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G = C42.424C23order 128 = 27

285th non-split extension by C42 of C23 acting via C23/C2=C22

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

Aliases: C42.424C23, C4.662- 1+4, C8⋊Q820C2, C4⋊C4.136D4, Q8.Q825C2, C8.5Q86C2, Q8⋊Q811C2, C4.Q1628C2, C4⋊C8.76C22, C2.32(Q8○D8), C22⋊C4.28D4, C4⋊C4.181C23, (C4×C8).118C22, (C2×C4).440C24, (C2×C8).336C23, C4.SD1618C2, C23.303(C2×D4), C4⋊Q8.124C22, C4.Q8.43C22, C8⋊C4.33C22, C2.D8.41C22, C2.50(D4○SD16), C22⋊C8.67C22, (C2×Q8).171C23, (C4×Q8).118C22, C22⋊Q8.47C22, (C22×C4).313C23, Q8⋊C4.53C22, C23.48D4.4C2, C23.20D4.4C2, C23.47D4.3C2, C22.700(C22×D4), C42.C2.27C22, C42.30C227C2, C42.7C22.2C2, C42⋊C2.170C22, C22.35C24.3C2, C23.41C23.10C2, C2.88(C23.38C23), (C2×C4).564(C2×D4), (C2×C4⋊C4).655C22, SmallGroup(128,1974)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.424C23
C1C2C4C2×C4C22×C4C2×C4⋊C4C23.41C23 — C42.424C23
C1C2C2×C4 — C42.424C23
C1C22C42⋊C2 — C42.424C23
C1C2C2C2×C4 — C42.424C23

Generators and relations for C42.424C23
 G = < a,b,c,d,e | a4=b4=e2=1, c2=b2, d2=a2, ab=ba, cac-1=dad-1=a-1, eae=ab2, cbc-1=dbd-1=b-1, be=eb, dcd-1=bc, ece=a2c, de=ed >

Subgroups: 260 in 154 conjugacy classes, 84 normal (all characteristic)
C1, C2 [×3], C2, C4 [×2], C4 [×13], C22, C22 [×3], C8 [×4], C2×C4 [×6], C2×C4 [×11], Q8 [×6], C23, C42 [×2], C42 [×3], C22⋊C4 [×2], C22⋊C4 [×3], C4⋊C4 [×8], C4⋊C4 [×15], C2×C8 [×4], C22×C4, C22×C4, C2×Q8 [×2], C2×Q8 [×2], C4×C8, C8⋊C4, C22⋊C8 [×2], Q8⋊C4 [×8], C4⋊C8 [×2], C4.Q8 [×4], C2.D8 [×4], C2×C4⋊C4, C42⋊C2 [×2], C4×Q8 [×2], C22⋊Q8 [×2], C22⋊Q8 [×2], C42.C2 [×3], C42.C2 [×3], C422C2 [×2], C4⋊Q8 [×3], C4⋊Q8, C42.7C22, Q8⋊Q8, C4.Q16, Q8.Q8 [×2], C23.47D4, C23.48D4, C23.20D4 [×2], C4.SD16, C42.30C22, C8.5Q8, C8⋊Q8, C22.35C24, C23.41C23, C42.424C23
Quotients: C1, C2 [×15], C22 [×35], D4 [×4], C23 [×15], C2×D4 [×6], C24, C22×D4, 2- 1+4 [×2], C23.38C23, D4○SD16, Q8○D8, C42.424C23

