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## G = C4×C4⋊C4order 64 = 26

### Direct product of C4 and C4⋊C4

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

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C4×C4⋊C4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C4×C4⋊C4
 Lower central C1 — C2 — C4×C4⋊C4
 Upper central C1 — C22×C4 — C4×C4⋊C4
 Jennings C1 — C23 — C4×C4⋊C4

Generators and relations for C4×C4⋊C4
G = < a,b,c | a4=b4=c4=1, ab=ba, ac=ca, cbc-1=b-1 >

Subgroups: 121 in 97 conjugacy classes, 73 normal (13 characteristic)
C1, C2 [×3], C2 [×4], C4 [×8], C4 [×10], C22 [×3], C22 [×4], C2×C4 [×22], C2×C4 [×10], C23, C42 [×4], C42 [×4], C4⋊C4 [×8], C22×C4 [×3], C22×C4 [×4], C2.C42 [×2], C2×C42, C2×C42 [×2], C2×C4⋊C4 [×2], C4×C4⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×2], Q8 [×2], C23, C42 [×4], C4⋊C4 [×4], C22×C4 [×3], C2×D4, C2×Q8, C4○D4 [×2], C2×C42, C2×C4⋊C4, C42⋊C2, C4×D4 [×2], C4×Q8 [×2], C4×C4⋊C4

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

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

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

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

40 conjugacy classes

 class 1 2A ··· 2G 4A ··· 4H 4I ··· 4AF order 1 2 ··· 2 4 ··· 4 4 ··· 4 size 1 1 ··· 1 1 ··· 1 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 type + + + + + - image C1 C2 C2 C2 C4 C4 D4 Q8 C4○D4 kernel C4×C4⋊C4 C2.C42 C2×C42 C2×C4⋊C4 C42 C4⋊C4 C2×C4 C2×C4 C22 # reps 1 2 3 2 8 16 2 2 4

Matrix representation of C4×C4⋊C4 in GL4(𝔽5) generated by

 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
,
 4 0 0 0 0 1 0 0 0 0 4 1 0 0 3 1
,
 4 0 0 0 0 2 0 0 0 0 3 2 0 0 0 2
G:=sub<GL(4,GF(5))| [2,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1],[4,0,0,0,0,1,0,0,0,0,4,3,0,0,1,1],[4,0,0,0,0,2,0,0,0,0,3,0,0,0,2,2] >;

C4×C4⋊C4 in GAP, Magma, Sage, TeX

C_4\times C_4\rtimes C_4
% in TeX

G:=Group("C4xC4:C4");
// GroupNames label

G:=SmallGroup(64,59);
// by ID

G=gap.SmallGroup(64,59);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,122]);
// Polycyclic

G:=Group<a,b,c|a^4=b^4=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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