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G = C4×C22⋊C4order 64 = 26

Direct product of C4 and C22⋊C4

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

Aliases: C4×C22⋊C4, C22⋊C42, C24.22C22, C23.53C23, C2.1(C4×D4), (C22×C4)⋊6C4, (C2×C42)⋊1C2, (C2×C4).142D4, (C23×C4).2C2, C2.3(C2×C42), C23.23(C2×C4), C22.27(C2×D4), C42(C2.C42), C2.C4213C2, C2.2(C42⋊C2), C22.13(C4○D4), (C22×C4).85C22, C22.15(C22×C4), (C2×C4)⋊6(C2×C4), C2.2(C2×C22⋊C4), (C2×C22⋊C4).14C2, (C2×C4)(C2.C42), (C22×C4)(C2.C42), (C2×C4)(C2×C22⋊C4), SmallGroup(64,58)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C22⋊C4
C1C2C22C23C22×C4C2×C42 — C4×C22⋊C4
C1C2 — C4×C22⋊C4
C1C22×C4 — C4×C22⋊C4
C1C23 — C4×C22⋊C4

Generators and relations for C4×C22⋊C4
 G = < a,b,c,d | a4=b2=c2=d4=1, ab=ba, ac=ca, ad=da, dbd-1=bc=cb, cd=dc >

Subgroups: 185 in 129 conjugacy classes, 73 normal (11 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×10], C22, C22 [×10], C22 [×12], C2×C4 [×16], C2×C4 [×18], C23, C23 [×6], C23 [×4], C42 [×4], C22⋊C4 [×8], C22×C4 [×2], C22×C4 [×8], C22×C4 [×4], C24, C2.C42 [×2], C2×C42 [×2], C2×C22⋊C4 [×2], C23×C4, C4×C22⋊C4
Quotients: C1, C2 [×7], C4 [×12], C22 [×7], C2×C4 [×18], D4 [×4], C23, C42 [×4], C22⋊C4 [×4], C22×C4 [×3], C2×D4 [×2], C4○D4 [×2], C2×C42, C2×C22⋊C4, C42⋊C2, C4×D4 [×4], C4×C22⋊C4

Smallest permutation representation of C4×C22⋊C4
On 32 points
Generators in S32
(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)
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 24)(10 21)(11 22)(12 23)(25 32)(26 29)(27 30)(28 31)
(1 5)(2 6)(3 7)(4 8)(9 30)(10 31)(11 32)(12 29)(13 17)(14 18)(15 19)(16 20)(21 28)(22 25)(23 26)(24 27)
(1 24 13 30)(2 21 14 31)(3 22 15 32)(4 23 16 29)(5 27 17 9)(6 28 18 10)(7 25 19 11)(8 26 20 12)

G:=sub<Sym(32)| (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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12)>;

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), (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,24)(10,21)(11,22)(12,23)(25,32)(26,29)(27,30)(28,31), (1,5)(2,6)(3,7)(4,8)(9,30)(10,31)(11,32)(12,29)(13,17)(14,18)(15,19)(16,20)(21,28)(22,25)(23,26)(24,27), (1,24,13,30)(2,21,14,31)(3,22,15,32)(4,23,16,29)(5,27,17,9)(6,28,18,10)(7,25,19,11)(8,26,20,12) );

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)], [(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,24),(10,21),(11,22),(12,23),(25,32),(26,29),(27,30),(28,31)], [(1,5),(2,6),(3,7),(4,8),(9,30),(10,31),(11,32),(12,29),(13,17),(14,18),(15,19),(16,20),(21,28),(22,25),(23,26),(24,27)], [(1,24,13,30),(2,21,14,31),(3,22,15,32),(4,23,16,29),(5,27,17,9),(6,28,18,10),(7,25,19,11),(8,26,20,12)])

