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G = C42.302D4  order 128 = 27

284th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.302D4, C42.436C23, C4.102- (1+4), C4.292+ (1+4), C8⋊D4⋊21C2, D4⋊Q8⋊31C2, C4.Q16⋊31C2, C4⋊SD16⋊14C2, C4⋊C8.82C22, (C2×C8).78C23, D4.D4⋊14C2, C4⋊C4.193C23, (C2×C4).452C24, (C22×C4).529D4, C23.309(C2×D4), C4⋊Q8.330C22, C4.107(C8⋊C22), C4⋊M4(2)⋊13C2, (C2×D4).194C23, (C4×D4).132C22, (C2×Q8).181C23, (C4×Q8).128C22, C2.D8.113C22, D4⋊C4.59C22, C4⋊1D4.179C22, C4⋊D4.214C22, C4.102(C8.C22), (C2×C42).909C22, Q8⋊C4.56C22, (C2×SD16).42C22, C22.712(C22×D4), C22⋊Q8.218C22, (C22×C4).1107C23, (C2×M4(2)).90C22, C23.37C23⋊27C2, C22.26C24.49C2, C2.71(C22.31C24), (C2×C4).576(C2×D4), C2.69(C2×C8⋊C22), C2.68(C2×C8.C22), SmallGroup(128,1986)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C2×C4 — C42.302D4
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×D4 — C22.26C24 — C42.302D4
Lower central
C1 — C2 — C2×C4 — C42.302D4
Upper central
C1 — C22 — C2×C42 — C42.302D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.302D4

Subgroups: 388 in 193 conjugacy classes, 88 normal (30 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×9], C8 [×4], C2×C4 [×6], C2×C4 [×16], D4 [×12], Q8 [×8], C23, C23 [×2], C42 [×4], C42 [×2], C22⋊C4 [×6], C4⋊C4 [×4], C4⋊C4 [×8], C2×C8 [×4], M4(2) [×2], SD16 [×4], C22×C4 [×3], C22×C4 [×2], C2×D4 [×2], C2×D4 [×4], C2×Q8 [×2], C2×Q8 [×2], C4○D4 [×4], D4⋊C4 [×4], Q8⋊C4 [×4], C4⋊C8 [×4], C2.D8 [×4], C2×C42, C42⋊C2, C4×D4 [×2], C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C22⋊Q8, C4.4D4, C42.C2, C4⋊1D4, C4⋊Q8 [×3], C2×M4(2) [×2], C2×SD16 [×4], C2×C4○D4, C4⋊M4(2), C4⋊SD16 [×2], D4.D4 [×2], C8⋊D4 [×4], D4⋊Q8 [×2], C4.Q16 [×2], C22.26C24, C23.37C23, C42.302D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8⋊C22 [×2], C8.C22 [×2], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C2×C8⋊C22, C2×C8.C22, C42.302D4

Generators and relations
 G = < a,b,c,d | a4=b4=1, c4=d2=a2, ab=ba, cac-1=a-1, dad-1=ab2, cbc-1=dbd-1=b-1, dcd-1=c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

101500000
001610000
001600000
011600000
00000010
00000001
000016000
000001600
,
115000000
116000000
016010000
1161600000
000016000
000001600
000000160
000000016
,
40090000
004130000
040130000
000130000
000015288
0000151598
000088215
00009822
,
40090000
004130000
4130130000
400130000
000015288
00002289
000099152
00009822

G:=sub<GL(8,GF(17))| [1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0],[1,1,0,1,0,0,0,0,15,16,16,16,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[4,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,9,13,13,13,0,0,0,0,0,0,0,0,15,15,8,9,0,0,0,0,2,15,8,8,0,0,0,0,8,9,2,2,0,0,0,0,8,8,15,2],[4,0,4,4,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,9,13,13,13,0,0,0,0,0,0,0,0,15,2,9,9,0,0,0,0,2,2,9,8,0,0,0,0,8,8,15,2,0,0,0,0,8,9,2,2] >;
 

Character table of C42.302D4

 class 12A2B2C2D2E2F4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D
 size 11114882222224448888888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-111-1111-1-11-1-1-11111-1-1    linear of order 2
ρ31111-1111-1111-1-11-1-1-11-1-11-111-1    linear of order 2
ρ411111-11-1-11-11-1-1-11-11-11-11-11-11    linear of order 2
ρ51111-1111-1111-1-11-1-1-1-111-11-1-11    linear of order 2
ρ611111-11-1-11-11-1-1-11-111-11-11-11-1    linear of order 2
ρ7111111111111111111-1-1-1-1-1-1-1-1    linear of order 2
ρ81111-1-11-111-1111-1-11-111-1-1-1-111    linear of order 2
ρ911111-1-1111111111-1-1-1-1-1-11111    linear of order 2
ρ101111-11-1-111-1111-1-1-1111-1-111-1-1    linear of order 2
ρ111111-1-1-11-1111-1-11-111-111-1-111-1    linear of order 2
ρ12111111-1-1-11-11-1-1-111-11-11-1-11-11    linear of order 2
ρ131111-1-1-11-1111-1-11-1111-1-111-1-11    linear of order 2
ρ14111111-1-1-11-11-1-1-111-1-11-111-11-1    linear of order 2
ρ1511111-1-1111111111-1-11111-1-1-1-1    linear of order 2
ρ161111-11-1-111-1111-1-1-11-1-111-1-111    linear of order 2
ρ172222-200-2-2-2-2-2-22220000000000    orthogonal lifted from D4
ρ182222200-22-2-2-22-22-20000000000    orthogonal lifted from D4
ρ1922222002-2-22-2-22-2-20000000000    orthogonal lifted from D4
ρ202222-20022-22-22-2-220000000000    orthogonal lifted from D4
ρ214-4-440000-400040000000000000    orthogonal lifted from C8⋊C22
ρ224-44-400000-40400000000000000    orthogonal lifted from 2+ (1+4)
ρ234-4-4400004000-40000000000000    orthogonal lifted from C8⋊C22
ρ244-44-40000040-400000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2544-4-4000-4004000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2644-4-4000400-4000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{302}D_4
 
% in TeX
 
G:=Group("C4^2.302D4");
 
// GroupNames label
 
G:=SmallGroup(128,1986);
 
// by ID
 
G=gap.SmallGroup(128,1986);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,675,80,4037,1027,124]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^4=b^4=1,c^4=d^2=a^2,a*b=b*a,c*a*c^-1=a^-1,d*a*d^-1=a*b^2,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=c^3>;
 
// generators/relations
 

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