direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.8Q8, C24.21Q8, C24.165D4, C25.86C22, C23.168C24, C24.531C23, C23⋊6(C4⋊C4), (C24×C4).6C2, C24.96(C2×C4), C23.88(C2×Q8), C23.824(C2×D4), C22.106(C4×D4), (C22×C4).702D4, C22.59(C23×C4), C22.66(C22×D4), C22.106C22≀C2, C23.357(C4○D4), C22.21(C22×Q8), C23.116(C22×C4), (C22×C4).446C23, (C23×C4).279C22, C22.86(C22⋊Q8), C2.C42⋊58C22, C22.97(C22.D4), C2.5(C2×C4×D4), C22⋊3(C2×C4⋊C4), (C22×C4⋊C4)⋊4C2, (C2×C4)⋊4(C22×C4), C2.8(C22×C4⋊C4), C22⋊C4⋊34(C2×C4), (C2×C22⋊C4)⋊19C4, (C2×C4⋊C4)⋊98C22, (C22×C4)⋊15(C2×C4), C2.2(C2×C22⋊Q8), C2.2(C2×C22≀C2), (C2×C4).1184(C2×D4), C22.60(C2×C4○D4), C2.2(C2×C22.D4), (C2×C2.C42)⋊15C2, (C22×C22⋊C4).15C2, (C2×C22⋊C4).477C22, SmallGroup(128,1018)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.8Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=e2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=ce-1 >
Subgroups: 1020 in 616 conjugacy classes, 236 normal (16 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C24, C24, C24, C2.C42, C2×C22⋊C4, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C23×C4, C23×C4, C23×C4, C25, C2×C2.C42, C23.8Q8, C22×C22⋊C4, C22×C4⋊C4, C24×C4, C2×C23.8Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C4⋊C4, C4×D4, C22≀C2, C22⋊Q8, C22.D4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C23.8Q8, C22×C4⋊C4, C2×C4×D4, C2×C22≀C2, C2×C22⋊Q8, C2×C22.D4, C2×C23.8Q8
►On 64 points
(1 5)(2 6)(3 7)(4 8)(9 21)(10 22)(11 23)(12 24)(13 30)(14 31)(15 32)(16 29)(17 47)(18 48)(19 45)(20 46)(25 40)(26 37)(27 38)(28 39)(33 44)(34 41)(35 42)(36 43)(49 58)(50 59)(51 60)(52 57)(53 62)(54 63)(55 64)(56 61)
(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 50)(26 51)(27 52)(28 49)(29 31)(30 32)(33 56)(34 53)(35 54)(36 55)(37 60)(38 57)(39 58)(40 59)(41 62)(42 63)(43 64)(44 61)(45 47)(46 48)
(1 11)(2 12)(3 9)(4 10)(5 23)(6 24)(7 21)(8 22)(13 18)(14 19)(15 20)(16 17)(25 61)(26 62)(27 63)(28 64)(29 47)(30 48)(31 45)(32 46)(33 59)(34 60)(35 57)(36 58)(37 53)(38 54)(39 55)(40 56)(41 51)(42 52)(43 49)(44 50)
(1 31)(2 32)(3 29)(4 30)(5 14)(6 15)(7 16)(8 13)(9 47)(10 48)(11 45)(12 46)(17 21)(18 22)(19 23)(20 24)(25 52)(26 49)(27 50)(28 51)(33 54)(34 55)(35 56)(36 53)(37 58)(38 59)(39 60)(40 57)(41 64)(42 61)(43 62)(44 63)
(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 54 3 56)(2 37 4 39)(5 63 7 61)(6 26 8 28)(9 40 11 38)(10 55 12 53)(13 51 15 49)(14 44 16 42)(17 52 19 50)(18 41 20 43)(21 25 23 27)(22 64 24 62)(29 35 31 33)(30 60 32 58)(34 46 36 48)(45 59 47 57)
G:=sub<Sym(64)| (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,30)(14,31)(15,32)(16,29)(17,47)(18,48)(19,45)(20,46)(25,40)(26,37)(27,38)(28,39)(33,44)(34,41)(35,42)(36,43)(49,58)(50,59)(51,60)(52,57)(53,62)(54,63)(55,64)(56,61), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,50)(26,51)(27,52)(28,49)(29,31)(30,32)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(41,62)(42,63)(43,64)(44,61)(45,47)(46,48), (1,11)(2,12)(3,9)(4,10)(5,23)(6,24)(7,21)(8,22)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(29,47)(30,48)(31,45)(32,46)(33,59)(34,60)(35,57)(36,58)(37,53)(38,54)(39,55)(40,56)(41,51)(42,52)(43,49)(44,50), (1,31)(2,32)(3,29)(4,30)(5,14)(6,15)(7,16)(8,13)(9,47)(10,48)(11,45)(12,46)(17,21)(18,22)(19,23)(20,24)(25,52)(26,49)(27,50)(28,51)(33,54)(34,55)(35,56)(36,53)(37,58)(38,59)(39,60)(40,57)(41,64)(42,61)(43,62)(44,63), (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), (1,54,3,56)(2,37,4,39)(5,63,7,61)(6,26,8,28)(9,40,11,38)(10,55,12,53)(13,51,15,49)(14,44,16,42)(17,52,19,50)(18,41,20,43)(21,25,23,27)(22,64,24,62)(29,35,31,33)(30,60,32,58)(34,46,36,48)(45,59,47,57)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,21)(10,22)(11,23)(12,24)(13,30)(14,31)(15,32