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G = C42.469C23order 128 = 27

330th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.469C23, C4.492+ 1+4, (C8×D4)⋊22C2, C88D441C2, C4⋊C4.410D4, (C4×Q16)⋊13C2, C4⋊SD1643C2, Q8⋊Q842C2, (C2×D4).239D4, C4.90(C4○D8), C4.4D816C2, (C4×C8).86C22, Q86D4.5C2, D4.7D412C2, C4⋊C8.345C22, C4⋊C4.404C23, (C2×C8).186C23, (C2×C4).496C24, Q8.19(C4○D4), C22⋊C4.108D4, C23.112(C2×D4), C4⋊Q8.145C22, C2.71(D4○SD16), (C4×D4).337C22, (C2×D4).226C23, C41D4.84C22, C4⋊D4.76C22, C23.19D46C2, (C4×Q8).152C22, (C2×Q8).212C23, C2.132(D45D4), C4.Q8.103C22, C2.D8.192C22, C22⋊Q8.76C22, D4⋊C4.13C22, C23.24D429C2, C22⋊C8.205C22, (C22×C8).357C22, (C2×Q16).134C22, (C2×SD16).98C22, C22.756(C22×D4), C22.50C243C2, (C22×C4).1140C23, Q8⋊C4.180C22, C42⋊C2.184C22, C2.64(C2×C4○D8), C4.221(C2×C4○D4), (C2×C4).925(C2×D4), (C2×C4○D4).202C22, SmallGroup(128,2036)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.469C23
C1C2C4C2×C4C22×C4C2×C4○D4Q86D4 — C42.469C23
C1C2C2×C4 — C42.469C23
C1C22C4×D4 — C42.469C23
C1C2C2C2×C4 — C42.469C23

Generators and relations for C42.469C23
 G = < a,b,c,d,e | a4=b4=1, c2=a2, d2=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: 400 in 198 conjugacy classes, 88 normal (38 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×10], C22, C22 [×12], C8 [×4], C2×C4 [×3], C2×C4 [×2], C2×C4 [×14], D4 [×15], Q8 [×2], Q8 [×5], C23 [×2], C23 [×2], C42, C42 [×4], C22⋊C4 [×2], C22⋊C4 [×6], C4⋊C4 [×3], C4⋊C4 [×2], C4⋊C4 [×5], C2×C8 [×2], C2×C8 [×2], C2×C8 [×2], SD16 [×2], Q16 [×2], C22×C4 [×2], C22×C4 [×2], C2×D4, C2×D4 [×2], C2×D4 [×6], C2×Q8 [×2], C2×Q8, C4○D4 [×6], C4×C8, C22⋊C8 [×2], D4⋊C4 [×6], Q8⋊C4 [×2], Q8⋊C4 [×2], C4⋊C8, C4.Q8 [×2], C2.D8, C42⋊C2 [×2], C4×D4, C4×D4, C4×Q8 [×2], C4×Q8, C4⋊D4 [×2], C4⋊D4 [×2], C22⋊Q8 [×2], C4.4D4, C422C2 [×2], C41D4, C41D4, C4⋊Q8, C22×C8 [×2], C2×SD16 [×2], C2×Q16, C2×C4○D4 [×2], C23.24D4 [×2], C8×D4, C4×Q16, D4.7D4 [×2], C4⋊SD16, C88D4 [×2], Q8⋊Q8, C23.19D4 [×2], C4.4D8, Q86D4, C22.50C24, C42.469C23
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, D4○SD16, C42.469C23

Smallest permutation representation of C42.469C23
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 49 12)(2 43 50 9)(3 44 51 10)(4 41 52 11)(5 31 40 56)(6 32 37 53)(7 29 38 54)(8 30 39 55)(13 20 46 21)(14 17 47 22)(15 18 48 23)(16 19 45 24)(25 59 33 62)(26 60 34 63)(27 57 35 64)(28 58 36 61)
(1 37 3 39)(2 40 4 38)(5 52 7 50)(6 51 8 49)(9 56 11 54)(10 55 12 53)(13 36 15 34)(14 35 16 33)(17 57 19 59)(18 60 20 58)(21 61 23 63)(22 64 24 62)(25 47 27 45)(26 46 28 48)(29 43 31 41)(30 42 32 44)
(1 21 49 20)(2 22 50 17)(3 23 51 18)(4 24 52 19)(5 27 40 35)(6 28 37 36)(7 25 38 33)(8 26 39 34)(9 14 43 47)(10 15 44 48)(11 16 41 45)(12 13 42 46)(29 62 54 59)(30 63 55 60)(31 64 56 57)(32 61 53 58)
(1 24 49 19)(2 23 50 18)(3 22 51 17)(4 21 52 20)(5 58 40 61)(6 57 37 64)(7 60 38 63)(8 59 39 62)(9 48 43 15)(10 47 44 14)(11 46 41 13)(12 45 42 16)(25 30 33 55)(26 29 34 54)(27 32 35 53)(28 31 36 56)

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

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

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

35 conjugacy classes

class 1 2A2B2C2D2E2F2G4A···4H4I···4M4N4O4P4Q8A8B8C8D8E···8J
order122222224···44···4444488888···8
size111144882···24···4888822224···4

35 irreducible representations

dim1111111111112222244
type++++++++++++++++
imageC1C2C2C2C2C2C2C2C2C2C2C2D4D4D4C4○D4C4○D82+ 1+4D4○SD16
kernelC42.469C23C23.24D4C8×D4C4×Q16D4.7D4C4⋊SD16C88D4Q8⋊Q8C23.19D4C4.4D8Q86D4C22.50C24C22⋊C4C4⋊C4C2×D4Q8C4C4C2
# reps1211212121112114812

Matrix representation of C42.469C23 in GL4(𝔽17) generated by

1000
0100
00215
001115
,
0100
16000
0010
0001
,
14300
3300
00130
0094
,
4000
01300
00160
00016
,
4000
0400
00152
0072
G:=sub<GL(4,GF(17))| [1,0,0,0,0,1,0,0,0,0,2,11,0,0,15,15],[0,16,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[14,3,0,0,3,3,0,0,0,0,13,9,0,0,0,4],[4,0,0,0,0,13,0,0,0,0,16,0,0,0,0,16],[4,0,0,0,0,4,0,0,0,0,15,7,0,0,2,2] >;

C42.469C23 in GAP, Magma, Sage, TeX

C_4^2._{469}C_2^3
% in TeX

G:=Group("C4^2.469C2^3");
// GroupNames label

G:=SmallGroup(128,2036);
// by ID

G=gap.SmallGroup(128,2036);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,112,253,456,758,352,346,248,4037,1027,124]);
// Polycyclic

G:=Group<a,b,c,d,e|a^4=b^4=1,c^2=a^2,d^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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