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G = C42.300C23order 128 = 27

161st non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.300C23, C4.1742+ 1+4, C4⋊Q8.33C4, C89D444C2, C41D4.20C4, C4⋊D4.28C4, C4⋊C8.236C22, C42.225(C2×C4), (C2×C4).677C24, (C2×C8).438C23, C4.4D4.21C4, (C4×D4).64C22, C8⋊C4.97C22, C42.6C452C2, C23.44(C22×C4), C2.31(Q8○M4(2)), C22⋊C8.145C22, (C22×C8).450C22, (C2×C42).784C22, C22.201(C23×C4), (C22×C4).944C23, C2.51(C22.11C24), (C2×M4(2)).247C22, C22.26C24.28C2, C4⋊C4.120(C2×C4), (C2×D4).144(C2×C4), C22⋊C4.21(C2×C4), (C2×Q8).126(C2×C4), (C22×C8)⋊C236C2, (C22×C4).357(C2×C4), (C2×C4).277(C22×C4), (C2×C4○D4).97C22, SmallGroup(128,1712)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.300C23
C1C2C4C2×C4C22×C4C2×C4○D4C22.26C24 — C42.300C23
C1C22 — C42.300C23
C1C2×C4 — C42.300C23
C1C2C2C2×C4 — C42.300C23

Generators and relations for C42.300C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1b2, dad=a-1, ae=ea, bc=cb, bd=db, be=eb, dcd=ece=a2c, ede=b2d >

Subgroups: 332 in 197 conjugacy classes, 124 normal (14 characteristic)
C1, C2, C2 [×2], C2 [×5], C4 [×2], C4 [×9], C22, C22 [×15], C8 [×8], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×10], Q8 [×2], C23, C23 [×4], C42 [×2], C42 [×2], C22⋊C4 [×8], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4, C22×C4 [×6], C2×D4 [×10], C2×Q8 [×2], C4○D4 [×4], C8⋊C4 [×4], C22⋊C8 [×12], C4⋊C8 [×4], C2×C42, C4×D4 [×4], C4⋊D4 [×4], C4.4D4 [×2], C41D4, C4⋊Q8, C22×C8 [×4], C2×M4(2) [×4], C2×C4○D4 [×2], (C22×C8)⋊C2 [×4], C42.6C4 [×2], C89D4 [×8], C22.26C24, C42.300C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C24, C23×C4, 2+ 1+4 [×2], C22.11C24, Q8○M4(2) [×2], C42.300C23

Smallest permutation representation of C42.300C23
On 64 points
Generators in S64
(1 47 55 33)(2 38 56 44)(3 41 49 35)(4 40 50 46)(5 43 51 37)(6 34 52 48)(7 45 53 39)(8 36 54 42)(9 22 64 26)(10 31 57 19)(11 24 58 28)(12 25 59 21)(13 18 60 30)(14 27 61 23)(15 20 62 32)(16 29 63 17)
(1 3 5 7)(2 4 6 8)(9 11 13 15)(10 12 14 16)(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 51 53 55)(50 52 54 56)(57 59 61 63)(58 60 62 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 17)(2 30)(3 19)(4 32)(5 21)(6 26)(7 23)(8 28)(9 48)(10 35)(11 42)(12 37)(13 44)(14 39)(15 46)(16 33)(18 56)(20 50)(22 52)(24 54)(25 51)(27 53)(29 55)(31 49)(34 64)(36 58)(38 60)(40 62)(41 57)(43 59)(45 61)(47 63)
(1 19)(2 32)(3 21)(4 26)(5 23)(6 28)(7 17)(8 30)(9 40)(10 47)(11 34)(12 41)(13 36)(14 43)(15 38)(16 45)(18 54)(20 56)(22 50)(24 52)(25 49)(27 51)(29 53)(31 55)(33 57)(35 59)(37 61)(39 63)(42 60)(44 62)(46 64)(48 58)

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

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

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

38 conjugacy classes

class 1 2A2B2C2D···2H4A4B4C4D4E···4M8A···8P
order12222···244444···48···8
size11114···411114···44···4

38 irreducible representations

dim11111111144
type++++++
imageC1C2C2C2C2C4C4C4C42+ 1+4Q8○M4(2)
kernelC42.300C23(C22×C8)⋊C2C42.6C4C89D4C22.26C24C4⋊D4C4.4D4C41D4C4⋊Q8C4C2
# reps14281842224

Matrix representation of C42.300C23 in GL8(𝔽17)

130090000
401340000
13130130000
00040000
0000161611
000000160
00000100
0000151501
,
130000000
013000000
001300000
000130000
00001000
00000100
00000010
00000001
,
160500000
77660000
160100000
11116100000
000001300
00004000
0000131344
000080913
,
48000000
1313000000
44040000
0131300000
00000100
00001000
0000161611
000022016
,
1615000000
01000000
16160160000
111600000
00000100
00001000
0000111616
00000001

G:=sub<GL(8,GF(17))| [13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,13,0,0,0,0,0,0,9,4,13,4,0,0,0,0,0,0,0,0,16,0,0,15,0,0,0,0,16,0,1,15,0,0,0,0,1,16,0,0,0,0,0,0,1,0,0,1],[13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,13,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[16,7,16,11,0,0,0,0,0,7,0,11,0,0,0,0,5,6,1,6,0,0,0,0,0,6,0,10,0,0,0,0,0,0,0,0,0,4,13,8,0,0,0,0,13,0,13,0,0,0,0,0,0,0,4,9,0,0,0,0,0,0,4,13],[4,13,4,0,0,0,0,0,8,13,4,13,0,0,0,0,0,0,0,13,0,0,0,0,0,0,4,0,0,0,0,0,0,0,0,0,0,1,16,2,0,0,0,0,1,0,16,2,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,16],[16,0,16,1,0,0,0,0,15,1,16,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,16,1] >;

C42.300C23 in GAP, Magma, Sage, TeX

C_4^2._{300}C_2^3
% in TeX

G:=Group("C4^2.300C2^3");
// GroupNames label

G:=SmallGroup(128,1712);
// by ID

G=gap.SmallGroup(128,1712);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,1430,219,675,1018,521,124]);
// Polycyclic

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

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