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G = C42.367C23order 128 = 27

228th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.367C23, C4⋊C4.239D4, C4⋊C4.90C23, C22⋊C4.79D4, C2.21(Q8○D8), (C4×C8).350C22, (C2×C4).335C24, (C2×C8).456C23, C4.SD1640C2, C23.453(C2×D4), C4⋊Q8.110C22, (C2×Q8).91C23, C82M4(2)⋊38C2, C2.34(D4○SD16), C4.46(C4.4D4), (C2×D4).103C23, C8⋊C4.168C22, C23.38D436C2, (C22×C8).462C22, C4.4D4.32C22, C22.595(C22×D4), C42.C2.16C22, D4⋊C4.132C22, C23.41C237C2, (C22×C4).1033C23, Q8⋊C4.203C22, C23.24D4.12C2, C23.36D4.14C2, C22.16(C4.4D4), (C22×Q8).301C22, C42.78C2226C2, C42.28C2230C2, C42.30C2217C2, C42⋊C2.140C22, (C2×M4(2)).372C22, C23.38C23.13C2, C4.44(C2×C4○D4), (C2×C4).137(C2×D4), (C2×Q8⋊C4)⋊58C2, C2.46(C2×C4.4D4), (C2×C4).490(C4○D4), (C2×C4⋊C4).625C22, (C2×C4○D4).150C22, SmallGroup(128,1869)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.367C23
C1C2C4C2×C4C22×C4C22×C8C82M4(2) — C42.367C23
C1C2C2×C4 — C42.367C23
C1C22C42⋊C2 — C42.367C23
C1C2C2C2×C4 — C42.367C23

Generators and relations for C42.367C23
 G = < a,b,c,d,e | a4=b4=1, c2=b2, d2=a2b-1, e2=a2b2, ab=ba, cac-1=a-1b2, ad=da, eae-1=ab2, cbc-1=b-1, bd=db, be=eb, dcd-1=bc, ece-1=a2b2c, de=ed >

Subgroups: 340 in 186 conjugacy classes, 92 normal (38 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×2], C4 [×11], C22, C22 [×2], C22 [×5], C8 [×4], C2×C4 [×2], C2×C4 [×8], C2×C4 [×12], D4 [×4], Q8 [×10], C23, C23, C42 [×2], C42, C22⋊C4 [×2], C22⋊C4 [×5], C4⋊C4 [×4], C4⋊C4 [×2], C4⋊C4 [×11], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], M4(2) [×2], C22×C4, C22×C4 [×3], C2×D4, C2×D4, C2×Q8, C2×Q8 [×2], C2×Q8 [×7], C4○D4 [×4], C4×C8 [×2], C8⋊C4 [×2], D4⋊C4 [×2], D4⋊C4 [×2], Q8⋊C4 [×2], Q8⋊C4 [×10], C2×C4⋊C4, C42⋊C2 [×2], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4 [×2], C42.C2 [×2], C42.C2, C4⋊Q8 [×4], C4⋊Q8, C22×C8, C2×M4(2), C22×Q8, C2×C4○D4, C82M4(2), C2×Q8⋊C4, C23.24D4, C23.36D4, C23.38D4, C4.SD16 [×2], C42.78C22 [×2], C42.28C22 [×2], C42.30C22 [×2], C23.38C23, C23.41C23, C42.367C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C4○D4 [×4], C24, C4.4D4 [×4], C22×D4, C2×C4○D4 [×2], C2×C4.4D4, D4○SD16, Q8○D8, C42.367C23

Smallest permutation representation of C42.367C23
On 64 points
Generators in S64
(1 23 13 43)(2 24 14 44)(3 17 15 45)(4 18 16 46)(5 19 9 47)(6 20 10 48)(7 21 11 41)(8 22 12 42)(25 55 35 58)(26 56 36 59)(27 49 37 60)(28 50 38 61)(29 51 39 62)(30 52 40 63)(31 53 33 64)(32 54 34 57)
(1 11 5 15)(2 12 6 16)(3 13 7 9)(4 14 8 10)(17 43 21 47)(18 44 22 48)(19 45 23 41)(20 46 24 42)(25 33 29 37)(26 34 30 38)(27 35 31 39)(28 36 32 40)(49 58 53 62)(50 59 54 63)(51 60 55 64)(52 61 56 57)
(1 43 5 47)(2 18 6 22)(3 41 7 45)(4 24 8 20)(9 19 13 23)(10 42 14 46)(11 17 15 21)(12 48 16 44)(25 58 29 62)(26 50 30 54)(27 64 31 60)(28 56 32 52)(33 49 37 53)(34 63 38 59)(35 55 39 51)(36 61 40 57)
(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 60 9 53)(2 61 10 54)(3 62 11 55)(4 63 12 56)(5 64 13 49)(6 57 14 50)(7 58 15 51)(8 59 16 52)(17 25 41 39)(18 26 42 40)(19 27 43 33)(20 28 44 34)(21 29 45 35)(22 30 46 36)(23 31 47 37)(24 32 48 38)

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

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

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

32 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D4E4F4G4H4I···4O8A8B8C8D8E···8J
order1222222444444444···488888···8
size1111228222244448···822224···4

32 irreducible representations

dim11111111111122244
type++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4C4○D4D4○SD16Q8○D8
kernelC42.367C23C82M4(2)C2×Q8⋊C4C23.24D4C23.36D4C23.38D4C4.SD16C42.78C22C42.28C22C42.30C22C23.38C23C23.41C23C22⋊C4C4⋊C4C2×C4C2C2
# reps11111122221122822

Matrix representation of C42.367C23 in GL6(𝔽17)

1380000
040000
004000
000400
0000130
0000013
,
100000
010000
000100
0016000
000001
0000160
,
1380000
1340000
0013000
000400
0000130
000004
,
400000
040000
003300
0014300
000033
0000143
,
490000
0130000
0000130
0000013
004000
000400

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

C42.367C23 in GAP, Magma, Sage, TeX

C_4^2._{367}C_2^3
% in TeX

G:=Group("C4^2.367C2^3");
// GroupNames label

G:=SmallGroup(128,1869);
// by ID

G=gap.SmallGroup(128,1869);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,232,758,100,1018,2804,172,4037,124]);
// Polycyclic

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

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