)(16,29)(17,47)(18,48)(19,45)(20,46)(25,40)(26,37)(27,38)(28,39)(33,44)(34,41)(35,42)(36,43)(49,58)(50,59)(51,60)(52,57)(53,62)(54,63)(55,64)(56,61), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,50)(26,51)(27,52)(28,49)(29,31)(30,32)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(41,62)(42,63)(43,64)(44,61)(45,47)(46,48), (1,11)(2,12)(3,9)(4,10)(5,23)(6,24)(7,21)(8,22)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(29,47)(30,48)(31,45)(32,46)(33,59)(34,60)(35,57)(36,58)(37,53)(38,54)(39,55)(40,56)(41,51)(42,52)(43,49)(44,50), (1,31)(2,32)(3,29)(4,30)(5,14)(6,15)(7,16)(8,13)(9,47)(10,48)(11,45)(12,46)(17,21)(18,22)(19,23)(20,24)(25,52)(26,49)(27,50)(28,51)(33,54)(34,55)(35,56)(36,53)(37,58)(38,59)(39,60)(40,57)(41,64)(42,61)(43,62)(44,63), (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), (1,54,3,56)(2,37,4,39)(5,63,7,61)(6,26,8,28)(9,40,11,38)(10,55,12,53)(13,51,15,49)(14,44,16,42)(17,52,19,50)(18,41,20,43)(21,25,23,27)(22,64,24,62)(29,35,31,33)(30,60,32,58)(34,46,36,48)(45,59,47,57) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,21),(10,22),(11,23),(12,24),(13,30),(14,31),(15,32),(16,29),(17,47),(18,48),(19,45),(20,46),(25,40),(26,37),(27,38),(28,39),(33,44),(34,41),(35,42),(36,43),(49,58),(50,59),(51,60),(52,57),(53,62),(54,63),(55,64),(56,61)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,50),(26,51),(27,52),(28,49),(29,31),(30,32),(33,56),(34,53),(35,54),(36,55),(37,60),(38,57),(39,58),(40,59),(41,62),(42,63),(43,64),(44,61),(45,47),(46,48)], [(1,11),(2,12),(3,9),(4,10),(5,23),(6,24),(7,21),(8,22),(13,18),(14,19),(15,20),(16,17),(25,61),(26,62),(27,63),(28,64),(29,47),(30,48),(31,45),(32,46),(33,59),(34,60),(35,57),(36,58),(37,53),(38,54),(39,55),(40,56),(41,51),(42,52),(43,49),(44,50)], [(1,31),(2,32),(3,29),(4,30),(5,14),(6,15),(7,16),(8,13),(9,47),(10,48),(11,45),(12,46),(17,21),(18,22),(19,23),(20,24),(25,52),(26,49),(27,50),(28,51),(33,54),(34,55),(35,56),(36,53),(37,58),(38,59),(39,60),(40,57),(41,64),(42,61),(43,62),(44,63)], [(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)], [(1,54,3,56),(2,37,4,39),(5,63,7,61),(6,26,8,28),(9,40,11,38),(10,55,12,53),(13,51,15,49),(14,44,16,42),(17,52,19,50),(18,41,20,43),(21,25,23,27),(22,64,24,62),(29,35,31,33),(30,60,32,58),(34,46,36,48),(45,59,47,57)]])
56 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | ··· | 2W | 4A | ··· | 4P | 4Q | ··· | 4AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
56 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | D4 | D4 | Q8 | C4○D4 |
| kernel | C2×C23.8Q8 | C2×C2.C42 | C23.8Q8 | C22×C22⋊C4 | C22×C4⋊C4 | C24×C4 | C2×C22⋊C4 | C22×C4 | C24 | C24 | C23 |
| # reps | 1 | 2 | 8 | 2 | 2 | 1 | 16 | 8 | 4 | 4 | 8 |
Matrix representation of C2×C23.8Q8 ►in GL6(𝔽5)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 0 | 4 | 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 2 | 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 |
| 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 2 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 4 | 0 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[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],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,2,0,0,0,0,0,2,3,0,0,0,0,0,0,2,0,0,0,0,0,0,3],[3,0,0,0,0,0,0,2,0,0,0,0,0,0,3,4,0,0,0,0,0,2,0,0,0,0,0,0,0,4,0,0,0,0,1,0] >;
C2×C23.8Q8 in GAP, Magma, Sage, TeX
C_2\times C_2^3._8Q_8
% in TeX
G:=Group("C2xC2^3.8Q8"); // GroupNames label
G:=SmallGroup(128,1018);
// by ID
G=gap.SmallGroup(128,1018);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,456,758]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,b*c=c*b,f*b*f^-1=b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=c*e^-1>;
// generators/relations