C4×C22⋊C4 is a maximal subgroup of
C23.21C42  C24.46D4  C24.48D4  C24.53D4  C24.54D4  C24.55D4  C23.36C42  C23.17C42  C24.66D4  C24.70D4  C23.21M4(2)  (C2×C8).195D4  C24.72D4  C22⋊C44C8  C23.9M4(2)  C24.175C23  C24.176C23  D4×C42  C23⋊C42  C24.524C23  C25.85C22  C23.165C24  C4242D4  C439C2  C23.179C24  C432C2  C23.194C24  C23.195C24  C24.547C23  C23.203C24  C24.195C23  C23.211C24  C23.214C24  C23.215C24  C24.203C23  C24.204C23  C24.205C23  C24.549C23  C23.223C24  C23.224C24  C23.225C24  C23.226C24  C23.227C24  C24.208C23  C23.229C24  C23.234C24  C23.235C24  C24.212C23  C23.240C24  C23.241C24  C24.558C23  C24.215C23  C23.244C24  C24.217C23  C24.218C23  C24.220C23  C23.250C24  C24.221C23  C23.255C24  C24.223C23  C23.288C24  C4215D4  C23.295C24  C42.162D4  C42.163D4  C23.301C24  C24.254C23  C23.321C24  C23.322C24  C23.323C24  C24.259C23  C23.327C24  C23.328C24  C23.329C24  C24.263C23  C24.264C23  C23.333C24  C23.334C24  C23.335C24  C24.565C23  C24.267C23  C24.568C23  C24.268C23  C24.569C23  C24.278C23  C24.279C23  C23.359C24  C23.360C24  C23.364C24  C24.285C23  C24.286C23  C23.367C24  C24.289C23  C24.290C23  C23.372C24  C24.572C23  C24.293C23  C23.377C24  C23.380C24  C24.573C23  C23.388C24  C24.301C23  C23.390C24  C23.391C24  C23.392C24  C24.577C23  C24.304C23  C23.395C24  C23.396C24  C23.397C24  C24.308C23  C23.400C24  C23.401C24  C23.402C24  C24.579C23  C23.404C24  C23.405C24  C23.410C24  C24.309C23  C23.416C24  C23.417C24  C23.418C24  C24.311C23  C23.422C24  C24.313C23  C23.426C24  C24.315C23  C23.430C24  C23.431C24  C23.439C24  C23.449C24  C24.326C23  C24.327C23  C23.455C24  C23.456C24  C23.457C24  C23.458C24  C24.331C23  C24.332C23  C23.461C24  C24.584C23  C23.472C24  C23.473C24  C24.338C23  C24.339C23  C24.340C23  C24.341C23  C23.478C24  C23.483C24  C24.345C23  C24.346C23  C23.491C24  C24.347C23  C24.348C23  C23.502C24  C24.355C23  C23.508C24  C23.548C24  C24.375C23  C24.376C23  C23.553C24  C24.377C23  C24.378C23  C24.379C23  C23.567C24
 D2p⋊C42: D44C42  D6⋊C42  D102C42  D14⋊C42 ...
C4×C22⋊C4 is a maximal quotient of
C24.17Q8  C24.624C23  C24.626C23  C232C42  C42.378D4  C42.379D4  C23.36C42  C23.17C42  C23.5C42  D4.C42  Q8⋊C42  Q8.C42  D4.3C42
 D2p⋊C42: D4⋊C42  D6⋊C42  D102C42  D14⋊C42 ...

40 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4AB
order12···222224···44···4
size11···122221···12···2

40 irreducible representations

dim111111122
type++++++
imageC1C2C2C2C2C4C4D4C4○D4
kernelC4×C22⋊C4C2.C42C2×C42C2×C22⋊C4C23×C4C22⋊C4C22×C4C2×C4C22
# reps1222116844

Matrix representation of C4×C22⋊C4 in GL4(𝔽5) generated by

3000
0100
0040
0004
,
4000
0100
0040
0021
,
1000
0100
0040
0004
,
1000
0300
0033
0002
G:=sub<GL(4,GF(5))| [3,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[4,0,0,0,0,1,0,0,0,0,4,2,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,3,0,0,0,0,3,0,0,0,3,2] >;

C4×C22⋊C4 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC2^2:C4");
// GroupNames label

G:=SmallGroup(64,58);
// by ID

G=gap.SmallGroup(64,58);
# by ID

G:=PCGroup([6,-2,2,2,-2,2,2,96,121,199,86]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,d*b*d^-1=b*c=c*b,c*d=d*c>;
// generators/relations

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