direct product, p-group, metabelian, nilpotent (class 2), monomial, rational
Aliases: C2×C22.31C24, C23.17C24, C22.37C25, C24.612C23, C22.742- 1+4, C22.1012+ 1+4, (C22×C4)⋊43D4, (C2×C4).40C24, C2.16(D4×C23), C4⋊C4.281C23, C4⋊D4⋊62C22, C4.172(C22×D4), C23.409(C2×D4), C22⋊C4.5C23, C22⋊Q8⋊73C22, C22.2(C22×D4), (C2×D4).290C23, (C2×Q8).422C23, C2.6(C2×2+ 1+4), C2.5(C2×2- 1+4), (C23×C4).583C22, (C22×C4).1178C23, (C22×D4).419C22, (C22×Q8).488C22, (C2×C4)⋊11(C2×D4), (C2×C4⋊D4)⋊55C2, (C22×C4⋊C4)⋊41C2, (C2×C22⋊Q8)⋊62C2, (C2×C4⋊C4)⋊128C22, (C22×C4○D4)⋊14C2, (C2×C4○D4)⋊68C22, (C2×C22⋊C4).374C22, SmallGroup(128,2180)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C22.31C24
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=1, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >
Subgroups: 1324 in 820 conjugacy classes, 436 normal (10 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C4○D4, C24, C24, C2×C22⋊C4, C2×C4⋊C4, C4⋊D4, C22⋊Q8, C23×C4, C23×C4, C22×D4, C22×Q8, C2×C4○D4, C2×C4○D4, C22×C4⋊C4, C2×C4⋊D4, C2×C22⋊Q8, C22.31C24, C22×C4○D4, C2×C22.31C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4, 2+ 1+4, 2- 1+4, C25, C22.31C24, D4×C23, C2×2+ 1+4, C2×2- 1+4, C2×C22.31C24
►On 64 points
Generators in S64
(1 9)(2 10)(3 11)(4 12)(5 24)(6 21)(7 22)(8 23)(13 51)(14 52)(15 49)(16 50)(17 40)(18 37)(19 38)(20 39)(25 30)(26 31)(27 32)(28 29)(33 56)(34 53)(35 54)(36 55)(41 46)(42 47)(43 48)(44 45)(57 62)(58 63)(59 64)(60 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 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 23)(2 24)(3 21)(4 22)(5 10)(6 11)(7 12)(8 9)(13 59)(14 60)(15 57)(16 58)(17 30)(18 31)(19 32)(20 29)(25 40)(26 37)(27 38)(28 39)(33 46)(34 47)(35 48)(36 45)(41 56)(42 53)(43 54)(44 55)(49 62)(50 63)(51 64)(52 61)
(1 5)(2 8)(3 7)(4 6)(9 24)(10 23)(11 22)(12 21)(13 52)(14 51)(15 50)(16 49)(17 39)(18 38)(19 37)(20 40)(25 29)(26 32)(27 31)(28 30)(33 42)(34 41)(35 44)(36 43)(45 54)(46 53)(47 56)(48 55)(57 63)(58 62)(59 61)(60 64)
(1 34)(2 35)(3 36)(4 33)(5 43)(6 44)(7 41)(8 42)(9 53)(10 54)(11 55)(12 56)(13 30)(14 31)(15 32)(16 29)(17 59)(18 60)(19 57)(20 58)(21 45)(22 46)(23 47)(24 48)(25 51)(26 52)(27 49)(28 50)(37 61)(38 62)(39 63)(40 64)
(1 18)(2 19)(3 20)(4 17)(5 27)(6 28)(7 25)(8 26)(9 37)(10 38)(11 39)(12 40)(13 48)(14 45)(15 46)(16 47)(21 29)(22 30)(23 31)(24 32)(33 57)(34 58)(35 59)(36 60)(41 49)(42 50)(43 51)(44 52)(53 63)(54 64)(55 61)(56 62)
(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)
G:=sub<Sym(64)| (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,51)(14,52)(15,49)(16,50)(17,40)(18,37)(19,38)(20,39)(25,30)(26,31)(27,32)(28,29)(33,56)(34,53)(35,54)(36,55)(41,46)(42,47)(43,48)(44,45)(57,62)(58,63)(59,64)(60,61), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(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), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,59)(14,60)(15,57)(16,58)(17,30)(18,31)(19,32)(20,29)(25,40)(26,37)(27