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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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C2×C2.C42
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C2×C2.C42
 Lower central C1 — C2 — C2×C2.C42
 Upper central C1 — C24 — C2×C2.C42
 Jennings C1 — C23 — 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)])

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

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