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G = D4.5C42order 128 = 27

2nd non-split extension by D4 of C42 acting through Inn(D4)

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

Aliases: D4.5C42, Q8.5C42, C42.590C23, C8○D48C4, D4(C8⋊C4), Q8(C8⋊C4), (C4×D4).21C4, (C4×Q8).20C4, C4.61(C23×C4), C8.49(C22×C4), C4.13(C2×C42), (C4×M4(2))⋊28C2, M4(2)⋊25(C2×C4), M4(2)(C8⋊C4), (C2×C8).612C23, (C4×C8).324C22, (C2×C4).625C24, C42.199(C2×C4), C22.3(C2×C42), C82M4(2)⋊27C2, C2.2(Q8○M4(2)), C22.36(C23×C4), C2.17(C22×C42), C8⋊C4.173C22, (C22×C8).422C22, C23.137(C22×C4), (C2×C42).750C22, (C22×C4).1490C23, C42⋊C2.348C22, (C2×M4(2)).382C22, (C2×C8)⋊26(C2×C4), C4○D4(C8⋊C4), (C2×Q8)(C8⋊C4), (C4×C4○D4).9C2, (C2×C8⋊C4)⋊29C2, C4⋊C4.245(C2×C4), (C2×C8○D4).20C2, C4○D4.37(C2×C4), C8⋊C4(C2×M4(2)), (C2×D4).246(C2×C4), C22⋊C4.88(C2×C4), (C2×Q8).222(C2×C4), C8⋊C4(C42⋊C2), (C2×C4).455(C22×C4), (C22×C4).135(C2×C4), (C2×C4○D4).338C22, C8⋊C4(C2×C4○D4), SmallGroup(128,1607)

Series: Derived Chief Lower central Upper central Jennings

C1C2 — D4.5C42
C1C2C22C2×C4C22×C4C2×C4○D4C4×C4○D4 — D4.5C42
C1C2 — D4.5C42
C1C2×C4 — D4.5C42
C1C2C2C2×C4 — D4.5C42

Generators and relations for D4.5C42
 G = < a,b,c,d | a4=b2=d4=1, c4=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=a2c >

Subgroups: 316 in 278 conjugacy classes, 252 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C4 [×8], C4 [×8], C22, C22 [×6], C22 [×6], C8 [×16], C2×C4, C2×C4 [×23], C2×C4 [×6], D4 [×12], Q8 [×4], C23 [×3], C42, C42 [×9], C22⋊C4 [×6], C4⋊C4 [×6], C2×C8 [×32], M4(2) [×24], C22×C4 [×9], C2×D4 [×3], C2×Q8, C4○D4 [×8], C4×C8 [×6], C8⋊C4, C8⋊C4 [×9], C2×C42 [×3], C42⋊C2 [×3], C4×D4 [×6], C4×Q8 [×2], C22×C8 [×6], C2×M4(2) [×6], C8○D4 [×16], C2×C4○D4, C2×C8⋊C4 [×3], C4×M4(2) [×3], C82M4(2) [×6], C4×C4○D4, C2×C8○D4 [×2], D4.5C42
Quotients: C1, C2 [×15], C4 [×24], C22 [×35], C2×C4 [×84], C23 [×15], C42 [×16], C22×C4 [×42], C24, C2×C42 [×12], C23×C4 [×3], C22×C42, Q8○M4(2) [×2], D4.5C42

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

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

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

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

68 conjugacy classes

class 1 2A2B2C2D···2I4A4B4C4D4E···4Z8A···8AF
order12222···244444···48···8
size11112···211112···22···2

68 irreducible representations

dim1111111114
type++++++
imageC1C2C2C2C2C2C4C4C4Q8○M4(2)
kernelD4.5C42C2×C8⋊C4C4×M4(2)C82M4(2)C4×C4○D4C2×C8○D4C4×D4C4×Q8C8○D4C2
# reps133612124324

Matrix representation of D4.5C42 in GL5(𝔽17)

160000
013000
00400
000130
00004
,
160000
00400
013000
00004
000130
,
10000
09050
00905
05080
00508
,
40000
00010
00001
016000
001600

G:=sub<GL(5,GF(17))| [16,0,0,0,0,0,13,0,0,0,0,0,4,0,0,0,0,0,13,0,0,0,0,0,4],[16,0,0,0,0,0,0,13,0,0,0,4,0,0,0,0,0,0,0,13,0,0,0,4,0],[1,0,0,0,0,0,9,0,5,0,0,0,9,0,5,0,5,0,8,0,0,0,5,0,8],[4,0,0,0,0,0,0,0,16,0,0,0,0,0,16,0,1,0,0,0,0,0,1,0,0] >;

D4.5C42 in GAP, Magma, Sage, TeX

D_4._5C_4^2
% in TeX

G:=Group("D4.5C4^2");
// GroupNames label

G:=SmallGroup(128,1607);
// by ID

G=gap.SmallGroup(128,1607);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,925,232,521,172]);
// Polycyclic

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

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