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G = C42.303D4  order 128 = 27

285th non-split extension by C42 of D4 acting via D4/C2=C22

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

Aliases: C42.303D4, C42.437C23, C4.112- (1+4), C4.302+ (1+4), Q8⋊Q8⋊14C2, C4⋊2Q16⋊31C2, C4⋊C8.83C22, (C2×C8).79C23, C8.D4.6C2, C4⋊C4.194C23, (C2×C4).453C24, C23.310(C2×D4), (C22×C4).530D4, C4⋊Q8.331C22, C4.Q8.48C22, C4⋊M4(2).3C2, (C2×Q16).75C22, (C2×Q8).182C23, (C4×Q8).129C22, C4.103(C8.C22), (C2×C42).910C22, Q8⋊C4.57C22, C22.713(C22×D4), C22⋊Q8.219C22, (C22×C4).1108C23, (C2×M4(2)).91C22, C23.37C23.44C2, C2.72(C22.31C24), (C2×C4).577(C2×D4), C2.69(C2×C8.C22), SmallGroup(128,1987)

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

Derived series
C1 — C2×C4 — C42.303D4
Chief series
C1 — C2 — C4 — C2×C4 — C42 — C4×Q8 — C23.37C23 — C42.303D4
Lower central
C1 — C2 — C2×C4 — C42.303D4
Upper central
C1 — C22 — C2×C42 — C42.303D4
Jennings
C1 — C2 — C2 — C2×C4 — C42.303D4

Subgroups: 292 in 171 conjugacy classes, 88 normal (14 characteristic)
C1, C2, C2 [×2], C2, C4 [×2], C4 [×4], C4 [×11], C22, C22 [×3], C8 [×4], C2×C4 [×2], C2×C4 [×4], C2×C4 [×14], Q8 [×12], C23, C42 [×2], C42 [×2], C42 [×4], C22⋊C4 [×4], C4⋊C4 [×4], C4⋊C4 [×14], C2×C8 [×4], M4(2) [×2], Q16 [×4], C22×C4, C22×C4 [×2], C2×Q8 [×4], C2×Q8 [×2], Q8⋊C4 [×8], C4⋊C8 [×4], C4.Q8 [×4], C2×C42, C42⋊C2 [×2], C4×Q8 [×4], C4×Q8 [×2], C22⋊Q8 [×4], C22⋊Q8 [×2], C42.C2 [×2], C4⋊Q8 [×4], C2×M4(2) [×2], C2×Q16 [×4], C4⋊M4(2), C4⋊2Q16 [×4], C8.D4 [×4], Q8⋊Q8 [×4], C23.37C23 [×2], C42.303D4

Quotients:
C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C8.C22 [×4], C22×D4, 2+ (1+4), 2- (1+4), C22.31C24, C2×C8.C22 [×2], C42.303D4

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=a2c3 >

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

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

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

Matrix representation ►G ⊆ GL8(𝔽17)

162000000
161000000
001620000
001610000
000016000
000001600
00000010
00000001
,
115000000
116000000
001150000
001160000
0000161300
00009100
0000001613
00000091
,
00920000
001080000
1114000000
16000000
000041600
000001300
000000131
00000004
,
63000000
1611000000
00920000
001080000
000000131
00000004
000041600
000001300

G:=sub<GL(8,GF(17))| [16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,16,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1,0,0,0,0,0,0,0,0,16,9,0,0,0,0,0,0,13,1],[0,0,11,1,0,0,0,0,0,0,14,6,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,1,4],[6,16,0,0,0,0,0,0,3,11,0,0,0,0,0,0,0,0,9,10,0,0,0,0,0,0,2,8,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,0,0,0,16,13,0,0,0,0,13,0,0,0,0,0,0,0,1,4,0,0] >;
 

Character table of C42.303D4

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q8A8B8C8D
 size 11114222222444888888888888
ρ111111111111111111111111111    trivial
ρ21111-111-111-1-1-11111-1-11-1-1-111-1    linear of order 2
ρ311111111111111-1-1-1-1-1-1-1-11111    linear of order 2
ρ41111-111-111-1-1-11-1-1-111-111-111-1    linear of order 2
ρ51111-1-1111-111-1-1-11-1-1-111111-1-1    linear of order 2
ρ611111-11-11-1-1-11-1-11-1111-1-1-11-11    linear of order 2
ρ71111-1-1111-111-1-11-1111-1-1-111-1-1    linear of order 2
ρ811111-11-11-1-1-11-11-11-1-1-111-11-11    linear of order 2
ρ91111-1-1111-111-1-111-1-11-1-11-1-111    linear of order 2
ρ1011111-11-11-1-1-11-111-11-1-11-11-11-1    linear of order 2
ρ111111-1-1111-111-1-1-1-111-111-1-1-111    linear of order 2
ρ1211111-11-11-1-1-11-1-1-11-111-111-11-1    linear of order 2
ρ1311111111111111-1111-1-1-11-1-1-1-1    linear of order 2
ρ141111-111-111-1-1-11-111-11-11-11-1-11    linear of order 2
ρ15111111111111111-1-1-1111-1-1-1-1-1    linear of order 2
ρ161111-111-111-1-1-111-1-11-11-111-1-11    linear of order 2
ρ1722222-2-22-2-22-2-22000000000000    orthogonal lifted from D4
ρ182222-2-2-2-2-2-2-2222000000000000    orthogonal lifted from D4
ρ19222222-2-2-22-22-2-2000000000000    orthogonal lifted from D4
ρ202222-22-22-222-22-2000000000000    orthogonal lifted from D4
ρ214-44-40040-400000000000000000    orthogonal lifted from 2+ (1+4)
ρ224-4-44000400-4000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ234-4-44000-4004000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ2444-4-40-400040000000000000000    symplectic lifted from C8.C22, Schur index 2
ρ254-44-400-40400000000000000000    symplectic lifted from 2- (1+4), Schur index 2
ρ2644-4-404000-40000000000000000    symplectic lifted from C8.C22, Schur index 2

In GAP, Magma, Sage, TeX 

C_4^2._{303}D_4
 
% in TeX
 
G:=Group("C4^2.303D4");
 
// GroupNames label
 
G:=SmallGroup(128,1987);
 
// by ID
 
G=gap.SmallGroup(128,1987);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,456,758,891,352,675,80,4037,1027,124]);
 
// 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=a^2*c^3>;
 
// generators/relations
 

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