Character table of C42.424C23

 class 12A2B2C2D4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O8A8B8C8D8E8F
 size 11114224444488888888444488
ρ111111111111111111111111111    trivial
ρ21111111-1-11-1-1-11-1-111-111-11-11-1    linear of order 2
ρ31111-111-11-1-11-11-111-11-1-11-111-1    linear of order 2
ρ41111-1111-1-11-1111-11-1-1-1-1-1-1-111    linear of order 2
ρ51111-111-11-1-11-111-1-111-11-11-1-11    linear of order 2
ρ61111-1111-1-11-111-11-11-1-11111-1-1    linear of order 2
ρ711111111111111-1-1-1-111-1-1-1-1-1-1    linear of order 2
ρ81111111-1-11-1-1-1111-1-1-11-11-11-11    linear of order 2
ρ91111111-1-11-1-11-1-1-1111-1-11-11-11    linear of order 2
ρ10111111111111-1-11111-1-1-1-1-1-1-1-1    linear of order 2
ρ111111-1111-1-11-1-1-11-11-1111111-1-1    linear of order 2
ρ121111-111-11-1-111-1-111-1-111-11-1-11    linear of order 2
ρ131111-1111-1-11-1-1-1-11-1111-1-1-1-111    linear of order 2
ρ141111-111-11-1-111-11-1-11-11-11-111-1    linear of order 2
ρ151111111-1-11-1-11-111-1-11-11-11-11-1    linear of order 2
ρ16111111111111-1-1-1-1-1-1-1-1111111    linear of order 2
ρ172222-2-2-22-22-2200000000000000    orthogonal lifted from D4
ρ1822222-2-222-2-2-200000000000000    orthogonal lifted from D4
ρ1922222-2-2-2-2-22200000000000000    orthogonal lifted from D4
ρ202222-2-2-2-2222-200000000000000    orthogonal lifted from D4
ρ214-44-40-440000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ224-44-404-40000000000000000000    symplectic lifted from 2- 1+4, Schur index 2
ρ2344-4-400000000000000000220-2200    symplectic lifted from Q8○D8, Schur index 2
ρ2444-4-400000000000000000-2202200    symplectic lifted from Q8○D8, Schur index 2
ρ254-4-440000000000000000-2-202-2000    complex lifted from D4○SD16
ρ264-4-4400000000000000002-20-2-2000    complex lifted from D4○SD16

Smallest permutation representation of C42.424C23
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 39 29 9)(2 40 30 10)(3 37 31 11)(4 38 32 12)(5 60 44 23)(6 57 41 24)(7 58 42 21)(8 59 43 22)(13 49 35 25)(14 50 36 26)(15 51 33 27)(16 52 34 28)(17 46 54 62)(18 47 55 63)(19 48 56 64)(20 45 53 61)
(1 63 29 47)(2 62 30 46)(3 61 31 45)(4 64 32 48)(5 52 44 28)(6 51 41 27)(7 50 42 26)(8 49 43 25)(9 18 39 55)(10 17 40 54)(11 20 37 53)(12 19 38 56)(13 22 35 59)(14 21 36 58)(15 24 33 57)(16 23 34 60)
(1 27 3 25)(2 26 4 28)(5 54 7 56)(6 53 8 55)(9 15 11 13)(10 14 12 16)(17 42 19 44)(18 41 20 43)(21 64 23 62)(22 63 24 61)(29 51 31 49)(30 50 32 52)(33 37 35 39)(34 40 36 38)(45 59 47 57)(46 58 48 60)
(2 30)(4 32)(5 42)(6 8)(7 44)(10 40)(12 38)(14 36)(16 34)(17 56)(18 20)(19 54)(21 60)(22 24)(23 58)(26 50)(28 52)(41 43)(45 47)(46 64)(48 62)(53 55)(57 59)(61 63)

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

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

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

Matrix representation of C42.424C23 in GL8(𝔽17)

00100000
00010000
10000000
01000000
00007100
000011000
0000168112
000051076
,
01000000
160000000
00010000
001600000
000016000
000001600
000000160
000000016
,
512000000
1212000000
005120000
0012120000
0000111310
00004151615
000015229
00002126
,
160000000
01000000
001600000
00010000
00000100
000016000
00009812
0000801616
,
10000000
01000000
001600000
000160000
00001000
00000100
0000128160
00001511016

G:=sub<GL(8,GF(17))| [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,7,1,16,5,0,0,0,0,1,10,8,10,0,0,0,0,0,0,11,7,0,0,0,0,0,0,2,6],[0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,5,12,0,0,0,0,0,0,12,12,0,0,0,0,0,0,0,0,11,4,15,2,0,0,0,0,13,15,2,1,0,0,0,0,1,16,2,2,0,0,0,0,0,15,9,6],[16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,16,9,8,0,0,0,0,1,0,8,0,0,0,0,0,0,0,1,16,0,0,0,0,0,0,2,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,1,0,12,15,0,0,0,0,0,1,8,11,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16] >;

C42.424C23 in GAP, Magma, Sage, TeX

C_4^2._{424}C_2^3
% in TeX

G:=Group("C4^2.424C2^3");
// GroupNames label

G:=SmallGroup(128,1974);
// by ID

G=gap.SmallGroup(128,1974);
# by ID

G:=PCGroup([7,-2,2,2,2,-2,2,-2,448,253,120,758,219,100,675,1018,4037,1027,124]);
// Polycyclic

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

Export

Character table of C42.424C23 in TeX

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