,38)(28,39)(33,46)(34,47)(35,48)(36,45)(41,56)(42,53)(43,54)(44,55)(49,62)(50,63)(51,64)(52,61), (1,5)(2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,52)(14,51)(15,50)(16,49)(17,39)(18,38)(19,37)(20,40)(25,29)(26,32)(27,31)(28,30)(33,42)(34,41)(35,44)(36,43)(45,54)(46,53)(47,56)(48,55)(57,63)(58,62)(59,61)(60,64), (1,34)(2,35)(3,36)(4,33)(5,43)(6,44)(7,41)(8,42)(9,53)(10,54)(11,55)(12,56)(13,30)(14,31)(15,32)(16,29)(17,59)(18,60)(19,57)(20,58)(21,45)(22,46)(23,47)(24,48)(25,51)(26,52)(27,49)(28,50)(37,61)(38,62)(39,63)(40,64), (1,18)(2,19)(3,20)(4,17)(5,27)(6,28)(7,25)(8,26)(9,37)(10,38)(11,39)(12,40)(13,48)(14,45)(15,46)(16,47)(21,29)(22,30)(23,31)(24,32)(33,57)(34,58)(35,59)(36,60)(41,49)(42,50)(43,51)(44,52)(53,63)(54,64)(55,61)(56,62), (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)>;
G:=Group( (1,9)(2,10)(3,11)(4,12)(5,24)(6,21)(7,22)(8,23)(13,51)(14,52)(15,49)(16,50)(17,40)(18,37)(19,38)(20,39)(25,30)(26,31)(27,32)(28,29)(33,56)(34,53)(35,54)(36,55)(41,46)(42,47)(43,48)(44,45)(57,62)(58,63)(59,64)(60,61), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(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), (1,23)(2,24)(3,21)(4,22)(5,10)(6,11)(7,12)(8,9)(13,59)(14,60)(15,57)(16,58)(17,30)(18,31)(19,32)(20,29)(25,40)(26,37)(27,38)(28,39)(33,46)(34,47)(35,48)(36,45)(41,56)(42,53)(43,54)(44,55)(49,62)(50,63)(51,64)(52,61), (1,5)(2,8)(3,7)(4,6)(9,24)(10,23)(11,22)(12,21)(13,52)(14,51)(15,50)(16,49)(17,39)(18,38)(19,37)(20,40)(25,29)(26,32)(27,31)(28,30)(33,42)(34,41)(35,44)(36,43)(45,54)(46,53)(47,56)(48,55)(57,63)(58,62)(59,61)(60,64), (1,34)(2,35)(3,36)(4,33)(5,43)(6,44)(7,41)(8,42)(9,53)(10,54)(11,55)(12,56)(13,30)(14,31)(15,32)(16,29)(17,59)(18,60)(19,57)(20,58)(21,45)(22,46)(23,47)(24,48)(25,51)(26,52)(27,49)(28,50)(37,61)(38,62)(39,63)(40,64), (1,18)(2,19)(3,20)(4,17)(5,27)(6,28)(7,25)(8,26)(9,37)(10,38)(11,39)(12,40)(13,48)(14,45)(15,46)(16,47)(21,29)(22,30)(23,31)(24,32)(33,57)(34,58)(35,59)(36,60)(41,49)(42,50)(43,51)(44,52)(53,63)(54,64)(55,61)(56,62), (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) );
G=PermutationGroup([[(1,9),(2,10),(3,11),(4,12),(5,24),(6,21),(7,22),(8,23),(13,51),(14,52),(15,49),(16,50),(17,40),(18,37),(19,38),(20,39),(25,30),(26,31),(27,32),(28,29),(33,56),(34,53),(35,54),(36,55),(41,46),(42,47),(43,48),(44,45),(57,62),(58,63),(59,64),(60,61)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(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)], [(1,23),(2,24),(3,21),(4,22),(5,10),(6,11),(7,12),(8,9),(13,59),(14,60),(15,57),(16,58),(17,30),(18,31),(19,32),(20,29),(25,40),(26,37),(27,38),(28,39),(33,46),(34,47),(35,48),(36,45),(41,56),(42,53),(43,54),(44,55),(49,62),(50,63),(51,64),(52,61)], [(1,5),(2,8),(3,7),(4,6),(9,24),(10,23),(11,22),(12,21),(13,52),(14,51),(15,50),(16,49),(17,39),(18,38),(19,37),(20,40),(25,29),(26,32),(27,31),(28,30),(33,42),(34,41),(35,44),(36,43),(45,54),(46,53),(47,56),(48,55),(57,63),