48,56),(36,50,45,53)]])
32 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 4A | 4B | 4C | 4D | 4E | ··· | 4N | 4O | ··· | 4T |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | ··· | 8 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | D4 | C4○D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C23.524C24 | C23.7Q8 | C23.65C23 | C24.3C22 | C23.10D4 | C23.11D4 | C23.4Q8 | C2×C42⋊C2 | C2×C4⋊D4 | C22×C4 | C2×C4 | C22 | C22 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 2 | 1 | 1 | 4 | 8 | 3 | 1 |
Matrix representation of C23.524C24 ►in GL8(𝔽5)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 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 | 3 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 2 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 3 | 3 |
| 1 | 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 | 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 | 4 |
| 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 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 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 | 4 |
| 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 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 |
| 1 | 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 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 4 | 3 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 |
G:=sub<GL(8,GF(5))| [0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,3,2,0,0,0,0,0,0,1,2,0,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,4,3],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[3,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1,0,0,0,0,0,0,0,0,4,1,0,0,0,0,0,0,3,1] >;
C23.524C24 in GAP, Magma, Sage, TeX
C_2^3._{524}C_2^4 % in TeX
G:=Group("C2^3.524C2^4"); // GroupNames label
G:=SmallGroup(128,1356);
// by ID
G=gap.SmallGroup(128,1356);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,232,758,723,185,80]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=f^2=1,e^2=c,g^2=b,e*a*e^-1=a*b=b*a,f*a*f=a*c=c*a,a*d=d*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,g*f*g^-1=b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f=d*e=e*d,d*f=f*d,d*g=g*d,e*g=g*e>;
// generators/relations