direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.Q8, C24.13Q8, C25.29C22, C23.289C24, C24.234C23, C23.58(C2×Q8), C23.832(C2×D4), (C22×C4).365D4, C23.368(C4○D4), C22.56(C22×Q8), (C22×C4).779C23, (C23×C4).320C22, C22.172(C22×D4), C22.92(C22⋊Q8), C22.162(C4⋊D4), C2.C42⋊53C22, C22.32(C42⋊2C2), C2.9(C2×C4⋊D4), (C22×C4⋊C4)⋊13C2, C2.9(C2×C22⋊Q8), (C2×C4).290(C2×D4), (C2×C4⋊C4)⋊106C22, C2.5(C2×C42⋊2C2), C22.169(C2×C4○D4), (C2×C2.C42)⋊10C2, (C22×C22⋊C4).19C2, (C2×C22⋊C4).484C22, SmallGroup(128,1121)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.Q8
G = < a,b,c,d,e,f | a2=b2=c2=d2=e4=1, f2=ce2, ab=ba, ac=ca, ad=da, ae=ea, af=fa, fbf-1=bc=cb, ebe-1=bd=db, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 868 in 454 conjugacy classes, 164 normal (8 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C2×C4, C2×C4, C23, C23, C23, 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, C25, C2×C2.C42, C23.Q8, C22×C22⋊C4, C22×C4⋊C4, C2×C23.Q8
Quotients: C1, C2, C22, D4, Q8, C23, C2×D4, C2×Q8, C4○D4, C24, C4⋊D4, C22⋊Q8, C42⋊2C2, C22×D4, C22×Q8, C2×C4○D4, C23.Q8, C2×C4⋊D4, C2×C22⋊Q8, C2×C42⋊2C2, C2×C23.Q8
►On 64 points
Generators in S64
(1 49)(2 50)(3 51)(4 52)(5 25)(6 26)(7 27)(8 28)(9 19)(10 20)(11 17)(12 18)(13 45)(14 46)(15 47)(16 48)(21 41)(22 42)(23 43)(24 44)(29 64)(30 61)(31 62)(32 63)(33 55)(34 56)(35 53)(36 54)(37 59)(38 60)(39 57)(40 58)
(2 48)(4 46)(5 62)(6 35)(7 64)(8 33)(10 22)(12 24)(14 52)(16 50)(18 44)(20 42)(25 31)(26 53)(27 29)(28 55)(30 59)(32 57)(34 38)(36 40)(37 61)(39 63)(54 58)(56 60)
(1 17)(2 18)(3 19)(4 20)(5 34)(6 35)(7 36)(8 33)(9 51)(10 52)(11 49)(12 50)(13 21)(14 22)(15 23)(16 24)(25 56)(26 53)(27 54)(28 55)(29 58)(30 59)(31 60)(32 57)(37 61)(38 62)(39 63)(40 64)(41 45)(42 46)(43 47)(44 48)
(1 47)(2 48)(3 45)(4 46)(5 38)(6 39)(7 40)(8 37)(9 21)(10 22)(11 23)(12 24)(13 51)(14 52)(15 49)(16 50)(17 43)(18 44)(19 41)(20 42)(25 60)(26 57)(27 58)(28 59)(29 54)(30 55)(31 56)(32 53)(33 61)(34 62)(35 63)(36 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 37 19 63)(2 40 20 62)(3 39 17 61)(4 38 18 64)(5 44 36 46)(6 43 33 45)(7 42 34 48)(8 41 35 47)(9 32 49 59)(10 31 50 58)(11 30 51 57)(12 29 52 60)(13 26 23 55)(14 25 24 54)(15 28 21 53)(16 27 22 56)
G:=sub<Sym(64)| (1,49)(2,50)(3,51)(4,52)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(13,45)(14,46)(15,47)(16,48)(21,41)(22,42)(23,43)(24,44)(29,64)(30,61)(31,62)(32,63)(33,55)(34,56)(35,53)(36,54)(37,59)(38,60)(39,57)(40,58), (2,48)(4,46)(5,62)(6,35)(7,64)(8,33)(10,22)(12,24)(14,52)(16,50)(18,44)(20,42)(25,31)(26,53)(27,29)(28,55)(30,59)(32,57)(34,38)(36,40)(37,61)(39,63)(54,58)(56,60), (1,17)(2,18)(3,19)(4,20)(5,34)(6,35)(7,36)(8,33)(9,51)(10,52)(11,49)(12,50)(13,21)(14,22)(15,23)(16,24)(25,56)(26,53)(27,54)(28,55)(29,58)(30,59)(31,60)(32,57)(37,61)(38,62)(39,63)(40,64)(41,45)(42,46)(43,47)(44,48), (1,47)(2,48)(3,45)(4,46)(5,38)(6,39)(7,40)(8,37)(9,21)(10,22)(11,23)(12,24)(13,51)(14,52)(15,49)(16,50)(17,43)(18,44)(19,41)(20,42)(25,60)(26,57)(27,58)(28,59)(29,54)(30,55)(31,56)(32,53)(33,61)(34,62)(35,63)(36,64), (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), (1,37,19,63)(2,40,20,62)(3,39,17,61)(4,38,18,64)(5,44,36,46)(6,43,33,45)(7,42,34,48)(8,41,35,47)(9,32,49,59)(10,31,50,58)(11,30,51,57)(12,29,52,60)(13,26,23,55)(14,25,24,54)(15,28,21,53)(16,27,22,56)>;
