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G = C42.489C23order 128 = 27

350th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.489C23, C4.752- 1+4, (C8×D4)⋊26C2, C88D410C2, C4⋊C4.412D4, C83Q826C2, C4.Q1615C2, D4⋊Q815C2, (C4×SD16)⋊44C2, (C2×D4).244D4, C4.74(C4○D8), C8.83(C4○D4), C4⋊C8.301C22, C4⋊C4.245C23, (C2×C8).199C23, (C2×C4).532C24, (C4×C8).277C22, C22⋊C4.116D4, C23.117(C2×D4), C4⋊Q8.164C22, C2.85(D46D4), C2.88(D4○SD16), (C4×D4).345C22, (C2×D4).252C23, (C2×Q8).237C23, (C4×Q8).174C22, C2.D8.196C22, C4.Q8.110C22, D4⋊C4.16C22, C23.20D410C2, C4⋊D4.101C22, C23.25D412C2, C23.19D410C2, C22⋊C8.210C22, (C22×C8).199C22, C22.792(C22×D4), C22⋊Q8.100C22, C22.50C247C2, (C22×C4).1164C23, Q8⋊C4.184C22, (C2×SD16).166C22, C42⋊C2.203C22, C22.49C24.4C2, C4⋊C4(C4.Q8), C2.69(C2×C4○D8), C4.114(C2×C4○D4), (C2×C4).934(C2×D4), SmallGroup(128,2072)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.489C23
C1C2C4C2×C4C22×C4C42⋊C2C22.49C24 — C42.489C23
C1C2C2×C4 — C42.489C23
C1C22C4×D4 — C42.489C23
C1C2C2C2×C4 — C42.489C23

Generators and relations for C42.489C23
 G = < a,b,c,d,e | a4=b4=1, c2=d2=a2, e2=b2, ab=ba, cac-1=eae-1=a-1, ad=da, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece-1=a2c, de=ed >

Subgroups: 320 in 176 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×9], C8 [×2], C8 [×3], C2×C4 [×3], C2×C4 [×2], C2×C4 [×13], D4 [×6], Q8 [×5], C23 [×2], C23, C42, C42 [×5], C22⋊C4 [×2], C22⋊C4 [×9], C4⋊C4 [×3], C4⋊C4 [×4], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], C2×C8 [×4], SD16 [×2], C22×C4 [×2], C22×C4, C2×D4 [×2], C2×D4 [×2], C2×Q8, C2×Q8 [×2], C4×C8, C22⋊C8 [×2], D4⋊C4, D4⋊C4 [×2], Q8⋊C4, Q8⋊C4 [×2], C4⋊C8, C4.Q8, C4.Q8 [×4], C2.D8 [×4], C42⋊C2 [×4], C42⋊C2, C4×D4 [×2], C4×Q8, C4×Q8, C4⋊D4 [×2], C4⋊D4, C22⋊Q8 [×2], C4.4D4 [×3], C422C2 [×2], C4⋊Q8 [×2], C22×C8 [×2], C2×SD16, C23.25D4 [×2], C8×D4, C4×SD16, C88D4 [×2], D4⋊Q8, C4.Q16, C23.19D4 [×2], C23.20D4 [×2], C83Q8, C22.49C24, C22.50C24, C42.489C23
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, D46D4, C2×C4○D8, D4○SD16, C42.489C23

Smallest permutation representation of C42.489C23
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 25 31 36)(2 26 32 33)(3 27 29 34)(4 28 30 35)(5 42 56 19)(6 43 53 20)(7 44 54 17)(8 41 55 18)(9 15 51 40)(10 16 52 37)(11 13 49 38)(12 14 50 39)(21 62 46 60)(22 63 47 57)(23 64 48 58)(24 61 45 59)
(1 57 3 59)(2 60 4 58)(5 52 7 50)(6 51 8 49)(9 55 11 53)(10 54 12 56)(13 43 15 41)(14 42 16 44)(17 39 19 37)(18 38 20 40)(21 35 23 33)(22 34 24 36)(25 47 27 45)(26 46 28 48)(29 61 31 63)(30 64 32 62)
(1 4 3 2)(5 18 7 20)(6 19 8 17)(9 10 11 12)(13 39 15 37)(14 40 16 38)(21 63 23 61)(22 64 24 62)(25 35 27 33)(26 36 28 34)(29 32 31 30)(41 54 43 56)(42 55 44 53)(45 60 47 58)(46 57 48 59)(49 50 51 52)
(1 10 31 52)(2 9 32 51)(3 12 29 50)(4 11 30 49)(5 57 56 63)(6 60 53 62)(7 59 54 61)(8 58 55 64)(13 35 38 28)(14 34 39 27)(15 33 40 26)(16 36 37 25)(17 45 44 24)(18 48 41 23)(19 47 42 22)(20 46 43 21)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F4A···4H4I···4M4N···4R8A8B8C8D8E···8J
order12222224···44···44···488888···8
size11114482···24···48···822224···4

35 irreducible representations

dim1111111111112222244
type+++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82- 1+4D4○SD16
kernelC42.489C23C23.25D4C8×D4C4×SD16C88D4D4⋊Q8C4.Q16C23.19D4C23.20D4C83Q8C22.49C24C22.50C24C22⋊C4C4⋊C4C2×D4C8C4C4C2
# reps1211211221112114812

Matrix representation of C42.489C23 in GL4(𝔽17) generated by

16000
01600
0040
00013
,
161500
1100
0010
0001
,
111100
3600
0004
0040
,
16000
1100
00130
00013
,
13000
01300
00013
0040
G:=sub<GL(4,GF(17))| [16,0,0,0,0,16,0,0,0,0,4,0,0,0,0,13],[16,1,0,0,15,1,0,0,0,0,1,0,0,0,0,1],[11,3,0,0,11,6,0,0,0,0,0,4,0,0,4,0],[16,1,0,0,0,1,0,0,0,0,13,0,0,0,0,13],[13,0,0,0,0,13,0,0,0,0,0,4,0,0,13,0] >;

C42.489C23 in GAP, Magma, Sage, TeX

C_4^2._{489}C_2^3
% in TeX

G:=Group("C4^2.489C2^3");
// GroupNames label

G:=SmallGroup(128,2072);
// by ID

G=gap.SmallGroup(128,2072);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,100,346,248,4037,1027,124]);
// Polycyclic

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

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