(58,62),(59,61),(60,64)], [(1,34),(2,35),(3,36),(4,33),(5,43),(6,44),(7,41),(8,42),(9,53),(10,54),(11,55),(12,56),(13,30),(14,31),(15,32),(16,29),(17,59),(18,60),(19,57),(20,58),(21,45),(22,46),(23,47),(24,48),(25,51),(26,52),(27,49),(28,50),(37,61),(38,62),(39,63),(40,64)], [(1,18),(2,19),(3,20),(4,17),(5,27),(6,28),(7,25),(8,26),(9,37),(10,38),(11,39),(12,40),(13,48),(14,45),(15,46),(16,47),(21,29),(22,30),(23,31),(24,32),(33,57),(34,58),(35,59),(36,60),(41,49),(42,50),(43,51),(44,52),(53,63),(54,64),(55,61),(56,62)], [(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)]])
44 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | 2I | 2J | 2K | 2L | ··· | 2S | 4A | ··· | 4H | 4I | ··· | 4X |
| order | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
| size | 1 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 2 | ··· | 2 | 4 | ··· | 4 |
44 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | 2+ 1+4 | 2- 1+4 |
| kernel | C2×C22.31C24 | C22×C4⋊C4 | C2×C4⋊D4 | C2×C22⋊Q8 | C22.31C24 | C22×C4○D4 | C22×C4 | C22 | C22 |
| # reps | 1 | 1 | 8 | 4 | 16 | 2 | 8 | 2 | 2 |
Matrix representation of C2×C22.31C24 ►in GL8(𝔽5)
| 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 |
| 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 | 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 |
| 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 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 | 0 | 4 | 2 | 0 |
| 0 | 0 | 0 | 0 | 4 | 3 | 2 | 1 |
| 0 | 0 | 0 | 0 | 0 | 4 | 1 | 3 |
| 0 | 0 | 0 | 0 | 3 | 3 | 4 | 1 |
| 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 |
| 4 | 0 | 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 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 2 | 1 |
| 0 | 0 | 0 | 0 | 4 | 4 | 2 | 3 |
| 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 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 | 4 | 2 |
| 0 | 0 | 0 | 0 | 0 | 1 | 4 | 1 |
G:=sub<GL(8,GF(5))| [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],[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],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4],[1,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,2,1],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,4,0,3,0,0,0,0,4,3,4,3,0,0,0,0,2,2,1,4,0,0,0,0,0,1,3,1],[0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,2,0,4,0,0,0,0,3,0,3,4,0,0,0,0,0,0,2,2,0,0,0,0,0,0,1,3],[4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,4,0,1,1,0,0,0,0,0,0,4,4,0,0,0,0,0,0,2,1] >;
C2×C22.31C24 in GAP, Magma, Sage, TeX
C_2\times C_2^2._{31}C_2^4 % in TeX
G:=Group("C2xC2^2.31C2^4"); // GroupNames label
G:=SmallGroup(128,2180);
// by ID
G=gap.SmallGroup(128,2180);
# by ID
G:=PCGroup([7,-2,2,2,2,2,-2,2,477,1430,387,1123,136]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=e^2=f^2=1,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,e*d*e=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
// generators/relations