G:=Group( (1,49)(2,50)(3,51)(4,52)(5,25)(6,26)(7,27)(8,28)(9,19)(10,20)(11,17)(12,18)(13,45)(14,46)(15,47)(16,48)(21,41)(22,42)(23,43)(24,44)(29,64)(30,61)(31,62)(32,63)(33,55)(34,56)(35,53)(36,54)(37,59)(38,60)(39,57)(40,58), (2,48)(4,46)(5,62)(6,35)(7,64)(8,33)(10,22)(12,24)(14,52)(16,50)(18,44)(20,42)(25,31)(26,53)(27,29)(28,55)(30,59)(32,57)(34,38)(36,40)(37,61)(39,63)(54,58)(56,60), (1,17)(2,18)(3,19)(4,20)(5,34)(6,35)(7,36)(8,33)(9,51)(10,52)(11,49)(12,50)(13,21)(14,22)(15,23)(16,24)(25,56)(26,53)(27,54)(28,55)(29,58)(30,59)(31,60)(32,57)(37,61)(38,62)(39,63)(40,64)(41,45)(42,46)(43,47)(44,48), (1,47)(2,48)(3,45)(4,46)(5,38)(6,39)(7,40)(8,37)(9,21)(10,22)(11,23)(12,24)(13,51)(14,52)(15,49)(16,50)(17,43)(18,44)(19,41)(20,42)(25,60)(26,57)(27,58)(28,59)(29,54)(30,55)(31,56)(32,53)(33,61)(34,62)(35,63)(36,64), (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), (1,37,19,63)(2,40,20,62)(3,39,17,61)(4,38,18,64)(5,44,36,46)(6,43,33,45)(7,42,34,48)(8,41,35,47)(9,32,49,59)(10,31,50,58)(11,30,51,57)(12,29,52,60)(13,26,23,55)(14,25,24,54)(15,28,21,53)(16,27,22,56) );
G=PermutationGroup([[(1,49),(2,50),(3,51),(4,52),(5,25),(6,26),(7,27),(8,28),(9,19),(10,20),(11,17),(12,18),(13,45),(14,46),(15,47),(16,48),(21,41),(22,42),(23,43),(24,44),(29,64),(30,61),(31,62),(32,63),(33,55),(34,56),(35,53),(36,54),(37,59),(38,60),(39,57),(40,58)], [(2,48),(4,46),(5,62),(6,35),(7,64),(8,33),(10,22),(12,24),(14,52),(16,50),(18,44),(20,42),(25,31),(26,53),(27,29),(28,55),(30,59),(32,57),(34,38),(36,40),(37,61),(39,63),(54,58),(56,60)], [(1,17),(2,18),(3,19),(4,20),(5,34),(6,35),(7,36),(8,33),(9,51),(10,52),(11,49),(12,50),(13,21),(14,22),(15,23),(16,24),(25,56),(26,53),(27,54),(28,55),(29,58),(30,59),(31,60),(32,57),(37,61),(38,62),(39,63),(40,64),(41,45),(42,46),(43,47),(44,48)], [(1,47),(2,48),(3,45),(4,46),(5,38),(6,39),(7,40),(8,37),(9,21),(10,22),(11,23),(12,24),(13,51),(14,52),(15,49),(16,50),(17,43),(18,44),(19,41),(20,42),(25,60),(26,57),(27,58),(28,59),(29,54),(30,55),(31,56),(32,53),(33,61),(34,62),(35,63),(36,64)], [(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)], [(1,37,19,63),(2,40,20,62),(3,39,17,61),(4,38,18,64),(5,44,36,46),(6,43,33,45),(7,42,34,48),(8,41,35,47),(9,32,49,59),(10,31,50,58),(11,30,51,57),(12,29,52,60),(13,26,23,55),(14,25,24,54),(15,28,21,53),(16,27,22,56)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2O | 2P | 2Q | 2R | 2S | 4A | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | D4 | Q8 | C4○D4 |
| kernel | C2×C23.Q8 | C2×C2.C42 | C23.Q8 | C22×C22⋊C4 | C22×C4⋊C4 | C22×C4 | C24 | C23 |
| # reps | 1 | 1 | 8 | 3 | 3 | 12 | 4 | 12 |
Matrix representation of C2×C23.Q8 ►in GL8(𝔽5)
| 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 | 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 |
| 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 | 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 | 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 | 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 |
| 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 | 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 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 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 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 4 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(8,GF(5))| [4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,4,2,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0] >;
C2×C23.Q8 in GAP, Magma, Sage, TeX
C_2\times C_2^3.Q_8
% in TeX
G:=Group("C2xC2^3.Q8"); // GroupNames label
G:=SmallGroup(128,1121);
// by ID
G=gap.SmallGroup(128,1121);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,336,253,120,758,723]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^4=1,f^2=c*e^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,f*b*f^-1=b*c=c*b,e*b*e^-1=b*d=d*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=e^-1>;
// generators/relations