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G = C42.100D4order 128 = 27

82nd non-split extension by C42 of D4 acting via D4/C2=C22

p-group, metabelian, nilpotent (class 3), monomial

Aliases: C42.100D4, (C4×D4)⋊14C4, D4.1(C4⋊C4), (C2×D4).23Q8, C428C43C2, (C2×D4).199D4, C2.1(D4.Q8), C22.4Q162C2, C42.138(C2×C4), C23.742(C2×D4), (C22×C4).268D4, C4.95(C22⋊Q8), C4.120(C4⋊D4), C2.1(D4.2D4), C22.41(C4○D8), (C22×C8).12C22, C4.33(C42⋊C2), C22.61(C8⋊C22), (C2×C42).249C22, C22.69(C22⋊Q8), C22.107(C4⋊D4), (C22×C4).1326C23, (C22×D4).457C22, C2.13(C23.7Q8), C2.18(C23.37D4), C2.21(C23.24D4), C4.3(C2×C4⋊C4), (C2×C4⋊C8)⋊12C2, (C2×C4×D4).16C2, C4⋊C4.191(C2×C4), (C2×C4).261(C2×Q8), (C2×D4⋊C4).2C2, (C2×D4).206(C2×C4), (C2×C4).1316(C2×D4), (C2×C4⋊C4).34C22, (C2×C4).862(C4○D4), (C2×C4).364(C22×C4), (C2×C4).331(C22⋊C4), C22.249(C2×C22⋊C4), SmallGroup(128,536)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.100D4
C1C2C22C2×C4C22×C4C22×D4C2×C4×D4 — C42.100D4
C1C2C2×C4 — C42.100D4
C1C23C2×C42 — C42.100D4
C1C2C2C22×C4 — C42.100D4

Generators and relations for C42.100D4
 G = < a,b,c,d | a4=b4=c4=1, d2=cbc-1=b-1, ab=ba, cac-1=dad-1=a-1b2, bd=db, dcd-1=b-1c-1 >

Subgroups: 388 in 178 conjugacy classes, 68 normal (28 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C4 [×4], C4 [×8], C22 [×3], C22 [×4], C22 [×16], C8 [×2], C2×C4 [×6], C2×C4 [×4], C2×C4 [×22], D4 [×4], D4 [×6], C23, C23 [×10], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×6], C22×C4 [×3], C22×C4 [×11], C2×D4 [×6], C2×D4 [×3], C24, C2.C42 [×2], D4⋊C4 [×4], C4⋊C8 [×2], C2×C42, C2×C22⋊C4, C2×C4⋊C4, C2×C4⋊C4 [×2], C4×D4 [×4], C4×D4 [×2], C22×C8 [×2], C23×C4, C22×D4, C22.4Q16 [×2], C428C4, C2×D4⋊C4 [×2], C2×C4⋊C8, C2×C4×D4, C42.100D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×6], Q8 [×2], C23, C22⋊C4 [×4], C4⋊C4 [×4], C22×C4, C2×D4 [×3], C2×Q8, C4○D4 [×2], C2×C22⋊C4, C2×C4⋊C4, C42⋊C2, C4⋊D4 [×2], C22⋊Q8 [×2], C4○D8 [×2], C8⋊C22 [×2], C23.7Q8, C23.24D4, C23.37D4, D4.2D4 [×2], D4.Q8 [×2], C42.100D4

Smallest permutation representation of C42.100D4
On 64 points
Generators in S64
(1 59 51 47)(2 44 52 64)(3 61 53 41)(4 46 54 58)(5 63 55 43)(6 48 56 60)(7 57 49 45)(8 42 50 62)(9 38 17 29)(10 26 18 35)(11 40 19 31)(12 28 20 37)(13 34 21 25)(14 30 22 39)(15 36 23 27)(16 32 24 33)
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 23 21 19)(18 24 22 20)(25 31 29 27)(26 32 30 28)(33 39 37 35)(34 40 38 36)(41 47 45 43)(42 48 46 44)(49 55 53 51)(50 56 54 52)(57 63 61 59)(58 64 62 60)
(1 36 32 54)(2 53 25 35)(3 34 26 52)(4 51 27 33)(5 40 28 50)(6 49 29 39)(7 38 30 56)(8 55 31 37)(9 18 48 61)(10 60 41 17)(11 24 42 59)(12 58 43 23)(13 22 44 57)(14 64 45 21)(15 20 46 63)(16 62 47 19)
(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)

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

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

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

38 conjugacy classes

class 1 2A···2G2H2I2J2K4A···4H4I···4N4O4P4Q4R8A···8H
order12···222224···44···444448···8
size11···144442···24···488884···4

38 irreducible representations

dim11111112222224
type+++++++++-+
imageC1C2C2C2C2C2C4D4D4D4Q8C4○D4C4○D8C8⋊C22
kernelC42.100D4C22.4Q16C428C4C2×D4⋊C4C2×C4⋊C8C2×C4×D4C4×D4C42C22×C4C2×D4C2×D4C2×C4C22C22
# reps12121182222482

Matrix representation of C42.100D4 in GL6(𝔽17)

1300000
0130000
0013000
0001300
00001613
000091
,
010000
1600000
0001600
001000
0000160
0000016
,
1430000
330000
0051200
00121200
000003
0000110
,
1430000
14140000
0051200
005500
0000014
000060

G:=sub<GL(6,GF(17))| [13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,16,9,0,0,0,0,13,1],[0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16],[14,3,0,0,0,0,3,3,0,0,0,0,0,0,5,12,0,0,0,0,12,12,0,0,0,0,0,0,0,11,0,0,0,0,3,0],[14,14,0,0,0,0,3,14,0,0,0,0,0,0,5,5,0,0,0,0,12,5,0,0,0,0,0,0,0,6,0,0,0,0,14,0] >;

C42.100D4 in GAP, Magma, Sage, TeX

C_4^2._{100}D_4
% in TeX

G:=Group("C4^2.100D4");
// GroupNames label

G:=SmallGroup(128,536);
// by ID

G=gap.SmallGroup(128,536);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,512,422,2804,718,172]);
// Polycyclic

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

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