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G = C42.467C23order 128 = 27

328th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.467C23, C4.472+ 1+4, (C8×D4)⋊20C2, C4⋊C4.408D4, D4⋊Q813C2, C42Q1615C2, (C4×SD16)⋊41C2, (C2×D4).237D4, D46D4.6C2, C4.72(C4○D8), C2.46(Q8○D8), D4.19(C4○D4), D4.7D411C2, C8.18D414C2, C4⋊C8.297C22, C4⋊C4.402C23, (C2×C4).494C24, (C4×C8).274C22, (C2×C8).350C23, C22⋊C4.106D4, C4.SD1630C2, C23.110(C2×D4), C4⋊Q8.143C22, C2.D8.56C22, (C2×D4).421C23, (C4×D4).335C22, C23.20D46C2, (C2×Q16).37C22, (C4×Q8).151C22, (C2×Q8).210C23, C2.130(D45D4), C4.Q8.166C22, C22⋊Q8.74C22, C23.24D412C2, C22⋊C8.203C22, (C22×C8).162C22, C22.754(C22×D4), D4⋊C4.167C22, C22.50C242C2, (C22×C4).1138C23, Q8⋊C4.114C22, (C2×SD16).157C22, C42⋊C2.182C22, C2.62(C2×C4○D8), C4.219(C2×C4○D4), (C2×C4).923(C2×D4), (C2×C4○D4).200C22, SmallGroup(128,2034)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.467C23
C1C2C4C2×C4C22×C4C2×C4○D4D46D4 — C42.467C23
C1C2C2×C4 — C42.467C23
C1C22C4×D4 — C42.467C23
C1C2C2C2×C4 — C42.467C23

Generators and relations for C42.467C23
 G = < a,b,c,d,e | a4=b4=d2=1, c2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd=b-1, be=eb, dcd=bc, ece-1=a2c, de=ed >

Subgroups: 360 in 193 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×10], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×17], D4 [×2], D4 [×7], Q8 [×8], C23 [×2], C23, C42, C42 [×3], C22⋊C4 [×2], C22⋊C4 [×7], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×8], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×3], C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C2×Q8, C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×6], C4⋊C8, C4.Q8, C2.D8 [×2], C2×C4⋊C4, C42⋊C2 [×2], C4×D4 [×2], C4×Q8, C4×Q8, C4⋊D4, C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C422C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16, C2×Q16 [×2], C2×C4○D4 [×2], C23.24D4 [×2], C8×D4, C4×SD16, D4.7D4 [×2], C42Q16, C8.18D4 [×2], D4⋊Q8, C23.20D4 [×2], C4.SD16, D46D4, C22.50C24, C42.467C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×2], C24, C4○D8 [×2], C22×D4, C2×C4○D4, 2+ 1+4, D45D4, C2×C4○D8, Q8○D8, C42.467C23

Smallest permutation representation of C42.467C23
On 64 points
Generators in S64
(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 42 46 17)(2 43 47 18)(3 44 48 19)(4 41 45 20)(5 62 33 39)(6 63 34 40)(7 64 35 37)(8 61 36 38)(9 15 23 49)(10 16 24 50)(11 13 21 51)(12 14 22 52)(25 56 60 31)(26 53 57 32)(27 54 58 29)(28 55 59 30)
(1 26 3 28)(2 25 4 27)(5 52 7 50)(6 51 8 49)(9 40 11 38)(10 39 12 37)(13 36 15 34)(14 35 16 33)(17 53 19 55)(18 56 20 54)(21 61 23 63)(22 64 24 62)(29 43 31 41)(30 42 32 44)(45 58 47 60)(46 57 48 59)
(1 3)(2 4)(5 64)(6 61)(7 62)(8 63)(9 11)(10 12)(13 49)(14 50)(15 51)(16 52)(17 44)(18 41)(19 42)(20 43)(21 23)(22 24)(25 29)(26 30)(27 31)(28 32)(33 37)(34 38)(35 39)(36 40)(45 47)(46 48)(53 59)(54 60)(55 57)(56 58)
(1 10 46 24)(2 9 47 23)(3 12 48 22)(4 11 45 21)(5 30 33 55)(6 29 34 54)(7 32 35 53)(8 31 36 56)(13 20 51 41)(14 19 52 44)(15 18 49 43)(16 17 50 42)(25 38 60 61)(26 37 57 64)(27 40 58 63)(28 39 59 62)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I4J4K4L···4Q8A8B8C8D8E···8J
order122222224···44444···488888···8
size111144442···24448···822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82+ 1+4Q8○D8
kernelC42.467C23C23.24D4C8×D4C4×SD16D4.7D4C42Q16C8.18D4D4⋊Q8C23.20D4C4.SD16D46D4C22.50C24C22⋊C4C4⋊C4C2×D4D4C4C4C2
# reps1211212121112114812

Matrix representation of C42.467C23 in GL4(𝔽17) generated by

1000
0100
0074
001310
,
0100
16000
0010
0001
,
31400
141400
0004
0040
,
1000
01600
00160
00016
,
13000
01300
001310
0074
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,7,13,0,0,4,10],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[3,14,0,0,14,14,0,0,0,0,0,4,0,0,4,0],[1,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[13,0,0,0,0,13,0,0,0,0,13,7,0,0,10,4] >;

C42.467C23 in GAP, Magma, Sage, TeX

C_4^2._{467}C_2^3
% in TeX

G:=Group("C4^2.467C2^3");
// GroupNames label

G:=SmallGroup(128,2034);
// by ID

G=gap.SmallGroup(128,2034);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,346,248,4037,1027,124]);
// Polycyclic

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

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