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G = C42.678C23order 128 = 27

93rd non-split extension by C42 of C23 acting via C23/C22=C2

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

Aliases: C42.678C23, (C4×D4).25C4, (C4×Q8).24C4, C4.48(C8○D4), C4⋊C8.231C22, C42.207(C2×C4), (C2×C4).644C24, (C2×C8).400C23, (C4×C8).327C22, C82M4(2)⋊29C2, C4⋊M4(2)⋊33C2, C42.6C446C2, C8⋊C4.154C22, C42.12C448C2, C4.44(C42⋊C2), C2.13(Q8○M4(2)), C22⋊C8.139C22, (C2×C42).757C22, C23.102(C22×C4), (C22×C4).914C23, (C22×C8).432C22, C22.172(C23×C4), C42.7C2222C2, C42⋊C2.350C22, C22.18(C42⋊C2), (C2×M4(2)).346C22, (C2×C4⋊C8)⋊46C2, C2.13(C2×C8○D4), C4⋊C4.220(C2×C4), C4⋊C8(C42⋊C2), (C4×C4○D4).13C2, C4.295(C2×C4○D4), (C2×D4).229(C2×C4), C22⋊C4.70(C2×C4), (C2×Q8).207(C2×C4), (C2×C4).829(C4○D4), (C2×C4).260(C22×C4), (C22×C4).338(C2×C4), C2.44(C2×C42⋊C2), (C22×C8)⋊C2.19C2, (C2×C4○D4).282C22, SmallGroup(128,1657)

Series: Derived Chief Lower central Upper central Jennings

C1C22 — C42.678C23
C1C2C4C2×C4C22×C4C42⋊C2C4×C4○D4 — C42.678C23
C1C22 — C42.678C23
C1C2×C4 — C42.678C23
C1C2C2C2×C4 — C42.678C23

Generators and relations for C42.678C23
 G = < a,b,c,d,e | a4=b4=d2=e2=1, c2=b, ab=ba, cac-1=a-1, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=a2c, ce=ec, ede=b2d >

Subgroups: 268 in 196 conjugacy classes, 134 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×4], C4 [×9], C22, C22 [×2], C22 [×8], C8 [×8], C2×C4 [×6], C2×C4 [×10], C2×C4 [×13], D4 [×6], Q8 [×2], C23, C23 [×2], C42 [×4], C42 [×6], C22⋊C4 [×6], C4⋊C4 [×2], C4⋊C4 [×4], C2×C8 [×8], C2×C8 [×4], M4(2) [×4], C22×C4 [×3], C22×C4 [×6], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×4], C4×C8 [×4], C8⋊C4 [×4], C22⋊C8 [×8], C4⋊C8 [×2], C4⋊C8 [×6], C2×C42, C2×C42 [×2], C42⋊C2, C42⋊C2 [×2], C4×D4 [×2], C4×D4 [×4], C4×Q8 [×2], C22×C8 [×2], C2×M4(2) [×2], C2×C4○D4, C82M4(2) [×2], (C22×C8)⋊C2 [×2], C2×C4⋊C8, C4⋊M4(2), C42.12C4 [×2], C42.6C4 [×2], C42.7C22 [×4], C4×C4○D4, C42.678C23
Quotients: C1, C2 [×15], C4 [×8], C22 [×35], C2×C4 [×28], C23 [×15], C22×C4 [×14], C4○D4 [×4], C24, C42⋊C2 [×4], C8○D4 [×2], C23×C4, C2×C4○D4 [×2], C2×C42⋊C2, C2×C8○D4, Q8○M4(2), C42.678C23

Smallest permutation representation of C42.678C23
On 64 points
Generators in S64
(1 63 55 10)(2 11 56 64)(3 57 49 12)(4 13 50 58)(5 59 51 14)(6 15 52 60)(7 61 53 16)(8 9 54 62)(17 39 31 42)(18 43 32 40)(19 33 25 44)(20 45 26 34)(21 35 27 46)(22 47 28 36)(23 37 29 48)(24 41 30 38)
(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)
(2 56)(4 50)(6 52)(8 54)(9 62)(11 64)(13 58)(15 60)(17 21)(18 28)(19 23)(20 30)(22 32)(24 26)(25 29)(27 31)(33 37)(34 41)(35 39)(36 43)(38 45)(40 47)(42 46)(44 48)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 41)(10 42)(11 43)(12 44)(13 45)(14 46)(15 47)(16 48)(25 49)(26 50)(27 51)(28 52)(29 53)(30 54)(31 55)(32 56)(33 57)(34 58)(35 59)(36 60)(37 61)(38 62)(39 63)(40 64)

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

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

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

50 conjugacy classes

class 1 2A2B2C2D2E2F2G4A4B4C4D4E···4R4S4T4U4V8A···8H8I···8T
order1222222244444···444448···88···8
size1111224411112···244442···24···4

50 irreducible representations

dim11111111111224
type+++++++++
imageC1C2C2C2C2C2C2C2C2C4C4C4○D4C8○D4Q8○M4(2)
kernelC42.678C23C82M4(2)(C22×C8)⋊C2C2×C4⋊C8C4⋊M4(2)C42.12C4C42.6C4C42.7C22C4×C4○D4C4×D4C4×Q8C2×C4C4C2
# reps122112241124882

Matrix representation of C42.678C23 in GL4(𝔽17) generated by

13000
13400
0010
0001
,
16000
01600
00130
00013
,
11500
11600
0080
0008
,
1000
11600
0010
00016
,
16000
01600
0001
0010
G:=sub<GL(4,GF(17))| [13,13,0,0,0,4,0,0,0,0,1,0,0,0,0,1],[16,0,0,0,0,16,0,0,0,0,13,0,0,0,0,13],[1,1,0,0,15,16,0,0,0,0,8,0,0,0,0,8],[1,1,0,0,0,16,0,0,0,0,1,0,0,0,0,16],[16,0,0,0,0,16,0,0,0,0,0,1,0,0,1,0] >;

C42.678C23 in GAP, Magma, Sage, TeX

C_4^2._{678}C_2^3
% in TeX

G:=Group("C4^2.678C2^3");
// GroupNames label

G:=SmallGroup(128,1657);
// by ID

G=gap.SmallGroup(128,1657);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,224,253,100,1018,521,124]);
// 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,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=a^2*c,c*e=e*c,e*d*e=b^2*d>;
// generators/relations

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