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G = C2×C2.C42order 64 = 26

Direct product of C2 and C2.C42

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

C1C2 — C2×C2.C42
C1C2C22C23C24C23×C4 — C2×C2.C42
C1C2 — C2×C2.C42
C1C24 — C2×C2.C42
C1C23 — C2×C2.C42

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 [×14], C4 [×12], C22, C22 [×34], C2×C4 [×12], C2×C4 [×36], C23, C23 [×14], C22×C4 [×18], C22×C4 [×12], C24, C2.C42 [×4], C23×C4 [×3], C2×C2.C42
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×6], Q8 [×2], C23, C42 [×4], C22⋊C4 [×12], C4⋊C4 [×12], C22×C4 [×3], C2×D4 [×3], C2×Q8, C2.C42 [×8], C2×C42, C2×C22⋊C4 [×3], C2×C4⋊C4 [×3], C2×C2.C42

Smallest permutation representation of C2×C2.C42
Regular action on 64 points
Generators in S64
(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  C232C42  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  D44C42  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···2O4A···4X
order12···24···4
size11···12···2

40 irreducible representations

dim111122
type++++-
imageC1C2C2C4D4Q8
kernelC2×C2.C42C2.C42C23×C4C22×C4C23C23
# reps1432462

Matrix representation of C2×C2.C42 in GL5(𝔽5)

40000
01000
00400
00040
00004
,
10000
01000
00100
00040
00004
,
20000
01000
00400
00010
00004
,
20000
03000
00400
00001
00010

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

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