direct product, p-group, metabelian, nilpotent (class 2), monomial
Aliases: C2×C2.C42, C23.53D4, C23.10Q8, C22.8C42, C23.51C23, C24.34C22, (C22×C4)⋊5C4, (C23×C4).1C2, C2.1(C2×C42), C22.7(C2×Q8), C23.37(C2×C4), C22.26(C2×D4), C22.17(C4⋊C4), (C22×C4).83C22, C22.13(C22×C4), C22.27(C22⋊C4), (C2×C4)⋊9(C2×C4), C2.1(C2×C4⋊C4), C2.1(C2×C22⋊C4), SmallGroup(64,56)
Series: Derived ►Chief ►Lower central ►Upper central ►Jennings
Generators and relations for C2×C2.C42
G = < a,b,c,d | a2=b2=c4=d4=1, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 225 in 165 conjugacy classes, 105 normal (6 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, C23, C23, C22×C4, C22×C4, C24, C2.C42, C23×C4, C2×C2.C42
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C2×Q8, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C2.C42
►Regular action on 64 points
(1 29)(2 30)(3 31)(4 32)(5 23)(6 24)(7 21)(8 22)(9 59)(10 60)(11 57)(12 58)(13 19)(14 20)(15 17)(16 18)(25 42)(26 43)(27 44)(28 41)(33 56)(34 53)(35 54)(36 55)(37 63)(38 64)(39 61)(40 62)(45 51)(46 52)(47 49)(48 50)
(1 47)(2 48)(3 45)(4 46)(5 9)(6 10)(7 11)(8 12)(13 36)(14 33)(15 34)(16 35)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 37)(26 38)(27 39)(28 40)(29 49)(30 50)(31 51)(32 52)(41 62)(42 63)(43 64)(44 61)
(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 59 55 27)(2 24 56 40)(3 57 53 25)(4 22 54 38)(5 13 61 49)(6 33 62 30)(7 15 63 51)(8 35 64 32)(9 36 44 29)(10 14 41 50)(11 34 42 31)(12 16 43 52)(17 37 45 21)(18 26 46 58)(19 39 47 23)(20 28 48 60)
G:=sub<Sym(64)| (1,29)(2,30)(3,31)(4,32)(5,23)(6,24)(7,21)(8,22)(9,59)(10,60)(11,57)(12,58)(13,19)(14,20)(15,17)(16,18)(25,42)(26,43)(27,44)(28,41)(33,56)(34,53)(35,54)(36,55)(37,63)(38,64)(39,61)(40,62)(45,51)(46,52)(47,49)(48,50), (1,47)(2,48)(3,45)(4,46)(5,9)(6,10)(7,11)(8,12)(13,36)(14,33)(15,34)(16,35)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,37)(26,38)(27,39)(28,40)(29,49)(30,50)(31,51)(32,52)(41,62)(42,63)(43,64)(44,61), (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), (1,59,55,27)(2,24,56,40)(3,57,53,25)(4,22,54,38)(5,13,61,49)(6,33,62,30)(7,15,63,51)(8,35,64,32)(9,36,44,29)(10,14,41,50)(11,34,42,31)(12,16,43,52)(17,37,45,21)(18,26,46,58)(19,39,47,23)(20,28,48,60)>;
G:=Group( (1,29)(2,30)(3,31)(4,32)(5,23)(6,24)(7,21)(8,22)(9,59)(10,60)(11,57)(12,58)(13,19)(14,20)(15,17)(16,18)(25,42)(26,43)(27,44)(28,41)(33,56)(34,53)(35,54)(36,55)(37,63)(38,64)(39,61)(40,62)(45,51)(46,52)(47,49)(48,50), (1,47)(2,48)(3,45)(4,46)(5,9)(6,10)(7,11)(8,12)(13,36)(14,33)(15,34)(16,35)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,37)(26,38)(27,39)(28,40)(29,49)(30,50)(31,51)(32,52)(41,62)(42,63)(43,64)(44,61), (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), (1,59,55,27)(2,24,56,40)(3,57,53,25)(4,22,54,38)(5,13,61,49)(6,33,62,30)(7,15,63,51)(8,35,64,32)(9,36,44,29)(10,14,41,50)(11,34,42,31)(12,16,43,52)(17,37,45,21)(18,26,46,58)(19,39,47,23)(20,28,48,60) );
G=PermutationGroup([[(1,29),(2,30),(3,31),(4,32),(5,23),(6,24),(7,21),(8,22),(9,59),(10,60),(11,57),(12,58),(13,19),(14,20),(15,17),(16,18),(25,42),(26,43),(27,44),(28,41),(33,56),(34,53),(35,54),(36,55),(37,63),(38,64),(39,61),(40,62),(45,51),(46,52),(47,49),(48,50)], [(1,47),(2,48),(3,45),(4,46),(5,9),(6,10),(7,11),(8,12),(13,36),(14,33),(15,34),(16,35),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,37),(26,38),(27,39),(28,40),(29,49),(30,50),(31,51),(32,52),(41,62),(42,63),(43,64),(44,61)], [(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)], [(1,59,55,27),(2,24,56,40),(3,57,53,25),(4,22,54,38),(5,13,61,49),(6,33,62,30),(7,15,63,51),(8,35,64,32),(9,36,44,29),(10,14,41,50),(11,34,42,31),(12,16,43,52),(17,37,45,21),(18,26,46,58),(19,39,47,23),(20,28,48,60)]])
C2×C2.C42 is a maximal subgroup of
C23.19C42 C23.30D8 C24.17Q8 C24.624C23 C24.625C23 C24.626C23 C23⋊2C42 C24.50D4 C24.5Q8 C24.52D4 C24.631C23 C24.632C23 C24.633C23 C24.634C23 C24.635C23 C24.636C23 C24.150D4 C2×C4×C22⋊C4 C2×C4×C4⋊C4 D4⋊4C42 C24.547C23 C24.198C23 C23.211C24 C24.549C23 C23.225C24 C23.231C24 C23.235C24 C23.241C24 C24.218C23 C23.250C24 C24.563C23 C24.258C23 C24.262C23 C24.567C23 C23.344C24 C23.350C24 C23.388C24 C24.577C23 C23.398C24 C23.405C24 C23.410C24 C23.443C24 C23.449C24 C24.583C23 C24.584C23 C24.592C23 C23.543C24 C23.546C24 C23.556C24 C23.559C24
C2×C2.C42 is a maximal quotient of
C24.17Q8 C24.625C23 C23.28C42 C23.29C42 C24.63D4 C24.132D4 C24.152D4 C24.7Q8 C24.162C23 C23.15C42
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 | 1 | 2 | 2 |
| type | + | + | + | + | - | |
| image | C1 | C2 | C2 | C4 | D4 | Q8 |
| kernel | C2×C2.C42 | C2.C42 | C23×C4 | C22×C4 | C23 | C23 |
| # reps | 1 | 4 | 3 | 24 | 6 | 2 |
Matrix representation of C2×C2.C42 ►in GL5(𝔽5)
| 4 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 4 |
| 2 | 0 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 | 0 |
| 0 | 0 | 4 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(5,GF(5))| [4,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,4],[2,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,4],[2,0,0,0,0,0,3,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,1,0] >;
C2×C2.C42 in GAP, Magma, Sage, TeX
C_2\times C_2.C_4^2
% in TeX
G:=Group("C2xC2.C4^2"); // GroupNames label
G:=SmallGroup(64,56);
// by ID
G=gap.SmallGroup(64,56);
# by ID
G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^2=c^4=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations