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G = C42.439D4order 128 = 27

72nd non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.439D4, C24.597C23, C23.751C24, C428C475C2, (C22×C4).98Q8, C4.88(C22⋊Q8), C23.104(C2×Q8), (C22×C42).26C2, C221(C42.C2), C23.251(C4○D4), (C23×C4).651C22, (C22×C4).261C23, C22.461(C22×D4), C23.8Q8.70C2, C23.7Q8.78C2, C22.179(C22×Q8), (C2×C42).1012C22, C2.94(C22.19C24), C23.63C23202C2, C23.65C23167C2, C2.C42.448C22, C2.45(C23.37C23), C2.109(C23.36C23), (C2×C4).171(C2×Q8), C2.46(C2×C22⋊Q8), (C2×C4).1205(C2×D4), (C2×C42.C2)⋊29C2, C2.21(C2×C42.C2), (C2×C4).669(C4○D4), (C2×C4⋊C4).554C22, C22.592(C2×C4○D4), (C2×C22⋊C4).361C22, SmallGroup(128,1583)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.439D4
C1C2C22C23C24C23×C4C22×C42 — C42.439D4
C1C23 — C42.439D4
C1C23 — C42.439D4
C1C23 — C42.439D4

Generators and relations for C42.439D4
 G = < a,b,c,d | a4=b4=c4=1, d2=b2, ab=ba, cac-1=dad-1=ab2, cbc-1=dbd-1=a2b, dcd-1=b2c-1 >

Subgroups: 452 in 278 conjugacy classes, 120 normal (18 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×18], C22 [×3], C22 [×8], C22 [×12], C2×C4 [×16], C2×C4 [×58], C23, C23 [×6], C23 [×4], C42 [×4], C42 [×6], C22⋊C4 [×8], C4⋊C4 [×24], C22×C4 [×2], C22×C4 [×16], C22×C4 [×16], C24, C2.C42 [×12], C2×C42 [×2], C2×C42 [×2], C2×C42 [×4], C2×C22⋊C4 [×4], C2×C4⋊C4 [×12], C42.C2 [×4], C23×C4, C23×C4 [×2], C23.7Q8 [×2], C428C4, C23.8Q8 [×4], C23.63C23 [×4], C23.65C23 [×2], C22×C42, C2×C42.C2, C42.439D4
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], Q8 [×4], C23 [×15], C2×D4 [×6], C2×Q8 [×6], C4○D4 [×10], C24, C22⋊Q8 [×4], C42.C2 [×4], C22×D4, C22×Q8, C2×C4○D4 [×5], C2×C22⋊Q8, C22.19C24 [×2], C2×C42.C2, C23.36C23 [×2], C23.37C23, C42.439D4

Smallest permutation representation of C42.439D4
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 7 23 11)(2 8 24 12)(3 5 21 9)(4 6 22 10)(13 29 25 37)(14 30 26 38)(15 31 27 39)(16 32 28 40)(17 44 62 60)(18 41 63 57)(19 42 64 58)(20 43 61 59)(33 49 46 56)(34 50 47 53)(35 51 48 54)(36 52 45 55)
(1 57 15 45)(2 42 16 33)(3 59 13 47)(4 44 14 35)(5 18 29 55)(6 64 30 49)(7 20 31 53)(8 62 32 51)(9 63 37 52)(10 19 38 56)(11 61 39 50)(12 17 40 54)(21 43 25 34)(22 60 26 48)(23 41 27 36)(24 58 28 46)
(1 34 23 47)(2 48 24 35)(3 36 21 45)(4 46 22 33)(5 50 9 53)(6 54 10 51)(7 52 11 55)(8 56 12 49)(13 41 25 57)(14 58 26 42)(15 43 27 59)(16 60 28 44)(17 38 62 30)(18 31 63 39)(19 40 64 32)(20 29 61 37)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4X4Y···4AF
order12···222224···44···4
size11···122222···28···8

44 irreducible representations

dim111111112222
type+++++++++-
imageC1C2C2C2C2C2C2C2D4Q8C4○D4C4○D4
kernelC42.439D4C23.7Q8C428C4C23.8Q8C23.63C23C23.65C23C22×C42C2×C42.C2C42C22×C4C2×C4C23
# reps1214421144128

Matrix representation of C42.439D4 in GL6(𝔽5)

400000
010000
004000
000400
000030
000003
,
200000
020000
004000
000400
000040
000001
,
010000
100000
000100
004000
000004
000040
,
010000
400000
000100
001000
000001
000010

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

C42.439D4 in GAP, Magma, Sage, TeX

C_4^2._{439}D_4
% in TeX

G:=Group("C4^2.439D4");
// GroupNames label

G:=SmallGroup(128,1583);
// by ID

G=gap.SmallGroup(128,1583);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,112,253,232,758,100,2019]);
// Polycyclic

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

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