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G = C42.373D4order 128 = 27

6th non-split extension by C42 of D4 acting via D4/C22=C2

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

Aliases: C42.373D4, C42.603C23, D4⋊C84C2, Q8⋊C84C2, C4⋊D4.2C4, (C4×C8).5C22, C22⋊Q8.2C4, C4.26(C8○D4), C42.56(C2×C4), C4.4D4.3C4, (C4×D4).2C22, C42.C2.5C4, (C4×Q8).2C22, C4.119(C4○D8), C4⋊C8.249C22, (C22×C4).201D4, C42.12C49C2, C23.43(C22⋊C4), (C2×C42).159C22, C2.8(C42⋊C22), C2.5(C23.24D4), C23.36C23.1C2, C4⋊C4.49(C2×C4), (C2×D4).49(C2×C4), (C2×Q8).44(C2×C4), (C2×C4).1446(C2×D4), (C2×C4).76(C22⋊C4), (C22×C4).181(C2×C4), (C2×C4).308(C22×C4), C22.158(C2×C22⋊C4), C2.14((C22×C8)⋊C2), SmallGroup(128,214)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.373D4
C1C2C22C2×C4C42C2×C42C23.36C23 — C42.373D4
C1C2C2×C4 — C42.373D4
C1C2×C4C2×C42 — C42.373D4
C1C22C22C42 — C42.373D4

Generators and relations for C42.373D4
 G = < a,b,c,d | a4=b4=1, c4=a2b2, d2=b, ab=ba, cac-1=ab2, ad=da, bc=cb, bd=db, dcd-1=a2b-1c3 >

Subgroups: 204 in 106 conjugacy classes, 46 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×7], C22, C22 [×6], C8 [×4], C2×C4 [×6], C2×C4 [×10], D4 [×4], Q8 [×2], C23, C23, C42 [×4], C42, C22⋊C4 [×5], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×4], C22×C4 [×3], C22×C4, C2×D4, C2×D4, C2×Q8, C4×C8 [×2], C4×C8, C22⋊C8 [×2], C4⋊C8 [×2], C4⋊C8, C2×C42, C42⋊C2, C4×D4, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22.D4, C4.4D4, C42.C2, C422C2, D4⋊C8 [×2], Q8⋊C8 [×2], C42.12C4 [×2], C23.36C23, C42.373D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×4], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×2], C2×C22⋊C4, C8○D4 [×2], C4○D8 [×2], (C22×C8)⋊C2, C23.24D4, C42⋊C22, C42.373D4

Smallest permutation representation of C42.373D4
On 64 points
Generators in S64
(1 43 51 35)(2 40 52 48)(3 45 53 37)(4 34 54 42)(5 47 55 39)(6 36 56 44)(7 41 49 33)(8 38 50 46)(9 20 32 63)(10 60 25 17)(11 22 26 57)(12 62 27 19)(13 24 28 59)(14 64 29 21)(15 18 30 61)(16 58 31 23)
(1 22 55 61)(2 23 56 62)(3 24 49 63)(4 17 50 64)(5 18 51 57)(6 19 52 58)(7 20 53 59)(8 21 54 60)(9 45 28 33)(10 46 29 34)(11 47 30 35)(12 48 31 36)(13 41 32 37)(14 42 25 38)(15 43 26 39)(16 44 27 40)
(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 21 22 54 55 60 61 8)(2 53 23 59 56 7 62 20)(3 58 24 6 49 19 63 52)(4 5 17 18 50 51 64 57)(9 48 45 31 28 36 33 12)(10 30 46 35 29 11 34 47)(13 44 41 27 32 40 37 16)(14 26 42 39 25 15 38 43)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E···4L4M4N4O4P8A···8P
order12222244444···444448···8
size11114811112···248884···4

38 irreducible representations

dim11111111122224
type+++++++
imageC1C2C2C2C2C4C4C4C4D4D4C8○D4C4○D8C42⋊C22
kernelC42.373D4D4⋊C8Q8⋊C8C42.12C4C23.36C23C4⋊D4C22⋊Q8C4.4D4C42.C2C42C22×C4C4C4C2
# reps12221222222882

Matrix representation of C42.373D4 in GL4(𝔽17) generated by

4000
0400
0001
0010
,
1000
0100
00130
00013
,
31400
3300
0090
0008
,
31400
141400
0090
0009
G:=sub<GL(4,GF(17))| [4,0,0,0,0,4,0,0,0,0,0,1,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[3,3,0,0,14,3,0,0,0,0,9,0,0,0,0,8],[3,14,0,0,14,14,0,0,0,0,9,0,0,0,0,9] >;

C42.373D4 in GAP, Magma, Sage, TeX

C_4^2._{373}D_4
% in TeX

G:=Group("C4^2.373D4");
// GroupNames label

G:=SmallGroup(128,214);
// by ID

G=gap.SmallGroup(128,214);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,-2,2,112,141,723,184,1123,570,136,172]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=a^2*b^2,d^2=b,a*b=b*a,c*a*c^-1=a*b^2,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a^2*b^-1*c^3>;
// generators/relations

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