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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

Aliases: C4×C4⋊C4, C4⋊C42, C427C4, C23.54C23, C2.2(C4×D4), C2.1(C4×Q8), (C2×C4).24Q8, (C2×C4).143D4, C2.4(C2×C42), (C2×C42).7C2, C22.8(C2×Q8), C22.28(C2×D4), C43(C2.C42), C2.3(C42⋊C2), C22.14(C4○D4), C22.16(C22×C4), (C22×C4).86C22, C2.C42.12C2, C2.2(C2×C4⋊C4), (C2×C4⋊C4).21C2, (C2×C4).33(C2×C4), (C2×C4)2(C2.C42), SmallGroup(64,59)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C4⋊C4
C1C2C22C23C22×C4C2×C42 — C4×C4⋊C4
C1C2 — C4×C4⋊C4
C1C22×C4 — C4×C4⋊C4
C1C23 — 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)])

C4×C4⋊C4 is a maximal subgroup of
C42.46Q8  C42.4Q8  C42.10Q8  C42.403D4  C42.404D4  C42.57D4  C8⋊C42  C42.102D4  C4⋊C43C8  (C2×C8).Q8  C2.D84C4  C4.Q89C4  C4.Q810C4  C2.D85C4  C42.61Q8  C42.27Q8  M4(2)⋊7Q8  C42.121D4  C42.122D4  C42.123D4  D4×C42  Q8×C42  C24.524C23  D44C42  Q84C42  C23.165C24  C23.167C24  C4214Q8  C23.178C24  C432C2  C24.192C23  C23.201C24  C23.202C24  C42.33Q8  C424Q8  C23.214C24  C24.203C23  C24.204C23  C23.218C24  C24.205C23  C23.225C24  C23.226C24  C23.227C24  C24.208C23  C23.229C24  C23.231C24  C23.233C24  C23.237C24  C23.238C24  C24.212C23  C23.241C24  C24.215C23  C24.219C23  C23.250C24  C23.251C24  C23.252C24  C23.253C24  C23.255C24  C24.223C23  C4215D4  C23.295C24  C42.162D4  C425Q8  C23.301C24  C42.34Q8  C23.345C24  C23.346C24  C24.271C23  C23.348C24  C23.351C24  C23.352C24  C23.353C24  C23.354C24  C24.282C23  C23.362C24  C23.368C24  C23.369C24  C23.374C24  C23.375C24  C24.295C23  C23.379C24  C24.301C23  C23.390C24  C23.391C24  C23.392C24  C24.304C23  C23.395C24  C23.396C24  C23.397C24  C23.406C24  C23.407C24  C23.408C24  C23.409C24  C23.411C24  C23.412C24  C23.413C24  C23.414C24  C24.309C23  C23.416C24  C23.419C24  C23.420C24  C24.311C23  C23.422C24  C23.424C24  C23.425C24  C23.428C24  C23.429C24  C23.432C24  C23.433C24  C4219D4  C4220D4  C42.167D4  C4221D4  C42.168D4  C42.169D4  C42.170D4  C42.171D4  C426Q8  C427Q8  C42.35Q8  C24.327C23  C23.456C24  C23.458C24  C24.332C23  C42.172D4  C42.173D4  C42.174D4  C42.175D4  C42.176D4  C42.177D4  C42.36Q8  C42.37Q8  C23.473C24  C24.338C23  C24.339C23  C24.341C23  C42.181D4  C23.485C24  C23.486C24  C24.345C23  C23.488C24  C24.346C23  C23.490C24  C42.182D4  C23.493C24  C23.494C24  C24.347C23  C23.496C24  C24.348C23  C4223D4  C428Q8  C42.38Q8  C23.550C24  C23.551C24  C23.554C24  C23.555C24  C4232D4  C42.198D4  C4211Q8
 C2p.(C4×D4): D4⋊C42  Q8⋊C42  C4⋊C813C4  C4⋊C814C4  D4⋊C4⋊C4  C4.67(C4×D4)  C4.68(C4×D4)  C2.(C4×Q16) ...
C4×C4⋊C4 is a maximal quotient of
C24.624C23  C24.625C23  C24.626C23  C8⋊C42
 C2p.(C4×D4): C43.7C2  C42.45Q8  C4⋊C813C4  C4⋊C814C4  C8.14C42  C8.5C42  C8.6C42  Dic3⋊C42 ...

40 conjugacy classes

class 1 2A···2G4A···4H4I···4AF
order12···24···44···4
size11···11···12···2

40 irreducible representations

dim111111222
type+++++-
imageC1C2C2C2C4C4D4Q8C4○D4
kernelC4×C4⋊C4C2.C42C2×C42C2×C4⋊C4C42C4⋊C4C2×C4C2×C4C22
# reps1232816224

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

2000
0100
0010
0001
,
4000
0100
0041
0031
,
4000
0200
0032
0002
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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