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## G = C22×C4⋊C4order 64 = 26

### Direct product of C22 and C4⋊C4

direct product, p-group, metabelian, nilpotent (class 2), monomial

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×C4⋊C4
 Chief series C1 — C2 — C22 — C23 — C24 — C23×C4 — C22×C4⋊C4
 Lower central C1 — C2 — C22×C4⋊C4
 Upper central C1 — C24 — C22×C4⋊C4
 Jennings C1 — C22 — C22×C4⋊C4

Generators and relations for C22×C4⋊C4
G = < a,b,c,d | a2=b2=c4=d4=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 249 in 209 conjugacy classes, 169 normal (8 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, C2×C4, C23, C4⋊C4, C22×C4, C22×C4, C24, C2×C4⋊C4, C23×C4, C23×C4, C22×C4⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, Q8, C23, C4⋊C4, C22×C4, C2×D4, C2×Q8, C24, C2×C4⋊C4, C23×C4, C22×D4, C22×Q8, C22×C4⋊C4

Smallest permutation representation of C22×C4⋊C4
Regular action on 64 points
Generators in S64
(1 15)(2 16)(3 13)(4 14)(5 9)(6 10)(7 11)(8 12)(17 41)(18 42)(19 43)(20 44)(21 31)(22 32)(23 29)(24 30)(25 56)(26 53)(27 54)(28 55)(33 61)(34 62)(35 63)(36 64)(37 51)(38 52)(39 49)(40 50)(45 57)(46 58)(47 59)(48 60)
(1 44)(2 41)(3 42)(4 43)(5 22)(6 23)(7 24)(8 21)(9 32)(10 29)(11 30)(12 31)(13 18)(14 19)(15 20)(16 17)(25 61)(26 62)(27 63)(28 64)(33 56)(34 53)(35 54)(36 55)(37 60)(38 57)(39 58)(40 59)(45 52)(46 49)(47 50)(48 51)
(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 54 12 57)(2 53 9 60)(3 56 10 59)(4 55 11 58)(5 48 16 26)(6 47 13 25)(7 46 14 28)(8 45 15 27)(17 62 22 51)(18 61 23 50)(19 64 24 49)(20 63 21 52)(29 40 42 33)(30 39 43 36)(31 38 44 35)(32 37 41 34)

G:=sub<Sym(64)| (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,11)(8,12)(17,41)(18,42)(19,43)(20,44)(21,31)(22,32)(23,29)(24,30)(25,56)(26,53)(27,54)(28,55)(33,61)(34,62)(35,63)(36,64)(37,51)(38,52)(39,49)(40,50)(45,57)(46,58)(47,59)(48,60), (1,44)(2,41)(3,42)(4,43)(5,22)(6,23)(7,24)(8,21)(9,32)(10,29)(11,30)(12,31)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(45,52)(46,49)(47,50)(48,51), (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), (1,54,12,57)(2,53,9,60)(3,56,10,59)(4,55,11,58)(5,48,16,26)(6,47,13,25)(7,46,14,28)(8,45,15,27)(17,62,22,51)(18,61,23,50)(19,64,24,49)(20,63,21,52)(29,40,42,33)(30,39,43,36)(31,38,44,35)(32,37,41,34)>;

G:=Group( (1,15)(2,16)(3,13)(4,14)(5,9)(6,10)(7,11)(8,12)(17,41)(18,42)(19,43)(20,44)(21,31)(22,32)(23,29)(24,30)(25,56)(26,53)(27,54)(28,55)(33,61)(34,62)(35,63)(36,64)(37,51)(38,52)(39,49)(40,50)(45,57)(46,58)(47,59)(48,60), (1,44)(2,41)(3,42)(4,43)(5,22)(6,23)(7,24)(8,21)(9,32)(10,29)(11,30)(12,31)(13,18)(14,19)(15,20)(16,17)(25,61)(26,62)(27,63)(28,64)(33,56)(34,53)(35,54)(36,55)(37,60)(38,57)(39,58)(40,59)(45,52)(46,49)(47,50)(48,51), (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), (1,54,12,57)(2,53,9,60)(3,56,10,59)(4,55,11,58)(5,48,16,26)(6,47,13,25)(7,46,14,28)(8,45,15,27)(17,62,22,51)(18,61,23,50)(19,64,24,49)(20,63,21,52)(29,40,42,33)(30,39,43,36)(31,38,44,35)(32,37,41,34) );

G=PermutationGroup([[(1,15),(2,16),(3,13),(4,14),(5,9),(6,10),(7,11),(8,12),(17,41),(18,42),(19,43),(20,44),(21,31),(22,32),(23,29),(24,30),(25,56),(26,53),(27,54),(28,55),(33,61),(34,62),(35,63),(36,64),(37,51),(38,52),(39,49),(40,50),(45,57),(46,58),(47,59),(48,60)], [(1,44),(2,41),(3,42),(4,43),(5,22),(6,23),(7,24),(8,21),(9,32),(10,29),(11,30),(12,31),(13,18),(14,19),(15,20),(16,17),(25,61),(26,62),(27,63),(28,64),(33,56),(34,53),(35,54),(36,55),(37,60),(38,57),(39,58),(40,59),(45,52),(46,49),(47,50),(48,51)], [(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)], [(1,54,12,57),(2,53,9,60),(3,56,10,59),(4,55,11,58),(5,48,16,26),(6,47,13,25),(7,46,14,28),(8,45,15,27),(17,62,22,51),(18,61,23,50),(19,64,24,49),(20,63,21,52),(29,40,42,33),(30,39,43,36),(31,38,44,35),(32,37,41,34)]])

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 C22×C4⋊C4 C2×C4⋊C4 C23×C4 C22×C4 C23 C23 # reps 1 12 3 16 4 4

Matrix representation of C22×C4⋊C4 in GL5(𝔽5)

 1 0 0 0 0 0 1 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4 0 0 0 0 0 4
,
 4 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 4 0
,
 2 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 0 2 0 0 0 2 0

G:=sub<GL(5,GF(5))| [1,0,0,0,0,0,1,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4,0,0,0,0,0,4],[4,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,1,0],[2,0,0,0,0,0,4,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,2,0] >;

C22×C4⋊C4 in GAP, Magma, Sage, TeX

C_2^2\times C_4\rtimes C_4
% in TeX

G:=Group("C2^2xC4:C4");
// GroupNames label

G:=SmallGroup(64,194);
// by ID

G=gap.SmallGroup(64,194);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,192,217,103]);
// 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,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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