direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C23.7Q8, C24.20Q8, C24.137D4, C25.83C22, C23.160C24, C24.527C23, C23⋊5(C4⋊C4), (C23×C4)⋊16C4, (C24×C4).9C2, C24.120(C2×C4), (C22×C4).767D4, C23.820(C2×D4), C23.138(C2×Q8), C22.51(C23×C4), C22.60(C22×D4), C23.353(C4○D4), C22.17(C22×Q8), (C22×C4).438C23, C23.279(C22×C4), (C23×C4).676C22, C22.85(C22⋊Q8), C22.155(C4⋊D4), C2.C42⋊48C22, C22.63(C42⋊C2), C22⋊2(C2×C4⋊C4), C4⋊3(C2×C22⋊C4), (C22×C4⋊C4)⋊3C2, C2.1(C2×C4⋊D4), C2.4(C22×C4⋊C4), (C2×C4⋊C4)⋊96C22, (C22×C4)⋊51(C2×C4), C2.1(C2×C22⋊Q8), (C2×C4)⋊12(C22⋊C4), (C2×C4).1388(C2×D4), C2.6(C2×C42⋊C2), C22.53(C2×C4○D4), C2.5(C22×C22⋊C4), (C2×C2.C42)⋊6C2, (C2×C4).483(C22×C4), (C22×C22⋊C4).9C2, C22.131(C2×C22⋊C4), (C2×C22⋊C4).409C22, SmallGroup(128,1010)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C23.7Q8
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, bc=cb, fbf-1=bd=db, be=eb, cd=dc, ce=ec, cf=fc, de=ed, df=fd, fef-1=e-1 >
Subgroups: 1020 in 616 conjugacy classes, 260 normal (14 characteristic)
C1, C2, C2, C2, C4, 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, C23×C4, C23×C4, C25, C2×C2.C42, C23.7Q8, C22×C22⋊C4, C22×C4⋊C4, C24×C4, C2×C23.7Q8
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C4○D4, C24, C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4, C22⋊Q8, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C23.7Q8, C22×C22⋊C4, C22×C4⋊C4, C2×C42⋊C2, C2×C4⋊D4, C2×C22⋊Q8, C2×C23.7Q8
►On 64 points
(1 45)(2 46)(3 47)(4 48)(5 58)(6 59)(7 60)(8 57)(9 23)(10 24)(11 21)(12 22)(13 51)(14 52)(15 49)(16 50)(17 41)(18 42)(19 43)(20 44)(25 40)(26 37)(27 38)(28 39)(29 36)(30 33)(31 34)(32 35)(53 63)(54 64)(55 61)(56 62)
(1 49)(2 50)(3 51)(4 52)(5 56)(6 53)(7 54)(8 55)(9 43)(10 44)(11 41)(12 42)(13 47)(14 48)(15 45)(16 46)(17 21)(18 22)(19 23)(20 24)(25 34)(26 35)(27 36)(28 33)(29 38)(30 39)(31 40)(32 37)(57 61)(58 62)(59 63)(60 64)
(1 51)(2 52)(3 49)(4 50)(5 38)(6 39)(7 40)(8 37)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(17 23)(18 24)(19 21)(20 22)(25 60)(26 57)(27 58)(28 59)(29 56)(30 53)(31 54)(32 55)(33 63)(34 64)(35 61)(36 62)
(1 9)(2 10)(3 11)(4 12)(5 31)(6 32)(7 29)(8 30)(13 17)(14 18)(15 19)(16 20)(21 47)(22 48)(23 45)(24 46)(25 62)(26 63)(27 64)(28 61)(33 57)(34 58)(35 59)(36 60)(37 53)(38 54)(39 55)(40 56)(41 51)(42 52)(43 49)(44 50)
(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 63 49 35)(2 62 50 34)(3 61 51 33)(4 64 52 36)(5 24 40 20)(6 23 37 19)(7 22 38 18)(8 21 39 17)(9 26 43 59)(10 25 44 58)(11 28 41 57)(12 27 42 60)(13 30 47 55)(14 29 48 54)(15 32 45 53)(16 31 46 56)
G:=sub<Sym(64)| (1,45)(2,46)(3,47)(4,48)(5,58)(6,59)(7,60)(8,57)(9,23)(10,24)(11,21)(12,22)(13,51)(14,52)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(25,40)(26,37)(27,38)(28,39)(29,36)(30,33)(31,34)(32,35)(53,63)(54,64)(55,61)(56,62), (1,49)(2,50)(3,51)(4,52)(5,56)(6,53)(7,54)(8,55)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,21)(18,22)(19,23)(20,24)(25,34)(26,35)(27,36)(28,33)(29,38)(30,39)(31,40)(32,37)(57,61)(58,62)(59,63)(60,64), (1,51)(2,52)(3,49)(4,50)(5,38)(6,39)(7,40)(8,37)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,23)(18,24)(19,21)(20,22)(25,60)(26,57)(27,58)(28,59)(29,56)(30,53)(31,54)(32,55)(33,63)(34,64)(35,61)(36,62), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,17)(14,18)(15,19)(16,20)(21,47)(22,48)(23,45)(24,46)(25,62)(26,63)(27,64)(28,61)(33,57)(34,58)(35,59)(36,60)(37,53)(38,54)(39,55)(40,56)(41,51)(42,52)(43,49)(44,50), (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), (1,63,49,35)(2,62,50,34)(3,61,51,33)(4,64,52,36)(5,24,40,20)(6,23,37,19)(7,22,38,18)(8,21,39,17)(9,26,43,59)(10,25,44,58)(11,28,41,57)(12,27,42,60)(13,30,47,55)(14,29,48,54)(15,32,45,53)(16,31,46,56)>;
