direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: Q8×C23, C2.2C25, C4.11C24, C22.14C24, C23.74C23, C24.37C22, (C23×C4).12C2, (C2×C4).140C23, (C22×C4).133C22, SmallGroup(64,262)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for Q8×C23
G = < a,b,c,d,e | a2=b2=c2=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede-1=d-1 >
Subgroups: 425, all normal (4 characteristic)
C1, C2, C2, C4, C22, C2×C4, Q8, C23, C22×C4, C2×Q8, C24, C23×C4, C22×Q8, Q8×C23
Quotients: C1, C2, C22, Q8, C23, C2×Q8, C24, C22×Q8, C25, Q8×C23
►Regular action on 64 points
Generators in S64
(1 5)(2 6)(3 7)(4 8)(9 23)(10 24)(11 21)(12 22)(13 32)(14 29)(15 30)(16 31)(17 44)(18 41)(19 42)(20 43)(25 34)(26 35)(27 36)(28 33)(37 45)(38 46)(39 47)(40 48)(49 60)(50 57)(51 58)(52 59)(53 64)(54 61)(55 62)(56 63)
(1 11)(2 12)(3 9)(4 10)(5 21)(6 22)(7 23)(8 24)(13 20)(14 17)(15 18)(16 19)(25 63)(26 64)(27 61)(28 62)(29 44)(30 41)(31 42)(32 43)(33 55)(34 56)(35 53)(36 54)(37 58)(38 59)(39 60)(40 57)(45 51)(46 52)(47 49)(48 50)
(1 31)(2 32)(3 29)(4 30)(5 16)(6 13)(7 14)(8 15)(9 44)(10 41)(11 42)(12 43)(17 23)(18 24)(19 21)(20 22)(25 50)(26 51)(27 52)(28 49)(33 60)(34 57)(35 58)(36 59)(37 53)(38 54)(39 55)(40 56)(45 64)(46 61)(47 62)(48 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 36 3 34)(2 35 4 33)(5 27 7 25)(6 26 8 28)(9 56 11 54)(10 55 12 53)(13 51 15 49)(14 50 16 52)(17 48 19 46)(18 47 20 45)(21 61 23 63)(22 64 24 62)(29 57 31 59)(30 60 32 58)(37 41 39 43)(38 44 40 42)
G:=sub<Sym(64)| (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,32)(14,29)(15,30)(16,31)(17,44)(18,41)(19,42)(20,43)(25,34)(26,35)(27,36)(28,33)(37,45)(38,46)(39,47)(40,48)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (1,11)(2,12)(3,9)(4,10)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,63)(26,64)(27,61)(28,62)(29,44)(30,41)(31,42)(32,43)(33,55)(34,56)(35,53)(36,54)(37,58)(38,59)(39,60)(40,57)(45,51)(46,52)(47,49)(48,50), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,44)(10,41)(11,42)(12,43)(17,23)(18,24)(19,21)(20,22)(25,50)(26,51)(27,52)(28,49)(33,60)(34,57)(35,58)(36,59)(37,53)(38,54)(39,55)(40,56)(45,64)(46,61)(47,62)(48,63), (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), (1,36,3,34)(2,35,4,33)(5,27,7,25)(6,26,8,28)(9,56,11,54)(10,55,12,53)(13,51,15,49)(14,50,16,52)(17,48,19,46)(18,47,20,45)(21,61,23,63)(22,64,24,62)(29,57,31,59)(30,60,32,58)(37,41,39,43)(38,44,40,42)>;
G:=Group( (1,5)(2,6)(3,7)(4,8)(9,23)(10,24)(11,21)(12,22)(13,32)(14,29)(15,30)(16,31)(17,44)(18,41)(19,42)(20,43)(25,34)(26,35)(27,36)(28,33)(37,45)(38,46)(39,47)(40,48)(49,60)(50,57)(51,58)(52,59)(53,64)(54,61)(55,62)(56,63), (1,11)(2,12)(3,9)(4,10)(5,21)(6,22)(7,23)(8,24)(13,20)(14,17)(15,18)(16,19)(25,63)(26,64)(27,61)(28,62)(29,44)(30,41)(31,42)(32,43)(33,55)(34,56)(35,53)(36,54)(37,58)(38,59)(39,60)(40,57)(45,51)(46,52)(47,49)(48,50), (1,31)(2,32)(3,29)(4,30)(5,16)(6,13)(7,14)(8,15)(9,44)(10,41)(11,42)(12,43)(17,23)(18,24)(19,21)(20,22)(25,50)(26,51)(27,52)(28,49)(33,60)(34,57)(35,58)(36,59)(37,53)(38,54)(39,55)(40,56)(45,64)(46,61)(47,62)(48,63), (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), (1,36,3,34)(2,35,4,33)(5,27,7,25)(6,26,8,28)(9,56,11,54)(10,55,12,53)(13,51,15,49)(14,50,16,52)(17,48,19,46)(18,47,20,45)(21,61,23,63)(22,64,24,62)(29,57,31,59)(30,60,32,58)(37,41,39,43)(38,44,40,42) );
G=PermutationGroup([[(1,5),(2,6),(3,7),(4,8),(9,23),(10,24),(11,21),(12,22),(13,32),(14,29),(15,30),(16,31),(17,44),(18,41),(19,42),(20,43),(25,34),(26,35),(27,36),(28,33),(37,45),(38,46),(39,47),(40,48),(49,60),(50,57),(51,58),(52,59),(53,64),(54,61),(55,62),(56,63)], [(1,11),(2,12),(3,9),(4,10),(5,21),(6,22),(7,23),(8,24),(13,20),(14,17),(15,18),(16,19),(25,63),(26,64),(27,61),(28,62),(29,44),(30,41),(31,42),(32,43),(33,55),(34,56),(35,53),(36,54),(37,58),(38,59),(39,60),(40,57),(45,51),(46,52),(47,49),(48,50)], [(1,31),(2,32),(3,29),(4,30),(5,16),(6,13),(7,14),(8,15),(9,44),(10,41),(11,42),(12,43),(17,23),(18,24),(19,21),(20,22),(25,50),(26,51),(27,52),(28,49),(33,60),(34,57),(35,58),(36,59),(37,53),(38,54),(39,55),(40,56),(45,64),(46,61),(47,62),(48,63)], [(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)], [(1,36,3,34),(2,35,4,33),(5,27,7,25),(6,26,8,28),(9,56,11,54),(10,55,12,53),(13,51,15,49),(14,50,16,52),(17,48,19,46),(18,47,20,45),(21,61,23,63),(22,64,24,62),(29,57,31,59),(30,60,32,58),(37,41,39,43),(38,44,40,42)]])
Q8×C23 is a maximal subgroup of
C24.636C23 C4.C22≀C2 C24.155D4 C23.192C24 C23.309C24 C23.334C24 C24.565C23 C23.514C24 C24.178D4 C22.75C25
Q8×C23 is a maximal quotient of
C22.47C25 C22.90C25 C22.91C25 C22.92C25 C22.93C25
40 conjugacy classes
| class | 1 | 2A | ··· | 2O | 4A | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 |
40 irreducible representations
| dim | 1 | 1 | 1 | 2 |
| type | + | + | + | - |
| image | C1 | C2 | C2 | Q8 |
| kernel | Q8×C23 | C23×C4 | C22×Q8 | C23 |
| # reps | 1 | 3 | 28 | 8 |
Matrix representation of Q8×C23 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 4 | 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 | 4 | 0 | 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 | 3 |
| 0 | 0 | 0 | 1 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,1,1,0,0,0,3,4],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,3,0,0,0,0,4,2] >;
Q8×C23 in GAP, Magma, Sage, TeX
Q_8\times C_2^3
% in TeX
G:=Group("Q8xC2^3"); // GroupNames label
G:=SmallGroup(64,262);
// by ID
G=gap.SmallGroup(64,262);
# by ID
G:=PCGroup([6,-2,2,2,2,2,-2,192,409,199]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^2=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations