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G = C42.7D4order 128 = 27

7th non-split extension by C42 of D4 acting faithfully

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

Aliases: C42.7D4, C4⋊Q84C4, C4.8(C4×D4), C4.4D44C4, C42.9(C2×C4), (C2×Q8).74D4, (C22×C4).69D4, C4.9C4212C2, C23.135(C2×D4), C4.140(C4⋊D4), C22.35C22≀C2, (C22×C4).36C23, C42⋊C22.6C2, (C22×Q8).25C22, C42⋊C2.34C22, C23.32C233C2, C2.51(C23.23D4), (C2×M4(2)).195C22, C23.38C23.2C2, C22.10(C22.D4), (C2×D4).90(C2×C4), (C2×C4).247(C2×D4), (C2×Q8).78(C2×C4), (C2×C4).331(C4○D4), (C2×C4.10D4)⋊20C2, (C2×C4).19(C22⋊C4), (C2×C4).196(C22×C4), (C2×C4○D4).30C22, C22.50(C2×C22⋊C4), SmallGroup(128,644)

Series: Derived Chief Lower central Upper central Jennings

C1C2×C4 — C42.7D4
C1C2C4C2×C4C22×C4C22×Q8C23.32C23 — C42.7D4
C1C2C2×C4 — C42.7D4
C1C2C22×C4 — C42.7D4
C1C2C2C22×C4 — C42.7D4

Generators and relations for C42.7D4
 G = < a,b,c,d | a4=b4=1, c4=b2, d2=b, ab=ba, cac-1=dad-1=a-1b-1, cbc-1=b-1, bd=db, dcd-1=bc3 >

Subgroups: 300 in 157 conjugacy classes, 54 normal (18 characteristic)
C1, C2, C2 [×4], C4 [×4], C4 [×11], C22 [×3], C22 [×4], C8 [×2], C2×C4 [×2], C2×C4 [×8], C2×C4 [×13], D4 [×4], Q8 [×10], C23, C23, C42 [×6], C42 [×4], C22⋊C4 [×7], C4⋊C4 [×11], C2×C8 [×2], M4(2) [×4], C22×C4, C22×C4 [×2], C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8 [×4], C2×Q8 [×4], C4○D4 [×4], C4.10D4 [×2], C4≀C2 [×4], C42⋊C2, C42⋊C2 [×2], C42⋊C2 [×2], C4×Q8 [×4], C22⋊Q8 [×2], C22.D4 [×2], C4.4D4 [×2], C4⋊Q8 [×2], C2×M4(2) [×2], C22×Q8, C2×C4○D4, C4.9C42 [×2], C2×C4.10D4, C42⋊C22 [×2], C23.32C23, C23.38C23, C42.7D4
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C2×C4 [×6], D4 [×8], C23, C22⋊C4 [×4], C22×C4, C2×D4 [×4], C4○D4 [×2], C2×C22⋊C4, C4×D4 [×2], C22≀C2, C4⋊D4 [×2], C22.D4, C23.23D4, C42.7D4

Character table of C42.7D4

 class 12A2B2C2D2E4A4B4C4D4E4F4G4H4I4J4K4L4M4N4O4P4Q4R4S8A8B8C8D
 size 11222822224444444444448888888
ρ111111111111111111111111111111    trivial
ρ211111111111-1-1-1-1111-1-1-1-1-11-11-11-1    linear of order 2
ρ31111111111-11-1-1-1-1-1-1111-1-11-1-11-11    linear of order 2
ρ41111111111-1-1111-1-1-1-1-1-11111-1-1-1-1    linear of order 2
ρ511111-11111-11-1-1-1-1-1-1111-11-111-11-1    linear of order 2
ρ611111-11111-1-1111-1-1-1-1-1-11-1-1-11111    linear of order 2
ρ711111-11111111111111111-1-1-1-1-1-1-1    linear of order 2
ρ811111-111111-1-1-1-1111-1-1-1-11-11-11-11    linear of order 2
ρ911-11-1-111-1-1ii-11-1-ii-i-ii-i1-111ii-i-i    linear of order 4
ρ1011-11-1111-1-1-i-i-11-1i-iii-ii11-1-1ii-i-i    linear of order 4
ρ1111-11-1-111-1-1-ii1-11i-ii-ii-i-111-1-iii-i    linear of order 4
ρ1211-11-1111-1-1i-i1-11-ii-ii-ii-1-1-11-iii-i    linear of order 4
ρ1311-11-1-111-1-1i-i1-11-ii-ii-ii-111-1i-i-ii    linear of order 4
ρ1411-11-1111-1-1-ii1-11i-ii-ii-i-1-1-11i-i-ii    linear of order 4
ρ1511-11-1-111-1-1-i-i-11-1i-iii-ii1-111-i-iii    linear of order 4
ρ1611-11-1111-1-1ii-11-1-ii-i-ii-i11-1-1-i-iii    linear of order 4
ρ1722-2-2202-2-2202000000-2-2200000000    orthogonal lifted from D4
ρ18222220-2-2-2-200-2-2200000020000000    orthogonal lifted from D4
ρ19222220-2-2-2-20022-2000000-20000000    orthogonal lifted from D4
ρ2022-22-20-2-222002-2-200000020000000    orthogonal lifted from D4
ρ2122-22-20-2-22200-222000000-20000000    orthogonal lifted from D4
ρ2222-2-220-222-2-20000-22200000000000    orthogonal lifted from D4
ρ2322-2-2202-2-220-200000022-200000000    orthogonal lifted from D4
ρ2422-2-220-222-2200002-2-200000000000    orthogonal lifted from D4
ρ25222-2-20-22-22-2i00002i2i-2i00000000000    complex lifted from C4○D4
ρ26222-2-202-22-20-2i000000-2i2i2i00000000    complex lifted from C4○D4
ρ27222-2-20-22-222i0000-2i-2i2i00000000000    complex lifted from C4○D4
ρ28222-2-202-22-202i0000002i-2i-2i00000000    complex lifted from C4○D4
ρ298-8000000000000000000000000000    symplectic faithful, Schur index 2