G:=Group( (1,45)(2,46)(3,47)(4,48)(5,58)(6,59)(7,60)(8,57)(9,23)(10,24)(11,21)(12,22)(13,51)(14,52)(15,49)(16,50)(17,41)(18,42)(19,43)(20,44)(25,40)(26,37)(27,38)(28,39)(29,36)(30,33)(31,34)(32,35)(53,63)(54,64)(55,61)(56,62), (1,49)(2,50)(3,51)(4,52)(5,56)(6,53)(7,54)(8,55)(9,43)(10,44)(11,41)(12,42)(13,47)(14,48)(15,45)(16,46)(17,21)(18,22)(19,23)(20,24)(25,34)(26,35)(27,36)(28,33)(29,38)(30,39)(31,40)(32,37)(57,61)(58,62)(59,63)(60,64), (1,51)(2,52)(3,49)(4,50)(5,38)(6,39)(7,40)(8,37)(9,41)(10,42)(11,43)(12,44)(13,45)(14,46)(15,47)(16,48)(17,23)(18,24)(19,21)(20,22)(25,60)(26,57)(27,58)(28,59)(29,56)(30,53)(31,54)(32,55)(33,63)(34,64)(35,61)(36,62), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,17)(14,18)(15,19)(16,20)(21,47)(22,48)(23,45)(24,46)(25,62)(26,63)(27,64)(28,61)(33,57)(34,58)(35,59)(36,60)(37,53)(38,54)(39,55)(40,56)(41,51)(42,52)(43,49)(44,50), (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), (1,63,49,35)(2,62,50,34)(3,61,51,33)(4,64,52,36)(5,24,40,20)(6,23,37,19)(7,22,38,18)(8,21,39,17)(9,26,43,59)(10,25,44,58)(11,28,41,57)(12,27,42,60)(13,30,47,55)(14,29,48,54)(15,32,45,53)(16,31,46,56) );
G=PermutationGroup([[(1,45),(2,46),(3,47),(4,48),(5,58),(6,59),(7,60),(8,57),(9,23),(10,24),(11,21),(12,22),(13,51),(14,52),(15,49),(16,50),(17,41),(18,42),(19,43),(20,44),(25,40),(26,37),(27,38),(28,39),(29,36),(30,33),(31,34),(32,35),(53,63),(54,64),(55,61),(56,62)], [(1,49),(2,50),(3,51),(4,52),(5,56),(6,53),(7,54),(8,55),(9,43),(10,44),(11,41),(12,42),(13,47),(14,48),(15,45),(16,46),(17,21),(18,22),(19,23),(20,24),(25,34),(26,35),(27,36),(28,33),(29,38),(30,39),(31,40),(32,37),(57,61),(58,62),(59,63),(60,64)], [(1,51),(2,52),(3,49),(4,50),(5,38),(6,39),(7,40),(8,37),(9,41),(10,42),(11,43),(12,44),(13,45),(14,46),(15,47),(16,48),(17,23),(18,24),(19,21),(20,22),(25,60),(26,57),(27,58),(28,59),(29,56),(30,53),(31,54),(32,55),(33,63),(34,64),(35,61),(36,62)], [(1,9),(2,10),(3,11),(4,12),(5,31),(6,32),(7,29),(8,30),(13,17),(14,18),(15,19),(16,20),(21,47),(22,48),(23,45),(24,46),(25,62),(26,63),(27,64),(28,61),(33,57),(34,58),(35,59),(36,60),(37,53),(38,54),(39,55),(40,56),(41,51),(42,52),(43,49),(44,50)], [(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)], [(1,63,49,35),(2,62,50,34),(3,61,51,33),(4,64,52,36),(5,24,40,20),(6,23,37,19),(7,22,38,18),(8,21,39,17),(9,26,43,59),(10,25,44,58),(11,28,41,57),(12,27,42,60),(13,30,47,55),(14,29,48,54),(15,32,45,53),(16,31,46,56)]])
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×C23.7Q8 | C2×C2.C42 | C23.7Q8 | C22×C22⋊C4 | C22×C4⋊C4 | C24×C4 | C23×C4 | C22×C4 | C24 | C24 | C23 |
| # reps | 1 | 2 | 8 | 2 | 2 | 1 | 16 | 8 | 4 | 4 | 8 |
Matrix representation of C2×C23.7Q8 ►in GL6(𝔽5)
| 1 | 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 | 4 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 0 |
| 0 | 0 | 0 | 0 | 0 | 4 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 4 | 0 | 0 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 | 0 | 0 |
| 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 4 | 2 |
| 0 | 0 | 0 | 0 | 4 | 1 |
| 2 | 0 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 4 |
| 0 | 0 | 0 | 0 | 0 | 2 |
G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,3,0,0,0,0,0,0,4,4,0,0,0,0,2,1],[2,0,0,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,3,0,0,0,0,0,4,2] >;
C2×C23.7Q8 in GAP, Magma, Sage, TeX
C_2\times C_2^3._7Q_8
% in TeX
G:=Group("C2xC2^3.7Q8"); // GroupNames label
G:=SmallGroup(128,1010);
// by ID
G=gap.SmallGroup(128,1010);
# by ID
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758]);
// 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,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=e^-1>;
// generators/relations