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G = C42.162D4order 128 = 27

144th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.162D4, C23.296C24, C24.238C23, (C22×C4)⋊7Q8, C23.60(C2×Q8), C42(C23⋊Q8), C4.123(C22⋊Q8), C42(C23.4Q8), C42(C23.Q8), C23⋊Q8.36C2, C22.60(C22×Q8), (C23×C4).325C22, (C22×C4).782C23, (C2×C42).456C22, C22.179(C22×D4), C23.4Q8.36C2, C23.Q8.48C2, (C22×Q8).410C22, C42(C23.78C23), C42(C23.83C23), C43(C23.81C23), C23.78C2373C2, C2.12(C22.19C24), C23.81C23147C2, C23.83C23145C2, C2.C42.485C22, C2.11(C22.26C24), C2.7(C23.37C23), C2.14(C23.36C23), (C4×C4⋊C4)⋊53C2, (C2×C4×Q8)⋊11C2, (C2×C4).296(C2×D4), (C2×C4).354(C2×Q8), C2.13(C2×C22⋊Q8), (C2×C4).89(C4○D4), (C4×C22⋊C4).33C2, (C2×C4⋊C4).839C22, (C2×C4)2(C23⋊Q8), C22.176(C2×C4○D4), (C2×C42⋊C2).35C2, (C2×C4)2(C23.4Q8), (C2×C4)2(C23.Q8), (C2×C22⋊C4).488C22, (C2×C4)(C23.78C23), (C2×C4)(C23.83C23), (C2×C4)2(C23.81C23), (C22×C4)(C23.81C23), (C22×C4)(C23.78C23), SmallGroup(128,1128)

Series: Derived Chief Lower central Upper central Jennings

C1C23 — C42.162D4
C1C2C22C23C22×C4C2×C42C4×C22⋊C4 — C42.162D4
C1C23 — C42.162D4
C1C22×C4 — C42.162D4
C1C23 — C42.162D4

Generators and relations for C42.162D4
 G = < a,b,c,d | a4=b4=c4=d2=1, ab=ba, cac-1=a-1b2, dad=ab2, bc=cb, bd=db, dcd=a2c-1 >

Subgroups: 436 in 264 conjugacy classes, 116 normal (34 characteristic)
C1, C2 [×3], C2 [×4], C2 [×2], C4 [×4], C4 [×20], C22 [×3], C22 [×4], C22 [×10], C2×C4 [×20], C2×C4 [×40], Q8 [×8], C23, C23 [×2], C23 [×6], C42 [×4], C42 [×10], C22⋊C4 [×12], C4⋊C4 [×22], C22×C4 [×6], C22×C4 [×12], C22×C4 [×4], C2×Q8 [×6], C24, C2.C42 [×2], C2.C42 [×8], C2×C42 [×2], C2×C42 [×4], C2×C22⋊C4 [×2], C2×C22⋊C4 [×4], C2×C4⋊C4 [×3], C2×C4⋊C4 [×8], C42⋊C2 [×4], C4×Q8 [×4], C23×C4, C22×Q8, C4×C22⋊C4 [×2], C4×C4⋊C4, C4×C4⋊C4 [×2], C23⋊Q8, C23.78C23, C23.Q8 [×2], C23.81C23 [×2], C23.4Q8, C23.83C23, C2×C42⋊C2, C2×C4×Q8, C42.162D4
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], C22×D4, C22×Q8, C2×C4○D4 [×5], C2×C22⋊Q8, C22.19C24, C23.36C23 [×2], C22.26C24, C23.37C23 [×2], C42.162D4

Smallest permutation representation of C42.162D4
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 43 39 11)(2 44 40 12)(3 41 37 9)(4 42 38 10)(5 45 17 13)(6 46 18 14)(7 47 19 15)(8 48 20 16)(21 53 49 25)(22 54 50 26)(23 55 51 27)(24 56 52 28)(29 61 57 36)(30 62 58 33)(31 63 59 34)(32 64 60 35)
(1 13 49 59)(2 48 50 30)(3 15 51 57)(4 46 52 32)(5 25 34 43)(6 56 35 10)(7 27 36 41)(8 54 33 12)(9 19 55 61)(11 17 53 63)(14 24 60 38)(16 22 58 40)(18 28 64 42)(20 26 62 44)(21 31 39 45)(23 29 37 47)
(2 40)(4 38)(5 36)(6 62)(7 34)(8 64)(10 42)(12 44)(13 57)(14 30)(15 59)(16 32)(17 61)(18 33)(19 63)(20 35)(22 50)(24 52)(26 54)(28 56)(29 45)(31 47)(46 58)(48 60)

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

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

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

44 conjugacy classes

class 1 2A···2G2H2I4A···4H4I···4AH
order12···2224···44···4
size11···1441···14···4

44 irreducible representations

dim11111111111222
type++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2D4Q8C4○D4
kernelC42.162D4C4×C22⋊C4C4×C4⋊C4C23⋊Q8C23.78C23C23.Q8C23.81C23C23.4Q8C23.83C23C2×C42⋊C2C2×C4×Q8C42C22×C4C2×C4
# reps123112211114420

Matrix representation of C42.162D4 in GL6(𝔽5)

410000
310000
001100
000400
000001
000040
,
200000
020000
003000
000300
000040
000004
,
320000
120000
001000
003400
000030
000002
,
100000
240000
001000
003400
000010
000001

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

C42.162D4 in GAP, Magma, Sage, TeX

C_4^2._{162}D_4
% in TeX

G:=Group("C4^2.162D4");
// GroupNames label

G:=SmallGroup(128,1128);
// by ID

G=gap.SmallGroup(128,1128);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,2,224,253,758,723,184,80]);
// 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^-1*b^2,d*a*d=a*b^2,b*c=c*b,b*d=d*b,d*c*d=a^2*c^-1>;
// generators/relations

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