Smallest permutation representation of C42.7D4
On 32 points
Generators in S32
(1 28 31 2)(3 26 25 8)(4 7 30 29)(5 32 27 6)(9 19 22 10)(11 17 24 16)(12 15 21 20)(13 23 18 14)
(1 7 5 3)(2 4 6 8)(9 15 13 11)(10 12 14 16)(17 19 21 23)(18 24 22 20)(25 31 29 27)(26 28 30 32)
(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)
(1 12 7 14 5 16 3 10)(2 13 4 11 6 9 8 15)(17 29 19 27 21 25 23 31)(18 26 24 28 22 30 20 32)

G:=sub<Sym(32)| (1,28,31,2)(3,26,25,8)(4,7,30,29)(5,32,27,6)(9,19,22,10)(11,17,24,16)(12,15,21,20)(13,23,18,14), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32), (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), (1,12,7,14,5,16,3,10)(2,13,4,11,6,9,8,15)(17,29,19,27,21,25,23,31)(18,26,24,28,22,30,20,32)>;

G:=Group( (1,28,31,2)(3,26,25,8)(4,7,30,29)(5,32,27,6)(9,19,22,10)(11,17,24,16)(12,15,21,20)(13,23,18,14), (1,7,5,3)(2,4,6,8)(9,15,13,11)(10,12,14,16)(17,19,21,23)(18,24,22,20)(25,31,29,27)(26,28,30,32), (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), (1,12,7,14,5,16,3,10)(2,13,4,11,6,9,8,15)(17,29,19,27,21,25,23,31)(18,26,24,28,22,30,20,32) );

G=PermutationGroup([(1,28,31,2),(3,26,25,8),(4,7,30,29),(5,32,27,6),(9,19,22,10),(11,17,24,16),(12,15,21,20),(13,23,18,14)], [(1,7,5,3),(2,4,6,8),(9,15,13,11),(10,12,14,16),(17,19,21,23),(18,24,22,20),(25,31,29,27),(26,28,30,32)], [(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)], [(1,12,7,14,5,16,3,10),(2,13,4,11,6,9,8,15),(17,29,19,27,21,25,23,31),(18,26,24,28,22,30,20,32)])

Matrix representation of C42.7D4 in GL8(𝔽17)

00001000
00000100
00000010
413413116115
00010000
001600000
016000000
016100004
,
01000000
160000000
00010000
001600000
00000100
000016000
413413116115
404010116
,
00000100
00001000
413413116115
00000010
10000000
016000000
00100000
010100013
,
0000100160
1341341611162
000016070
1161167973
07010000
1001600000
010100000
7101161107

G:=sub<GL(8,GF(17))| [0,0,0,4,0,0,0,0,0,0,0,13,0,0,16,16,0,0,0,4,0,16,0,1,0,0,0,13,1,0,0,0,1,0,0,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,15,0,0,0,4],[0,16,0,0,0,0,4,4,1,0,0,0,0,0,13,0,0,0,0,16,0,0,4,4,0,0,1,0,0,0,13,0,0,0,0,0,0,16,1,1,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,16],[0,0,4,0,1,0,0,0,0,0,13,0,0,16,0,1,0,0,4,0,0,0,1,0,0,0,13,0,0,0,0,1,0,1,1,0,0,0,0,0,1,0,16,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,15,0,0,0,0,13],[0,13,0,11,0,10,0,7,0,4,0,6,7,0,1,1,0,13,0,11,0,16,0,0,0,4,0,6,1,0,10,11,10,16,16,7,0,0,0,6,0,11,0,9,0,0,0,11,16,16,7,7,0,0,0,0,0,2,0,3,0,0,0,7] >;

C42.7D4 in GAP, Magma, Sage, TeX

C_4^2._7D_4
% in TeX

G:=Group("C4^2.7D4");
// GroupNames label

G:=SmallGroup(128,644);
// by ID

G=gap.SmallGroup(128,644);
# by ID

G:=PCGroup([7,-2,2,2,-2,2,2,-2,224,141,288,422,352,2019,521,248,2804,1411,1027,124]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^4=1,c^4=b^2,d^2=b,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1*b^-1,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b*c^3>;
// generators/relations

Export

Character table of C42.7D4 in TeX

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