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G = C4224D4order 128 = 27

18th semidirect product of C42 and D4 acting via D4/C2=C22

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

Aliases: C4224D4, C23.503C24, C24.354C23, C22.2842+ 1+4, C424C426C2, C23.63(C4○D4), C232D4.15C2, C23.8Q880C2, C23.11D454C2, C23.10D450C2, C23.23D464C2, (C23×C4).133C22, (C22×C4).848C23, (C2×C42).590C22, C22.333(C22×D4), C24.3C2262C2, (C22×D4).185C22, C23.81C2353C2, C24.C22101C2, C2.29(C22.29C24), C2.76(C22.19C24), C2.69(C22.45C24), C2.C42.233C22, C2.46(C22.26C24), C2.79(C22.47C24), (C2×C4×D4)⋊51C2, (C2×C4).1200(C2×D4), (C2×C422C2)⋊14C2, (C2×C4).410(C4○D4), (C2×C4⋊C4).884C22, C22.379(C2×C4○D4), (C2×C22⋊C4).203C22, SmallGroup(128,1335)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C4224D4
C1C2C22C23C22×C4C2×C42C424C4 — C4224D4
C1C23 — C4224D4
C1C23 — C4224D4
C1C23 — C4224D4

Generators and relations for C4224D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=ab2, dad=a-1b2, bc=cb, dbd=a2b, dcd=c-1 >

Subgroups: 596 in 291 conjugacy classes, 100 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×5], C4 [×17], C22 [×3], C22 [×4], C22 [×27], C2×C4 [×10], C2×C4 [×39], D4 [×16], C23, C23 [×4], C23 [×19], C42 [×4], C42 [×4], C22⋊C4 [×22], C4⋊C4 [×10], C22×C4 [×6], C22×C4 [×6], C22×C4 [×8], C2×D4 [×16], C24, C24 [×2], C2.C42 [×2], C2.C42 [×6], C2×C42, C2×C42 [×2], C2×C22⋊C4 [×3], C2×C22⋊C4 [×10], C2×C4⋊C4 [×4], C2×C4⋊C4 [×2], C4×D4 [×4], C422C2 [×4], C23×C4 [×2], C22×D4, C22×D4 [×2], C424C4, C23.8Q8 [×2], C23.23D4 [×2], C24.C22 [×2], C24.3C22 [×2], C232D4, C23.10D4, C23.11D4, C23.81C23, C2×C4×D4, C2×C422C2, C4224D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×8], C24, C22×D4, C2×C4○D4 [×4], 2+ 1+4 [×2], C22.19C24, C22.26C24, C22.29C24, C22.45C24 [×2], C22.47C24 [×2], C4224D4

Smallest permutation representation of C4224D4
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 57 25 17)(2 58 26 18)(3 59 27 19)(4 60 28 20)(5 33 9 50)(6 34 10 51)(7 35 11 52)(8 36 12 49)(13 29 53 47)(14 30 54 48)(15 31 55 45)(16 32 56 46)(21 38 61 43)(22 39 62 44)(23 40 63 41)(24 37 64 42)
(1 21 13 51)(2 62 14 35)(3 23 15 49)(4 64 16 33)(5 20 37 46)(6 57 38 29)(7 18 39 48)(8 59 40 31)(9 60 42 32)(10 17 43 47)(11 58 44 30)(12 19 41 45)(22 54 52 26)(24 56 50 28)(25 61 53 34)(27 63 55 36)
(1 13)(2 56)(3 15)(4 54)(5 9)(6 8)(7 11)(10 12)(14 28)(16 26)(17 45)(18 30)(19 47)(20 32)(22 64)(24 62)(25 53)(27 55)(29 59)(31 57)(33 52)(35 50)(37 42)(38 40)(39 44)(41 43)(46 60)(48 58)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K2L4A···4H4I···4V4W4X4Y
order12···2222224···44···4444
size11···1444482···24···4888

38 irreducible representations

dim1111111111112224
type++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4C4○D4C4○D42+ 1+4
kernelC4224D4C424C4C23.8Q8C23.23D4C24.C22C24.3C22C232D4C23.10D4C23.11D4C23.81C23C2×C4×D4C2×C422C2C42C2×C4C23C22
# reps1122221111114882

Matrix representation of C4224D4 in GL6(𝔽5)

030000
300000
001000
000100
000013
000004
,
040000
400000
004000
000400
000020
000002
,
020000
200000
004300
001100
000013
000014
,
400000
010000
004000
001100
000040
000041

G:=sub<GL(6,GF(5))| [0,3,0,0,0,0,3,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,3,4],[0,4,0,0,0,0,4,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,2,0,0,0,0,0,0,2],[0,2,0,0,0,0,2,0,0,0,0,0,0,0,4,1,0,0,0,0,3,1,0,0,0,0,0,0,1,1,0,0,0,0,3,4],[4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,1,0,0,0,0,0,1,0,0,0,0,0,0,4,4,0,0,0,0,0,1] >;

C4224D4 in GAP, Magma, Sage, TeX

C_4^2\rtimes_{24}D_4
% in TeX

G:=Group("C4^2:24D4");
// GroupNames label

G:=SmallGroup(128,1335);
// by ID

G=gap.SmallGroup(128,1335);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,184,675,248]);
// Polycyclic

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

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