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G = C4×C42⋊C2order 128 = 27

Direct product of C4 and C42⋊C2

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

Aliases: C4×C42⋊C2, C431C2, C23.152C24, C24.521C23, (C2×C4)⋊6C42, C422(C4⋊C4), (C2×C42)⋊19C4, C4234(C2×C4), C4.28(C2×C42), C42(C424C4), C424C435C2, C422(C22⋊C4), C2.5(C22×C42), C42(C424C4), C22.13(C2×C42), C22.24(C23×C4), (C22×C42).17C2, C42(C2.C42), C23.203(C22×C4), (C23×C4).640C22, (C2×C42).1144C22, (C22×C4).1643C23, C2.C42.564C22, C42(C4×C4⋊C4), C42(C4×C4⋊C4), C42(C2×C4⋊C4), C4⋊C447(C2×C4), C42(C4×C22⋊C4), (C4×C4⋊C4)⋊117C2, C2.1(C4×C4○D4), C42(C4×C22⋊C4), C42(C2×C22⋊C4), C22⋊C4.82(C2×C4), (C4×C22⋊C4).77C2, C2.4(C2×C42⋊C2), C42(C2×C42⋊C2), C22.50(C2×C4○D4), (C2×C4)2(C424C4), (C2×C4).947(C4○D4), (C2×C4⋊C4).967C22, (C22×C4).450(C2×C4), (C2×C4).286(C22×C4), (C2×C42⋊C2).66C2, (C2×C42)(C424C4), (C2×C22⋊C4).549C22, SmallGroup(128,1002)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — C4×C42⋊C2
C1C2C22C23C22×C4C2×C42C43 — C4×C42⋊C2
C1C2 — C4×C42⋊C2
C1C2×C42 — C4×C42⋊C2
C1C23 — C4×C42⋊C2

Generators and relations for C4×C42⋊C2
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=bc2, cd=dc >

Subgroups: 476 in 376 conjugacy classes, 276 normal (12 characteristic)
C1, C2, C2 [×6], C2 [×4], C4 [×16], C4 [×20], C22, C22 [×10], C22 [×12], C2×C4 [×60], C2×C4 [×36], C23, C23 [×6], C23 [×4], C42 [×32], C42 [×16], C22⋊C4 [×16], C4⋊C4 [×16], C22×C4 [×2], C22×C4 [×32], C22×C4 [×8], C24, C2.C42 [×8], C2×C42 [×2], C2×C42 [×22], C2×C22⋊C4 [×4], C2×C4⋊C4 [×4], C42⋊C2 [×16], C23×C4, C23×C4 [×2], C43 [×2], C424C4 [×2], C4×C22⋊C4 [×4], C4×C4⋊C4 [×4], C22×C42, C2×C42⋊C2 [×2], C4×C42⋊C2
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C4○D4 [×8], C24, C2×C42 [×12], C42⋊C2 [×8], C23×C4 [×3], C2×C4○D4 [×4], C22×C42, C2×C42⋊C2 [×2], C4×C4○D4 [×4], C4×C42⋊C2

Smallest permutation representation of C4×C42⋊C2
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 19 13 42)(2 20 14 43)(3 17 15 44)(4 18 16 41)(5 48 29 38)(6 45 30 39)(7 46 31 40)(8 47 32 37)(9 58 35 56)(10 59 36 53)(11 60 33 54)(12 57 34 55)(21 62 27 51)(22 63 28 52)(23 64 25 49)(24 61 26 50)
(1 35 23 31)(2 36 24 32)(3 33 21 29)(4 34 22 30)(5 15 11 27)(6 16 12 28)(7 13 9 25)(8 14 10 26)(17 54 62 38)(18 55 63 39)(19 56 64 40)(20 53 61 37)(41 57 52 45)(42 58 49 46)(43 59 50 47)(44 60 51 48)
(1 23)(2 24)(3 21)(4 22)(5 11)(6 12)(7 9)(8 10)(13 25)(14 26)(15 27)(16 28)(29 33)(30 34)(31 35)(32 36)

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

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

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

80 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4BP
order12···222224···44···4
size11···122221···12···2

80 irreducible representations

dim1111111112
type+++++++
imageC1C2C2C2C2C2C2C4C4C4○D4
kernelC4×C42⋊C2C43C424C4C4×C22⋊C4C4×C4⋊C4C22×C42C2×C42⋊C2C2×C42C42⋊C2C2×C4
# reps1224412163216

Matrix representation of C4×C42⋊C2 in GL4(𝔽5) generated by

3000
0200
0020
0002
,
1000
0300
0021
0023
,
4000
0100
0020
0002
,
4000
0400
0042
0001
G:=sub<GL(4,GF(5))| [3,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[1,0,0,0,0,3,0,0,0,0,2,2,0,0,1,3],[4,0,0,0,0,1,0,0,0,0,2,0,0,0,0,2],[4,0,0,0,0,4,0,0,0,0,4,0,0,0,2,1] >;

C4×C42⋊C2 in GAP, Magma, Sage, TeX

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

G:=Group("C4xC4^2:C2");
// GroupNames label

G:=SmallGroup(128,1002);
// by ID

G=gap.SmallGroup(128,1002);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,456,184,80]);
// Polycyclic

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

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