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G = C4×C4⋊D4  order 128 = 27

Direct product of C4 and C4⋊D4

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

Aliases: C4×C4⋊D4, C42⋊45D4, C23.182C24, C24.187C23, C4⋊4(C4×D4), C22⋊1(C4×D4), (C22×C4)⋊46D4, (C22×C42)⋊14C2, C23.362(C2×D4), C22.73(C23×C4), C4○4(C23.7Q8), C22.78(C22×D4), C4○4(C23.23D4), C23.217(C4○D4), (C22×C4).453C23, C23.119(C22×C4), (C23×C4).679C22, (C2×C42).402C22, C23.7Q8⋊119C2, C23.23D4⋊117C2, C4○3(C24.3C22), C4○4(C24.C22), C2.6(C22.19C24), (C22×D4).468C22, C4○4(C23.65C23), C24.C22⋊190C2, C24.3C22⋊104C2, C2.4(C22.26C24), C23.65C23⋊174C2, C2.C42.515C22, C2.6(C23.36C23), (C2×C4×D4)⋊3C2, (C4×C4⋊C4)⋊19C2, C4⋊C4⋊24(C2×C4), C2.14(C2×C4×D4), (C2×D4)⋊29(C2×C4), C2.11(C4×C4○D4), C2.6(C2×C4⋊D4), C22⋊C4⋊25(C2×C4), (C4×C22⋊C4)⋊28C2, (C22×C4)⋊46(C2×C4), (C2×C4).1390(C2×D4), (C2×C4⋊D4).63C2, (C2×C4).18(C22×C4), C22.74(C2×C4○D4), (C2×C4).637(C4○D4), (C2×C4⋊C4).796C22, (C2×C4)○2(C23.7Q8), (C2×C22⋊C4).420C22, (C2×C4)○2(C23.65C23), (C22×C4)○(C23.65C23), (C2×C4)○(C2×C4⋊D4), (C22×C4)○(C2×C4⋊D4), SmallGroup(128,1032)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C22 — C4×C4⋊D4
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C23×C4 — C22×C42 — C4×C4⋊D4
Lower central
C1 — C22 — C4×C4⋊D4
Upper central
C1 — C22×C4 — C4×C4⋊D4
Jennings
C1 — C23 — C4×C4⋊D4

Generators and relations for C4×C4⋊D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 700 in 412 conjugacy classes, 172 normal (42 characteristic)
C1, C2, C2, C4, C4, C22, C22, C22, C2×C4, C2×C4, D4, C23, C23, C23, C42, C42, C22⋊C4, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C22×C4, C2×D4, C2×D4, C24, C24, C2.C42, C2×C42, C2×C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4, C4×D4, C4⋊D4, C23×C4, C23×C4, C22×D4, C22×D4, C4×C22⋊C4, C4×C4⋊C4, C23.7Q8, C23.23D4, C24.C22, C23.65C23, C24.3C22, C22×C42, C2×C4×D4, C2×C4×D4, C2×C4⋊D4, C4×C4⋊D4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C24, C4×D4, C4⋊D4, C23×C4, C22×D4, C2×C4○D4, C2×C4×D4, C4×C4○D4, C2×C4⋊D4, C22.19C24, C23.36C23, C22.26C24, C4×C4⋊D4

Smallest permutation representation of C4×C4⋊D4
►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 45 11 55)(2 46 12 56)(3 47 9 53)(4 48 10 54)(5 41 24 35)(6 42 21 36)(7 43 22 33)(8 44 23 34)(13 37 61 31)(14 38 62 32)(15 39 63 29)(16 40 64 30)(17 57 27 51)(18 58 28 52)(19 59 25 49)(20 60 26 50)

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

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

56 conjugacy classes

class 1 2A···2G2H2I2J2K2L2M2N2O4A···4H4I···4AB4AC···4AN
order12···2222222224···44···44···4
size11···1222244441···12···24···4

56 irreducible representations

dim1111111111112222
type+++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C4D4D4C4○D4C4○D4
kernelC4×C4⋊D4C4×C22⋊C4C4×C4⋊C4C23.7Q8C23.23D4C24.C22C23.65C23C24.3C22C22×C42C2×C4×D4C2×C4⋊D4C4⋊D4C42C22×C4C2×C4C23
# reps121122111311644124

Matrix representation of C4×C4⋊D4 ►in GL6(𝔽5)

200000
020000
001000
000100
000020
000002
,
040000
100000
000100
004000
000040
000004
,
040000
400000
000400
004000
000004
000010
,
040000
400000
000100
001000
000040
000001

G:=sub<GL(6,GF(5))| [2,0,0,0,0,0,0,2,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,1,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,4,0],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,1] >;
 

C4×C4⋊D4 in GAP, Magma, Sage, TeX 

C_4\times C_4\rtimes D_4
 
% in TeX
 
G:=Group("C4xC4:D4");
 
// GroupNames label
 
G:=SmallGroup(128,1032);
 
// by ID
 
G=gap.SmallGroup(128,1032);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,448,253,120,758,248]);
 
// 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,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
 
// generators/relations
 

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